Exotic compact object behavior in black hole analogues
EExotic compact object behavior in black hole analogues
Carlos A. R. Herdeiro and Nuno M. Santos
Centro de Astrofísica e Gravitação − CENTRA, Departamento de Física, Instituto Superior Técnico − IST,Universidade de Lisboa − UL, Avenida Rovisco Pais 1, 1049, Lisboa, Portugal (Dated: February 2019)Classical phenomenological aspects of acoustic perturbations on a draining bathtub geometry where a surfacewith reflectivity R is set at a small distance from the would-be acoustic horizon, which is excised, are addressed.Like most exotic compact objects featuring an ergoregion but not a horizon, this model is prone to instabilitieswhen | R | ≈
1. However, stability can be attained for sufficiently slow drains when | R | (cid:46) I. INTRODUCTION
Analogue models for gravity have proven to be a power-ful tool in understanding and probing several classical andquantum phenomena in curved spacetime, namely the emis-sion of Hawking radiation and the amplification of bosonicfield perturbations scattered off spinning objects, commonlydubbed superradiance. It was Unruh who first drew an ana-logue model for gravity relating the propagation of soundwaves in fluid flows with the kinematics of waves in a clas-sical gravitational field [1]. This seminal proposal unfolded acompletely unexpected way of exploring gravity and openeda track to test it in the laboratory. A summary of the historyand motivation behind analogue gravity can be found in [2].The analogy between the propagation of sound waves in anon-relativistic, irrotational, inviscid, barotropic fluid and thepropagation of a minimally coupled massless scalar field ina curved Lorentzian geometry was primarily established byVisser [3, 4], who also formulated the concepts of acoustichorizon, ergoregion and surface gravity in analogue models.The correspondence is purely kinematic, i.e. only kinematicaspects of general relativity, such as event horizons, appar-ent horizons and ergoregions, are carried over into fluid me-chanics. The effective geometry of the flow is mathematicallyencoded in a Lorentzian metric, commonly dubbed acousticmetric , governed by the fluid equations of motion and not byEinstein’s field equations. In other words, the dynamic as-pects of general relativity do not map into fluid mechanics.This partial isomorphism thus offers a rather simple way todisentangle the kinematic and dynamic contributions to someimportant phenomena in general relativity.Dumb holes, the acoustic analogues for black holes (BHs),bear several structural and phenomenological similarities toBHs. For instance, the draining bathtub vortex [3, 5], or sim-ply draining bathtub, which features both an ergoregion andan acoustic horizon, is a Kerr BH analogue. The model de-scribes the two-dimensional flow of a non-relativistic, locallyirrotational, inviscid, barotropic fluid swirling around a drain.The fluid velocity increases monotonically downstream. Theregion where the magnitude of the fluid velocity exceeds thespeed of sound defines the ergoregion. Nearer the drain is theacoustic horizon, which comprises the set of points in whichthe radial component of the fluid velocity equals the speed ofsound. Any sound wave produced inside the acoustic horizon cannot escape from the region around the drain.The phenomenology of the draining bathtub model hasbeen widely addressed over the last two decades. For instance,works on quasi-normal modes (QNMs) [6, 7], absorption pro-cesses [8] and superradiance [9–12] showed that this vortexgeometry shares many properties with Kerr spacetime.Kerr BHs are stable against linear bosonic perturbations[13–16]. The event horizon absorbs any negative-energyphysical states which may form inside the ergoregion andwould otherwise trigger an instability. In fact, as first shownby Friedman [17], asymptotically-flat, stationary solutions toEinstein’s field equations possessing an ergoregion but not anevent horizon may develop instabilities, usually called ergore-gion instabilities, when linearly interacting with scalar andelectromagnetic field perturbations, especially if rapidly spin-ning. Ergoregion instabilities are known to affect a plethoraof exotic compact objects (ECOs) [18–26]. These are looselydefined in the literature as objects without event horizon,more massive than neutron stars and suffciently dim not tohave been observed by state-of-the-art electromagnetic tele-scopes and detectors yet. Examples include boson stars[27], anisotropic stars [28], wormholes [29], gravastars [30],fuzzballs [31], black stars [32], superspinars [33], Proca stars[34], collapsed polymers [35], 2 − − not even those whichdo not feature an event horizon. There has been a renewed in-terest in these exotic alternatives over the last decades becausesome ECOs can mimic the physical behavior of BHs, namelythose whose near-horizon geometry slightly differs from thatof BHs, such as Kerr-like ECOs [25, 26]. These objects fea-ture an ergoregion and are endowed with a surface with reflec- a r X i v : . [ g r- q c ] A p r tive properties at a microscopic or Planck distance from thewould-be event horizon of the corresponding Kerr BH. Thepresence of an ergoregion and the absence of an event hori-zon are the two key ingredients for ergoregion instabilities todevelop. Indeed, perfectly-reflecting Kerr-like ECOs exhibitsome exponentially-growing QNMs and are unstable againstlinear perturbations in some region of parameter space. How-ever, the unstable modes can be mitigated or neutralized whenthe surface is not perfectly- but partially-reflecting, i.e. whendissipative effects are considered.This work focuses on classical phenomenological aspectsof a draining bathtub model whose acoustic horizon is re-placed by a surface with reflective properties. The physicalbehavior of this «toy model» is quite similar to that of Kerr-like ECOs. Although the experimental realization of suchsystem is not evident at present, it is still fruitful to explorethis theoretical setup, as it may provide some insights into thegeneric features of ECOs and even be useful for possible fu-ture experiments.The present paper is organized as follows. Sec. II coversin brief the main mathematical and physical features of thedraining bathtub geometry, introduces the equation governingacoustic perturbations and discusses the properties of its solu-tions. Numerical results regarding QNM frequencies and am-plification factors of the system are presented Sec. III, com-plemented with a low-frequency analytical treatment in theAppendices A and B. II. DRAINING BATHTUB MODELA. Acoustic metric
The draining bathtub geometry introduced by Visser [3] de-scribes the irrotational flow of a barotropic and inviscid fluidwith background density ρ in a plane with a sink at the origin.The irrotational nature of the flow together with the conserva-tion of angular momentum require the background density tobe constant, i.e. position-independent. As a result, the back-ground pressure p and the speed of sound c are also constantthroughout the flow. Furthermore, it follows from both theequation of continuity and the conservation of angular mo-mentum that the unperturbed velocity profile v of the flowingfluid is given in polar coordinates ( r , φ ) by v ≡ v ( r ) e r + v ( φ ) e φ , (1)with v ( r ) = − A / r and v ( φ ) = B / r , where A , B ∈ R + . e r and e φ are the radial and azimuthal unit vectors, respectively. Ageneral velocity profile v , allowing for perturbations, can bewritten as the gradient of a velocity potential Ψ , i.e. v = ∇Ψ .When v = v , Ψ = Ψ ( r , θ ) ≡ A log r + B φ (apart from a con-stant of integration).In polar coordinates ( t , r , φ ) , the line element of the drainingbathtub model has the form [5, 46]d s = − c d t + (cid:18) d r + Ar d t (cid:19) + (cid:18) r d φ − Br d t (cid:19) , (2) where the constant prefactor ( ρ / c ) has been omitted. Eq.(2) has the Painlevé-Gullstrand form [47].The flow is stationary and axisymmetric, i.e. the metrictensor in Eq. (2) does not depend explicitly on t nor on φ .In polar coordinates, the Killing vectors associated with thesecontinuous symmetries are ξ t ≡ ∂ t and ξ φ ≡ ∂ φ , respectively.This geometry has an acoustic horizon located at r = r H ≡ A / c and a region of transonic flow defined by r H < r < r E , where r E = √ A + B / c is the location of the ergosphere .Performing the coordinate transformationd˜ t = d t − Arc r − A d r , (3)d ˜ φ = d φ − ABr ( c r − A ) d r , (4)one can cast Eq. (2) in the form [10, 11, 48]d s = − f ( r ) d˜ t + [ g ( r )] − d r − Bc d˜ t d ˜ φ + r d ˜ φ , (5)where f ( r ) = − A + B c r and g ( r ) = − A c r . (6)The coordinates ( ˜ t , r , ˜ φ ) are equivalent to the Boyer-Lindquistcoordinates commonly used to write the Kerr metric. B. Acoustic perturbations
1. Model
Perturbations to the steady flow can be encoded in the ve-locity potential by adding a term to it, i.e. by considering Ψ = Ψ + Φ , where Φ is the perturbation. The equations ofmotion governing an acoustic perturbation in the velocity po-tential of an irrotational flow of a barotropic and inviscid fluidare the same as the Klein-Gordon equation for a minimallycoupled massless scalar field propagating in a Lorentzian ge-ometry [3]. In effect, the perturbation Φ in the velocity poten-tial satisfies the equation (cid:3) Φ = √− g ∂ µ (cid:0) √− g g µν ∂ ν Φ (cid:1) = , (7)where (cid:3) ≡ ∇ µ ∇ µ is the D’Alambert operator and ∇ µ denotesthe covariant derivative. In the present case, the index µ runsfrom 0 to 2, with 0 referring to the time coordinate t and 1 and2 to the spatial coordinates r and φ , respectively.One is interested in ( + ) − dimensional objects whose ge-ometry is described by Eq. (2) from r to infinity, where r is the location of a surface with reflectivity R . Such The speed of the flowing fluid is φ − independent and given by v ≡ (cid:107) v (cid:107) = √ A + B / r . It equals the speed of sound at r = r E and exceeds it every-where in the region r H < r < r E . objects will hereafter be referred to as ECO-like vortices.Perfectly-reflecting and perfectly-absorbing surfaces are de-fined by | R | = R =
0, respectively. The model sets thesurface at a small distance from the would-be acoustic hori-zon, i.e. at r = r H ( + δ ) , where 0 < δ (cid:28)
1. The requirementthat δ (cid:28) r < r E ) .For the sake of simplicity, the uniform scalings r → Ar / c and B → B / A will hereafter be adopted [6, 11]. Note thatthese linear transformations are equivalent to set A = c =
2. Perturbation equation
From the existence of the Killing vectors ξ t and ξ φ , onecan separate the t − and φ − dependence of the field Φ , whichin turn can be expressed as a superposition of modes with dif-ferent frequencies ω and periods in φ , i.e. Φ ( t , r , φ ) = + ∞ ∑ m = − ∞ (cid:90) + ∞ − ∞ d ω e − i ω t R ω m ( r ) e + im φ , (8)where R ω m ( r ) , dubbed radial function, is a function of theradial coordinate only and depends on B , ω and m . The radialfunction satisfies the ordinary differential equation (ODE) [9] (cid:20) d d r + K ( r ) dd r + K ( r ) (cid:21) R ω m ( r ) = , (9)where K ( r ) = + r + i ( mB − ω r ) r ( r − ) , (10) K ( r ) = ω r − m r + m B − mB ω r − imBr ( r − ) . (11)Defining a new radial function S ω m ( r ) as R ω m ( r ) = S ω m ( r ) e i [ ( ω − mB ) log ( r − )+ mB log ( r ) ] , (12)Eq. (9) becomes (cid:20) d d r + L ( r ) dd r + L ( r ) (cid:21) S ω m ( r ) = , (13)where L ( r ) = r + r ( r − ) , (14) L ( r ) = ω r − ( m + mB ω ) r + m ( + B )( r − ) . (15) ECO-like vortices reduce to the draining bathtub when δ = R = It is useful to introduce a tortoise coordinate defined by thecondition [9] d r ∗ d r = [ g ( r )] − = (cid:18) − r (cid:19) − . (16)Explicitly, the tortoise coordinate is given by r ∗ ( r ) = r +
12 log (cid:12)(cid:12)(cid:12)(cid:12) r − r + (cid:12)(cid:12)(cid:12)(cid:12) , (17)which maps the acoustic horizon at r H = r ∗ → − ∞ and r → + ∞ to r ∗ → + ∞ . Together with the definition [9] S ω m ( r ) = H ω m ( r ) √ r , (18)Eq. (13) can be written as (cid:20) d d r ∗ + V ( r ) (cid:21) H ω m ( r ) = , (19)where the effective potential V ( r ) is given by V ( r ) = (cid:18) ω − mBr (cid:19) − g ( r ) r (cid:18) m − + r (cid:19) , (20)which has the asymptotic behavior V ( r ) ∼ (cid:26) ω , r ∗ → + ∞ ϖ , r ∗ → − ∞ , (21)with ϖ ≡ ω − mB . B coincides with the angular velocity ofthe would-be acoustic horizon.
3. Asymptotic solutions
The presence of a surface with reflectivity R requires solu-tions to Eq. (19) to be a superposition of ingoing and outgoingwaves at r = r . Thus, following the notation in [49], generalsolutions have the asymptotics H ω m ( r ) ∼ e − i ϖ r ∗ + R e + i ϖ ( r ∗ − r ∗ ) , r ∗ → r ∗ A − s e − i ω r ∗ + A + s e + i ω r ∗ , r ∗ → + ∞ , (22)where r ∗ ≡ r ∗ ( r ) <
0. For the sake of simplicity, although R may depend on ω and/or r [49–51], this work will only focuson constant-valued ( ω − and r − independent) reflectivities.Perfectly-reflecting ( | R | =
1) boundary conditions (BCs),generically known as Robin BCs, are given by [52]cos ( ξ ) H ω m ( r ) + sin ( ξ ) H (cid:48) ω m ( r ) = , (23)where ξ ∈ [ , π ) and the prime denotes differentiation withrespect to r . Note that ξ = H ω m ( r ) =
0, whereas ξ = π / H ω m ( r ) at r = r , i.e. H (cid:48) ω m ( r ) =
0. Equivalently, DBCs (NBCs) can be defined by R = − R = . Perfectly-reflecting BCs will hereafter bespecialized to DBCs and NBCs only.The solution in Eq. (22) may be written as a superpositionof modes with asymptotics [18] H + ω m ( r ) ∼ e − i ϖ r ∗ , r ∗ → r ∗ A − ∞ e − i ω r ∗ + A + ∞ e + i ω r ∗ , r ∗ → + ∞ (24) H − ω m ( r ) ∼ A − h e − i ϖ r ∗ + A + h e + i ϖ r ∗ , r ∗ → r ∗ e + i ω r ∗ , r ∗ → + ∞ (25) H + ω m is a draining bathtub solution ( R =
0) of the scatteringproblem, whereas H − ω m is an ECO solution of the QNM eigen-value problem (since it is a superposition of ingoing and out-going waves at the reflective surface).From the constancy of the Wronskians of Eq. (19), one canwrite the following useful relations between the coefficients A ± s , A ± h and A ± ∞ : ω A − ∞ = ϖ A + h , (26) ω A + ∞ = − ϖ A −∗ h , (27) ω A − s = ϖ ( A + h − A − h R e − i ϖ r ∗ ) , (28) ω ( A + s A − ∞ − A − s A + ∞ ) = ϖ R e − i ϖ r ∗ , (29) ω ( | A − s | − | A + s | ) = ϖ ( − | R | ) . (30)
4. Superradiance
The amplification factor in a scattering process is definedby [53] Z ( ω , R ) = (cid:12)(cid:12)(cid:12)(cid:12) A + s A − s (cid:12)(cid:12)(cid:12)(cid:12) − = − (cid:18) − mB ω (cid:19) − | R | | A − s | , (31)where the last equality follows from the Wronskian relationin Eq. (30). The amplitude of the reflected wave is greaterthan that of the incident wave at infinity (i.e. Z >
0) when0 < ω < mB . Note that the superradiance condition does notdepend on R .
5. Quasi-normal modes
Once physical BCs at r ∗ → r ∗ and r ∗ → + ∞ are imposed,Eq. (19) defines an eigenvalue problem. If one requires purelyoutgoing waves at infinity, H ω m ( r ) ∼ e + i ω r ∗ , r ∗ → + ∞ , (32)the eigenvalues, the characteristic frequencies ω QNM , arecalled QNM frequencies and the corresponding perturbations Plugging H ω m ( r ) = ( + R ) e − i ϖ r ∗ and H (cid:48) ω m ( r ) = − i ϖ ( − R ) e − i ϖ r ∗ into Eq. (23), one can show that R = − [ cos ( ξ ) − i ϖ sin ( ξ )] / [ cos ( ξ ) + i ϖ sin ( ξ )] . Φ are dubbed QNMs [15]. The set of all eigenfrequencies isoften referred to as QNM spectrum. The QNM frequencies ω QNM are in general complex, i.e. ω QNM = ω R + i ω I , where ω R ≡ Re { ω QNM } and ω I ≡ Im { ω QNM } . The sign of ω I de-fines the stability of the QNM. According to the conventionfor the Fourier decomposition in Eq. (8), if: ω I <
0, the modeis stable and τ dam ≡ / | ω I | defines the damping e -foldingtimescale; ω I >
0, the mode is unstable and τ ins ≡ / ω I de-fines the instability e -folding timescale; ω I =
0, the modeis marginally stable or stationary. When analyzing unstableQNMs, one is commonly interested in those corresponding tothe shortest instability timescales. In the present case, theseare the fundamental m = A − s = A + h / A − h = R e − i ( ω QNM − mB ) r ∗ . (33)One can solve Eq. (33) for ω QNM , which yields ω R = mB + r ∗ (cid:2) arg ( R ) + arg ( A − h / A + h (cid:1) ] (34) ω I = − r ∗ (cid:0) log | R | + log | A − h / A + h | (cid:1) . (35)arg ( R ) dictates the difference in phase between ingoing andoutgoing waves. If R is a positive (negative) real number,the phase difference is an even (odd) multiple of π . Thus,NBCs (DBCs) refer to waves reflected in phase (antiphase).If R is a complex number, the phase difference is a multipleof some real number between 0 and π . Without loss of gen-erality (as far as QNM stability is concerned), the reflectivity R will hereafter be considered a real parameter. Note that ω R does not depend on the magnitude of R . Thus, once arg ( R ) is fixed, the introduction of dissipation ( | R | <
1) does not af-fect the real part of the QNM frequency. On the contrary, theimaginary part changes with changing | R | . It follows fromEq. (35) that | R | determines QNM stability. Such stabilityis achieved whenever ω I <
0, i.e. when | R | < (cid:12)(cid:12)(cid:12)(cid:12) A + h A − h (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) A − ∞ A + ∞ (cid:12)(cid:12)(cid:12)(cid:12) (36)where the last equality follows from the Wronskian relationsin Eqs. (26) and (27). The last term in Eq. (36) is the inverseof the superradiant coefficient for the draining bathtub ( R = | R | < + Z ( ω R ) , (37)where Z ( ω R ) ≡ Z ( ω R , ) . In other words, the upper boundon the range of values | R | can take to assure stability is afunction of the amplification factors for the draining bathtubonly. The same result can be derived from a «bounce-and-amplify» argument [26, 53]. When | R | =
1, the conditionin Eq. (37) is satisfied only when Z <
0, i.e. when the realpart of the QNM frequencies does not lie in the superradiantregime (0 < ω R < mB ), otherwise instabilities are triggered. III. NUMERICAL RESULTSA. Numerical method
The numerical results to be presented in the following wereobtained using a direct-integration method. The integrationof Eq. (19) is performed using the numerically convenientexpansions ˜ H h ( r ) = H h ( r , ϖ ) + R H h ( r , − ϖ ) , ˜ H ∞ ( r ) = A + s H ∞ ( r , ω ) + A − s H ∞ ( r , − ω ) for the radial function H ω m ( r ) in Eq. (22) in the near and inthe far regions, respectively, where H h ( r , ϖ ) = ( r − r H ) − i ϖ / N h ∑ n = c n ( r − r H ) n , H ∞ ( r , ω ) = e + i ω r ∗ N ∞ ∑ n = d n r − n . Note that ˜ H h ( r ) ≈ ( + R δ + i ϖ ) δ − i ϖ / . According to Eq.(22), H ω m ( r ) ∼ ( + R ) e − i ϖ r ∗ , meaning that one shouldrequire ˜ H h ( r ) ∼ ( + R ) δ − i ϖ / , from which follows that R = R δ − i ϖ . N h and N ∞ are the number of terms of the par-tial sums. The coefficients c n and d n are functions of ϖ and ω , respectively, and both depend on B and m . Inserting eachexpansion into Eq. (19) and equating coefficients order byorder, it is possible to write c , . . . , c N h ( d , . . . , d N ∞ ) in termsof c ( d ) . The latter is usually set to 1. The choice of N h and/or N ∞ should be a trade-off between computational timeand accuracy. If the goal is to compute QNM frequencies, oneassigns a guess value to ω and integrates Eq. (19) from r = r ∞ to r = r using the ansatz ˜ H ∞ with A − s = H ∞ = H ω m and d ˜ H ∞ / d r = d H ω m / d r at r = r ∞ , where r ∞ stands for the numerical value of infinity.The previous step is repeated for different guess values of ω until the solution satisfies the BC in Eq. (33) (one-parametershooting). If the algorithm is numerically stable, variations in N ∞ and/or r ∞ do yield similar results. On the other hand, if theaim is to estimate the amplification factors defined in Eq. (31)for a given frequency ω , one integrates Eq. (19) from r = r to r = r ∞ using the ansatz ˜ H h so that the solution satisfies therelations ˜ H h = H ω m and d ˜ H h / d r = d H ω m / d r at r = r and thenextracts the coefficients A ± s of the ansatz ˜ H ∞ at infinity.All numerical integrations were performed using the inte-gration parameters N h = N ∞ =
10. When computingQNM frequencies (amplification factors), r ∞ was set to 100(400). The guess values to the QNM frequencies were chosenaccording to the numerical results reported in [25]. Using Eq. (17), one can show that e − i ϖ r ∗ = e − i ϖ r (cid:16) δδ + (cid:17) − i ϖ / ∼ δ − i ϖ / ,where the last step holds as long as δ (cid:28) B. Quasi-normal modes
Fig. 1 shows the fundamental m = { δ , B } for DBCs ( R = −
1) and NBCs( R = ω I . As pointed out in Sec. II, ω R depends on arg ( R ) but not on | R | . This means that the top panels of Fig. 1 giveinformation about the real part of the QNM frequencies ofboth perfectly- and partially-reflecting ECO-like vortices.The results are qualitatively similar for both BCs. At firstorder in B (i.e. for B (cid:46) . ω R is a linear function of theangular velocity B , as one would expect from Eq. (34). Theinitial value (corresponding to B =
0) depends on r ∗ or, moreprecisely, on the the inverse of log δ , in accordance with Eq.(17). Similarly, ω I grows monotonically with increasing ro-tation. Both ω R and ω I change sign from negative to positiveas B increases. Within numerical accuracy, the sign changesoccur at the same critical value B c , meaning ω R , ω I < B < B c and ω R , ω I > B > B c . In other words, QNMsturn from stable to unstable as the fluid spins faster and fasterand, furthermore, perfectly-reflecting ECO-like vortices ad-mit zero-frequency ( ω =
0) QNMs. Similar phenomenologi-cal aspects regarding the interaction of massless bosonic fieldswith perfectly-reflecting Kerr-like ECOs have been reportedin [25, 26, 54]. It was shown in particular that some ECOsor ECO analogues can only support static configurations of ascalar field for a discrete set of critical radii [54–57]. This alsoholds true for the present case, as shown in the Appendix B.The aforementioned instability finds its origin in the pos-sible existence of negative-energy physical states inside theergoregion. In general, in BH physics, the absence of an eventhorizon turns horizonless rotating ECOs unstable [17, 25].The event horizon of Kerr BHs, which can be regarded as aperfectly-absorbing surface ( R = Z >
0) when scatteredoff perfectly-reflecting ECO-like vortices, thanks to the trans-fer of angular momentum and energy from the latter to the for-mer. According to Eq. (37), only the absence of superradiantamplification when R = Z <
0) guarantees exponentiallydecaying responses of perfectly-reflecting ( | R | =
1) ECOs toexternal linear perturbations. From a dynamical point of view,the instability is expected to result in the scatterer spinningslower and slower until its fundamental QNM mode, whosefrequency depends on B , starts decaying rather than growingover time, which occurs when B = B c (note that Z < B < B c ). The instability domain of ECO-like vortices is de-piected in Fig. 2. The threshold decreases monotonically as r → r H (i.e. as δ → Since δ (cid:28) δ is a more suitable parameter than r . - - - ω R δ = - δ = - δ = - δ = - Superradiant regime ℛ = - < ω < m Ω H - - - ω R δ = - δ = - δ = - δ = - ℛ = + - - - - - - B | ω I | Stable ω I < Unstable ω I > - - - - - B | ω I | Stable ω I < Unstable ω I > Figure 1. Real ( top ) and imaginary ( bottom ) parts of the fundamental m = r ≡ r H ( + δ ) , 0 < δ (cid:28)
1, where r H is the would-be acoustic horizon of the corresponding draining bathtub, as a function ofthe rotation parameter B , for DBCs ( left ) and NBCs ( right ). The left (right) arms of the interpolating functions refer to negative (positive)frequencies. The colored dots are zero-frequency (marginally stable) QNMs. - - - - ( δ ) B c ℛ = - ℛ = + Unstable
Figure 2. Ergoregion instability rotation parameter threshold ofthe fundamental m = R = −
1) and NBCs ( R = Is there any way of preventing ECO-like vortices from de-veloping instabilities without changing B ? The answer is af-firmative. In the presence of superradiance ( Z > B < B c , it is clear that only partially-reflecting ECO-like vor-tices may be stable. However, attention must be paid to thefact that an ECO with | R | < | R | can take toassure stability. Using the small-frequency approximations inEqs. (A15) and (A16), Eq. (37) takes the explicit form | R | < (cid:12)(cid:12)(cid:12)(cid:12) Γ ( m ) + i πψ i χ ( ω / ) m Γ ( m ) − i πψ i χ ( ω / ) m (cid:12)(cid:12)(cid:12)(cid:12) , (38)where χ = β ( β − m ) − ∏ m − n = ( β − m + n ) . The upper boundon | R | represents the threshold for the manifestation of er-goregion instabilities and depends both on ω and B , as shownin Fig. 3.One is interested in setting restrictions on | R | which arefrequency-independent, in order to avert exponentially grow-ing linear perturbations regardless of their frequency. The ab-solute upper bound on | R | from Eq. (37), herein dubbedmaximum reflectivity, is set by the maximum amplificationfactor when R =
0, i.e. to the minimum of the function ( + Z ) − , and is therefore independent of δ . For suf-ficiently slow drains ( B ≤ | R | (cid:46) Figure 3. Plot of ( + Z ) − as a function of the rotation param-eter B and of the frequency ω of acoustic perturbations. When | R | < ( + Z ) − , ECO-like vortices are dynamically stable.Note that the upper limit on | R | decreases as B increases. C. Superradiant scattering
Fig. 4 illustrates the amplification factors for superradi-ant m = δ = − , B = . ω = ω = mB )and have a maximum value. The graph referring to the drain-ing bathtub ( R =
0) is in agreement with numerical resultspreviously reported in the literature [6, 7]. The maximum am-plification is about 8% when R =
0, meaning the maximumreflectivity is approximately 92% when B = .
6. When | R | is nonvanishing, a resonance becomes noticeable around a fre-quency of about 0 .
372 (dotted vertical lines in Fig. 4), whichmatches the real part of a QNM frequency precisely. Like inclassical mechanics, an acoustic linear perturbation extractsmore angular momentum and energy when its frequency co-incides with the object’s proper frequencies of vibration. Thepeak’s height is determined by the imaginary part of ω QNM [49], being maximum when ω I =
0. As shown in Fig. 5, theQNM which sets the peaks in Fig. 4 is marginally-stable when R ≈ .
964 (dotted vertical line in Fig. 5). Fig. 6 confirms thatthe maximum value of the amplification factor occurs indeedfor a reflectivity around 0 . IV. CONCLUSION
GW astronomy opens a new window on the universe and isexpected to unveil spacetime features in the vicinity of com-pact objects, testing both general relativity and BH physicspredictions. However, present GW observations are not pre-cise enough to (indirectly) probe the true nature of BH candi-dates and do not rule out alternative scenarios. This has beenone of the strongest motivations behind ECO models. Theirphenomenology has been widely addressed in search of alter-natives to the BH paradigm.Following this trend, this work aimed to explore the phe-nomenology of acoustic perturbations of an analogue modelfor ECOs built from the draining bathtub geometry, namedherein ECO-like vortices. These objects feature an ergore-gion and are endowed with a surface with reflective propertiesrather than an acoustic horizon. Although ECO-like vorticesdo have the key ingredients to trigger ergoregion instabilities,it turns out that dissipation mitigates or even neutralizes expo-nentially growing modes.The analysis led to two main conclusions, supported by alow-frequency analysis and by direct-integration numericalcalculations. First, when the object’s surface is perfectly-reflecting ( | R | = ) , an instability develops when the vor-tex is spinning at a rate above some critical value of the ro-tation parameter. Despite the dependence of the instabilitydomain on the location of the surface, it generally occurswhen B > O ( . ) . The instability is intimately linked to theergoregion, where negative-energy physical states can form.These cannot be absorbed by the vortex’s surface and, there-fore, cause the exponential growth of acoustic perturbations.The ergoregion instability of ECO-like vortices is attenuatedor neutralized when its surface is not perfectly- but partially-reflecting. An absorption coefficient greater than approxi-mately 30% prevents unstable QNMs to develop in ECO-likevortices with B below unit. The results are similar to those re-ported in [25, 26, 49] and attests that a general way of prevent-ing such instabilities from arising is to incorporate dissipative-like effects into the surface.Second, the stimulation of exponentially growing QNMsis optimized for more reflective surfaces and, therefore, isexpected to generate narrower spectral lines in the emissioncross sections, similarly to those in the absorption cross sec-tions of spherically symmetric ECOs [49]. An interesting ex-tension of the work presented herein would precisely be tocompute the emission cross sections of ECO-like vortices, asit would be useful in probing the effects of dissipation uponsuperradiant scattering in possible future experiments. Imple-menting a setup which reproduces the ECO-like vortex intro-duced here appears to be a thorny issue. This would requireto place a right circular cylinder at the acoustic horizon of avortex flow.Moreover, the reflectivity R was assumed to be frequency-independent. A possible future extension may lift this as-sumption and consider ECOs with frequency-dependent re-flectivities. - - - - - Z NumericalAnalytical ℛ = ℛ = ℛ max ( Z ) % % % % - - - - - ω / mB Z ℛ = ω / mB ℛ = ω R = Figure 4. Numerical and analytical values of the amplification factors for superradiant (0 < ω < mB ) acoustic perturbations with m = B = . R at r = r H ( + δ ) , where r H is the would-be acoustichorizon of the corresponding draining bathtub and δ = − . The resonance matches the real part of the fundamental QNM frequency of thevortex-like ECO (dotted vertical lines). - - - - ℛ ω I ω R = Figure 5. Imaginary part of the QNM frequency which sets thepeaks of the amplification factors plotted in Fig. 4, as a function ofthe reflectivity R . ω I vanishes at R ≈ .
964 (dotted vertical line). ω / mB Z ℛ = ℛ = ℛ = ℛ = ℛ = Figure 6. Analytical approximation for the amplification factors plot-ted in Fig. 4 in the neighborhood of the fundamental m = R ≈ . ACKNOWLEDGMENTS
This work has been supported by Fundação para a Ciênciae a Tecnologia (FCT) grant PTDC/FIS-OUT/28407/2017 andby CENTRA (FCT) strategic project UID/FIS/00099/2013.This work has further been supported by the EuropeanUnion’s Horizon 2020 research and innovation (RISE) pro-grammes H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904 and H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. The authors would like to acknowledge networkingsupport by the COST Action CA16104.
Appendix A: Analytical analysis
A number of relevant quantities, such as QNM frequenciesand amplification factors, can be computed analytically in thelow-frequency regime ( ω (cid:28)
1) using matching-asymptotictechniques [22]. Contrarily to previous works [58], mostlyfocused on the BH case ( R = R . For thatpurpose, the spacetime region outside the reflective surface at r = r is split into a region near the would-be acoustic hori-zon, i.e. in the limit r − r H (cid:28) / ω , and a region far from it,i.e. at infinity, where r (cid:29) r H . In the following, besides thecondition B ω (cid:28)
1, the assumption ϖ (cid:28) (cid:28) r − r H (cid:28) / ω .To solve Eq. (13) in the far region, it is convenient to rewriteit in the form (cid:34) ∆ dd r (cid:18) ∆ dd r (cid:19) + (cid:18) ω r − mBr (cid:19) − ∆ m r (cid:35) S ω m ( r ) = , (A1)where ∆ ( r ) = rg ( r ) . When r (cid:29) r H and B ω (cid:28)
1, Eq. (A1)reduces to a Bessel ODE, (cid:20) r d d r + r dd r + ( ω r − m ) (cid:21) S ω m ( r ) = , (A2)whose general solution is a linear combination of Bessel func-tions of the first and second kinds, S ω m ( r ) = A J m ( ω r ) + B Y m ( ω r ) , (A3)where A , B ∈ C . The large- r behavior of the asymptotic so-lution in Eq. (A3) is S ω m ( r ) ∼ (cid:114) πω r [ A cos ( ω r − ς ) + B sin ( ω r − ς )] , (A4)where ς ≡ π ( m + / ) /
2, which can be written in terms of theradial function H ω m ( r ) and as a superposition of ingoing andoutgoing waves. In effect, H ω m ( r ) ∼ ( A + i B ) e − i ( ω r − ς ) + ( A − i B ) e + i ( ω r − ς ) √ πω . (A5) The amplitudes of the ingoing and outgoing waves are pro-portional to ( A + i B ) and ( A − i B ) , respectively. The maingoal of the asymptotic matching is to find expressions for A and B in terms of r (or δ ), R , B , m and ω .The small- r behavior of the asymptotic solution in Eq. (A3)is S ω m ( r ) ∼ A ( ω / ) m Γ ( m + ) r + m − B Γ ( m ) π ( ω / ) m r − m . (A6)In the near region, by defining S ω m ( r ) = r − m [ g ( r )] − i ϖ T ω m ( r ) , (A7)introducing the new coordinate z = r − and using the condi-tions B ω (cid:28) ϖ (cid:28)
1, one can bring Eq. (13) into thehypergeometric form (cid:20) z ( − z ) d d z + [ γ − ( α + β + ) z ] dd z − αβ (cid:21) T ω m ( z ) = , (A8)where 2 α = + m − i ϖ , 2 β = m − i ϖ and γ = m +
1. It iseasy to check that α , β and γ satisfy the relations α − β = α + β = α − = β + γ − α − β = i ϖ .The hypergeometric differential equation has three singularpoints: z = , , ∞ . Its most general solution in the neighbor-hood of z = T ω m ( z ) = C T i ω m ( z ) + D ( − z ) γ − α − β T o ω m ( z ) , (A9)where C , D ∈ C , T i ω m ( z ) = F ( α , β ; α + β − γ +
1; 1 − z ) and T o ω m ( z ) = F ( γ − β , γ − α ; γ − α − β +
1; 1 − z ) . F ( α , β ; γ ; z ) is the hypergeometric function. Since T i ω m ( ) = T o ω m ( ) =
1, one finds that the small- r behavior ofthe asymptotic solution in Eq. (A9) is T ω m ( z ) ∼ C + D ( − z ) + i ϖ , (A10)Using Eqs. (18) and (A7), one can show that H ω m ( r ) ∼ C e − i ϖ r ∗ + D e + i ϖ r ∗ (A11)near the would-be acoustic horizon, where C ≡ C (cid:18) r + re r (cid:19) − i ϖ , D ≡ D (cid:18) r + re r (cid:19) + i ϖ . (A12)Therefore, the BC (33) turns into DC = R (cid:18) r r − (cid:19) i ϖ . (A13)Given that γ ∈ Z , one must be careful when analyzing thelarge- r (small- z ) behavior of the asymptotic solution (A9) [59,60]. One can show that [58] T i ω m ( z ) ∼ T i (cid:20) ( − ) m − m ( α − m ) m ( β − m ) m z − m + ψ i (cid:21) , T o ω m ( z ) ∼ T o (cid:20) ( − ) m − m ( − α ) m ( − β ) m z − m + ψ o (cid:21) , T i ≡ ( − ) m + m ! Γ ( α + β − m ) Γ ( α − m ) Γ ( β − m ) , ψ i ≡ ψ ( α ) + ψ ( β ) − ψ ( m + ) − ψ ( ) , T o ≡ ( − ) m + m ! Γ ( m + − α − β ) Γ ( − α ) Γ ( − β ) , ψ o ≡ ψ ( m + − α ) + ψ ( m + − β ) − ψ ( m + ) − ψ ( ) . ( q ) n is the rising Pochhammer symbol and ψ ( · ) is thedigamma function [59, 60].Using Eq. (A7) and rearranging the terms, one gets S ω m ( r ) ∼ A r + m + B r − m , (A14)where A ≡ ( − ) m − m (cid:20) T i C ( α − m ) m ( β − m ) m + T o D ( − α ) m ( − β ) m (cid:21) , B ≡ ψ i T i C + ψ o T o D . Eq. (A14) exhibits the same dependence on r as Eq. (A6).Matching the two solutions, it is straightforward to show that A = Γ ( m + )( ω / ) m A , (A15) B = − π ( ω / ) m Γ ( m ) B . (A16)One can derive some relevant physical quantities from Eqs.(A15), (A16) and related, namely QNM frequencies andamplifications factors of acoustic perturbations scattered offECO-like vortices.The very definition of QNM requires the amplitude of theingoing wave at infinity to vanish. It then follows from Eq.(A5) that one must impose A + i B = . (A17)Eq. (A17) can be solved using a simple root-finding algo-rithm.On the other hand, according to Eq. (31), the amplificationfactors are given by Z m ( ω , R ) = (cid:12)(cid:12)(cid:12)(cid:12) A − i BA + i B (cid:12)(cid:12)(cid:12)(cid:12) − . (A18) Appendix B: Static configurations
In order to study the static configurations of Eq. (9), onedefines the radial function L ( r ) [9] R ω m ( r ) = r e i [ ( ω − mB ) log ( r − )+ mB log ( r ) ] L ω m ( r ) , (B1)introduces the new coordinate x = r − ω = (cid:20) d d x + W ( x ) dd x + W ( x ) (cid:21) L ω m ( x ) = , (B2) where W ( x ) = x + x + , (B3) W ( x ) = x ( x + ) (cid:20) m B x + x + + ( − m ) (cid:21) . (B4)Eq. (B2) is a standard Riemann-Papparitz ODE. Defining L ω m ( x ) = x σ ( x + ) − G ω m ( x ) , (B5)with 2 σ = − imB , and introducing the new coordinate y = − x ,one gets (cid:20) y ( − y ) d d y + ρ ( − y ) dd y − κλ (cid:21) G ω m ( y ) = , (B6)with 2 κ = − m ( + iB ) , 2 λ = − m ( − + iB ) and ρ = − imB .It is easy to check that κ , λ and ρ satisfy the relations κ + λ + = ρ , λ − κ = m and ρ = − ρ , where ρ is the complexconjugate of ρ .The general solution of Eq. (B6) reads [59] G ω m ( y ) = E F ( κ , λ , ρ , y )+ E y − ρ F ( − λ , − κ , ρ , y ) , (B7)where E , E ∈ C . Equivalently, L ω m ( x ) = ( x + ) − · [ E x σ F ( κ , λ , ρ , − x ) + E x − σ F ( − λ , − κ , ρ , − x )] , (B8)where E ≡ ( − ) − ρ E . The small- x behavior of Eq. (B8)must be written in terms of the radial function H ( r ) for theBC in Eq. (33) to be applied. One can show that H ω m ( r ) ∼ E e + imBr ∗ + E e − imBr ∗ (B9)near the would-be acoustic horizon, meaning the BC to beimposed at r = r is E / E = R e + imBr ∗ . (B10)The large- x behavior of Eq. (B8) can be analyzed by mak-ing use of the linear transformation formula 15 . . L ω m ( x ) = ( x + ) − (cid:104) F x + m + F x − m (cid:105) . (B11)with F = (cid:20) E Γ ( ρ ) λ Γ ( λ ) − E Γ ( ρ ) κ Γ ( − κ ) (cid:21) Γ ( λ − κ ) , (B12) F = (cid:20) E Γ ( ρ ) κ Γ ( κ ) − E Γ ( ρ ) λ Γ ( − λ ) (cid:21) Γ ( κ − λ ) . (B13)It follows from the explicit expression for the energy fluxacross an arbitrary surface at a constant radial coordinate r that the amplitudes of the ingoing wave and of the outgoing1wave are proportional to ( F − i F ) and ( F + i F ) , respec-tively [9]. The absence of ingoing waves at infinity requiresthat F − i F = , (B14)which implies that E E = Γ ( ρ ) Γ ( ρ ) (cid:20) Γ ( κ ) Γ ( − κ ) Γ ( λ ) Γ ( − λ ) (cid:21) · λ Γ ( − λ ) Γ ( λ − κ ) − i κ Γ ( − κ ) Γ ( κ − λ ) κ Γ ( κ ) Γ ( λ − κ ) − i λ Γ ( λ ) Γ ( κ − λ ) (B15)On the other hand, Eq. (23) can be written in terms of theradial function L ω m ( x ) ,˙ L ω m ( x ) L ω m ( x ) = − ( + x ) − ( ξ ) √ + x (B16) where the dot denotes differentiation with respect to x . LikeEq. (B14), Eq. (B16) is a condition on the amplitudes E and E . Explicitly, E E = ( r − ) − σ [( − σ ) r − ] tan ( ξ ) + r ( r − )[( + σ ) r − ] tan ( ξ ) + r ( r − ) . (B17)Equating the right-hand side of Eqs. 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