Exotic 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 surface with 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 instabilities 2 2 when |R| ≈ 1. However, stability can be attained for sufficiently slow drains when |R| . 70%. It is shown that the superradiant scattering of acoustic waves is more effective when their frequency approaches one of the system’s quasi-normal mode frequencies.
I. INTRODUCTION cannot escape from the region around the drain. The phenomenology of the draining bathtub model has been widely addressed over the last two decades. For instance, Analogue models for gravity have proven to be a power- works on quasi-normal modes (QNMs) [6,7], absorption pro- ful tool in understanding and probing several classical and cesses [8] and superradiance [9–12] showed that this vortex quantum phenomena in curved spacetime, namely the emis- geometry shares many properties with Kerr spacetime. sion of Hawking radiation and the amplification of bosonic Kerr BHs are stable against linear bosonic perturbations field perturbations scattered off spinning objects, commonly [13–16]. The event horizon absorbs any negative-energy dubbed superradiance. It was Unruh who first drew an ana- physical states which may form inside the ergoregion and logue model for gravity relating the propagation of sound would otherwise trigger an instability. In fact, as first shown waves in fluid flows with the kinematics of waves in a clas- by Friedman [17], asymptotically-flat, stationary solutions to sical gravitational field [1]. This seminal proposal unfolded a Einstein’s field equations possessing an ergoregion but not an completely unexpected way of exploring gravity and opened event horizon may develop instabilities, usually called ergore- a track to test it in the laboratory. A summary of the history gion instabilities, when linearly interacting with scalar and and motivation behind analogue gravity can be found in [2]. electromagnetic field perturbations, especially if rapidly spin- The analogy between the propagation of sound waves in a ning. Ergoregion instabilities are known to affect a plethora non-relativistic, irrotational, inviscid, barotropic fluid and the of exotic compact objects (ECOs) [18–26]. These are loosely propagation of a minimally coupled massless scalar field in defined in the literature as objects without event horizon, a curved Lorentzian geometry was primarily established by more massive than neutron stars and suffciently dim not to Visser [3,4], who also formulated the concepts of acoustic have been observed by state-of-the-art electromagnetic tele- horizon, ergoregion and surface gravity in analogue models. scopes and detectors yet. Examples include boson stars The correspondence is purely kinematic, i.e. only kinematic [27], anisotropic stars [28], wormholes [29], gravastars [30], aspects of general relativity, such as event horizons, appar- fuzzballs [31], black stars [32], superspinars [33], Proca stars ent horizons and ergoregions, are carried over into fluid me- [34], collapsed polymers [35], 2 − 2 holes [36] and AdS bub- chanics. The effective geometry of the flow is mathematically bles [37]. All these ECOs may be prone to light ring insta- encoded in a Lorentzian metric, commonly dubbed acoustic bilities [38]. The timescale of such instability, however, is metric, governed by the fluid equations of motion and not by unknown. An exhaustive review including references (if ex- Einstein’s field equations. In other words, the dynamic as- isting for a given ECO model) on the formation, stability and pects of general relativity do not map into fluid mechanics. electromagnetic and gravitational signatures of such objects This partial isomorphism thus offers a rather simple way to can be found in [39]. arXiv:1902.06748v2 [gr-qc] 1 Apr 2019 disentangle the kinematic and dynamic contributions to some The recent gravitational-wave (GW) detections [40–45] important phenomena in general relativity. from compact binary coalescences heralded the dawn of a Dumb holes, the acoustic analogues for black holes (BHs), brand new field in astronomy and astrophysics. The newborn bear several structural and phenomenological similarities to era of precision GW physics is expected to probe strong-field BHs. For instance, the draining bathtub vortex [3,5], or sim- gravity spacetime regions in the vicinity of compact objects ply draining bathtub, which features both an ergoregion and and, most importantly, to provide the strongest evidence of an acoustic horizon, is a Kerr BH analogue. The model de- event horizons. While current electromagnetic-wave and GW scribes the two-dimensional flow of a non-relativistic, locally observations do support the existence of BHs, some other ex- irrotational, inviscid, barotropic fluid swirling around a drain. otic alternatives are not excluded yet − not even those which The fluid velocity increases monotonically downstream. The do not feature an event horizon. There has been a renewed in- region where the magnitude of the fluid velocity exceeds the terest in these exotic alternatives over the last decades because speed of sound defines the ergoregion. Nearer the drain is the some ECOs can mimic the physical behavior of BHs, namely acoustic horizon, which comprises the set of points in which those whose near-horizon geometry slightly differs from that the radial component of the fluid velocity equals the speed of of BHs, such as Kerr-like ECOs [25, 26]. These objects fea- sound. Any sound wave produced inside the acoustic horizon ture an ergoregion and are endowed with a surface with reflec- 2
2 tive properties at a microscopic or Planck distance from the where the constant prefactor (ρ0/c) has been omitted. Eq. would-be event horizon of the corresponding Kerr BH. The (2) has the Painlevé-Gullstrand form [47]. presence of an ergoregion and the absence of an event hori- The flow is stationary and axisymmetric, i.e. the metric zon are the two key ingredients for ergoregion instabilities to tensor in Eq. (2) does not depend explicitly on t nor on φ. develop. Indeed, perfectly-reflecting Kerr-like ECOs exhibit In polar coordinates, the Killing vectors associated with these some exponentially-growing QNMs and are unstable against continuous symmetries are ξt ≡ ∂t and ξφ ≡ ∂φ , respectively. linear perturbations in some region of parameter space. How- This geometry has an acoustic horizon located at r = rH ≡ A/c ever, the unstable modes can be mitigated or neutralized when and a√ region of transonic flow defined by rH < r < rE , where 2 2 1 the surface is not perfectly- but partially-reflecting, i.e. when rE = A + B /c is the location of the ergosphere . dissipative effects are considered. Performing the coordinate transformation This work focuses on classical phenomenological aspects Ar of a draining bathtub model whose acoustic horizon is re- dt˜ = dt − dr, (3) placed by a surface with reflective properties. The physical c2r2 − A2 behavior of this «toy model» is quite similar to that of Kerr- AB dφ˜ = dφ − dr, (4) like ECOs. Although the experimental realization of such r(c2r2 − A2) system is not evident at present, it is still fruitful to explore this theoretical setup, as it may provide some insights into the one can cast Eq. (2) in the form [10, 11, 48] generic features of ECOs and even be useful for possible fu- 2B ture experiments. ds2 = − f (r)dt˜2 + [g(r)]−1dr2 − dt˜dφ˜ + r2dφ˜2, (5) The present paper is organized as follows. Sec. II covers c in brief the main mathematical and physical features of the where draining bathtub geometry, introduces the equation governing acoustic perturbations and discusses the properties of its solu- A2 + B2 A2 f (r) = 1 − and g(r) = 1 − . (6) tions. Numerical results regarding QNM frequencies and am- c2r2 c2r2 plification factors of the system are presented Sec. III, com- plemented with a low-frequency analytical treatment in the The coordinates (t˜,r,φ˜) are equivalent to the Boyer-Lindquist AppendicesA andB. coordinates commonly used to write the Kerr metric.
II. DRAINING BATHTUB MODEL B. Acoustic perturbations
A. Acoustic metric 1. Model
The draining bathtub geometry introduced by Visser [3] de- Perturbations to the steady flow can be encoded in the ve- scribes the irrotational flow of a barotropic and inviscid fluid locity potential by adding a term to it, i.e. by considering with background density ρ0 in a plane with a sink at the origin. Ψ = Ψ0 + Φ, where Φ is the perturbation. The equations of The irrotational nature of the flow together with the conserva- motion governing an acoustic perturbation in the velocity po- tion of angular momentum require the background density to tential of an irrotational flow of a barotropic and inviscid fluid be constant, i.e. position-independent. As a result, the back- are the same as the Klein-Gordon equation for a minimally ground pressure p0 and the speed of sound c are also constant coupled massless scalar field propagating in a Lorentzian ge- throughout the flow. Furthermore, it follows from both the ometry [3]. In effect, the perturbation Φ in the velocity poten- equation of continuity and the conservation of angular mo- tial satisfies the equation mentum that the unperturbed velocity profile v of the flowing 0 1 √ fluid is given in polar coordinates (r,φ) by Φ = √ ∂ −ggµν ∂ Φ = 0, (7) −g µ ν (r) (φ) v0 ≡ v er + v eφ , (1) µ 0 0 where ≡ ∇µ ∇ is the D’Alambert operator and ∇µ denotes (r) (φ) + the covariant derivative. In the present case, the index µ runs with v0 = −A/r and v0 = B/r, where A,B ∈ R . er and from 0 to 2, with 0 referring to the time coordinate t and 1 and eφ are the radial and azimuthal unit vectors, respectively. A 2 to the spatial coordinates r and φ, respectively. general velocity profile v, allowing for perturbations, can be One is interested in (2+1)−dimensional objects whose ge- written as the gradient of a velocity potential Ψ, i.e. v = ∇Ψ. ometry is described by Eq. (2) from r0 to infinity, where When v = v0, Ψ = Ψ0(r,θ) ≡ Alogr + Bφ (apart from a con- r0 is the location of a surface with reflectivity R. Such stant of integration). In polar coordinates (t,r,φ), the line element of the draining bathtub model has the form [5, 46] 1 2 2 √The speed of the flowing fluid is φ−independent and given by v0 ≡ kv0k = A B 2 2 ds2 = −c2dt2 + dr + dt + rdφ − dt , (2) A + B /r. It equals the speed of sound at r = rE and exceeds it every- r r where in the region rH < r < rE . 3 objects will hereafter be referred to as ECO-like vortices. It is useful to introduce a tortoise coordinate defined by the Perfectly-reflecting and perfectly-absorbing surfaces are de- condition [9] fined by |R| = 1 and R = 0, respectively. The model sets the −1 surface at a small distance from the would-be acoustic hori- dr∗ 1 = [g(r)]−1 = 1 − . (16) zon, i.e. at r0 = rH (1+δ), where 0 < δ 1. The requirement dr r2 that δ 1 allows ECO-like vortices to feature an ergoregion 2 (r0 < rE ) . Explicitly, the tortoise coordinate is given by For the sake of simplicity, the uniform scalings r → Ar/c B → B/A 1 r − 1 and will hereafter be adopted [6, 11]. Note that r∗(r) = r + log , (17) these linear transformations are equivalent to set A = c = 1 in 2 r + 1 Eqs. (2) and (5). which maps the acoustic horizon at rH = 1 to r∗ → −∞ and r → +∞ to r∗ → +∞. Together with the definition [9] 2. Perturbation equation H (r) S (r) = ω√m , (18) ωm r From the existence of the Killing vectors ξt and ξφ , one can separate the t− and φ−dependence of the field Φ, which Eq. (13) can be written as in turn can be expressed as a superposition of modes with dif- 2 ferent frequencies ω and periods in φ, i.e. d 2 + V (r) Hωm(r) = 0, (19) dr∗ +∞ Z +∞ −iωt +imφ Φ(t,r,φ) = ∑ dω e Rωm(r)e , (8) where the effective potential V (r) is given by m=−∞ −∞ 2 mB g(r) 2 5 where R m(r), dubbed radial function, is a function of the V (r) = ω − − 4m − 1 + , (20) ω r2 r2 r2 radial coordinate only and depends on B, ω and m. The radial 4 function satisfies the ordinary differential equation (ODE) [9] which has the asymptotic behavior
d2 d 2 + (r) + (r) R (r) = 0, (9) ω , r∗ → +∞ 2 K1 K2 ωm V (r) ∼ 2 , (21) dr dr ϖ , r∗ → −∞ where with ϖ ≡ ω − mB. B coincides with the angular velocity of the would-be acoustic horizon. 1 + r2 + 2i(mB − ωr2) K (r) = , (10) 1 r(r2 − 1) ω2r4 − m2r2 + m2B2 − 2mBωr2 − 2imB 3. Asymptotic solutions K (r) = . (11) 2 r2(r2 − 1) The presence of a surface with reflectivity R requires solu- Defining a new radial function Sωm(r) as tions to Eq. (19) to be a superposition of ingoing and outgoing waves at r = r0. Thus, following the notation in [49], general i [(ω−mB)log(r2−1)+2mBlog(r)] solutions have the asymptotics Rωm(r) = Sωm(r)e 2 , (12)
∗ −iϖr∗ +iϖ(r∗−2r ) ∗ Eq. (9) becomes e + Re 0 , r∗ → r0 Hωm(r) ∼ , (22) − −iωr + +iωr A e ∗ + A e ∗ , r∗ → +∞ d2 d s s + L1(r) + L2(r) Sωm(r) = 0, (13) dr2 dr ∗ where r0 ≡ r∗(r0) < 0. For the sake of simplicity, although R may depend on ω and/or r [49–51], this work will only focus where 0 on constant-valued (ω− and r0−independent) reflectivities. 2 r2 + 1 Perfectly-reflecting (|R| = 1) boundary conditions (BCs), L (r) = , (14) generically known as Robin BCs, are given by [52] 1 r(r2 − 1) 2 4 2 2 2 2 0 ω r − (m + 2mBω)r + m (1 + B ) cos(ξ)Hωm(r0) + sin(ξ)H (r0) = 0, (23) L (r) = . (15) ωm 2 (r2 − 1)2 where ξ ∈ [0,π) and the prime denotes differentiation with respect to r. Note that ξ = 0 corresponds to a Dirichlet BC (DBC), i.e. Hωm(r0) = 0, whereas ξ = π/2 refers to a Neumann BC (NBC) imposed on Hωm(r) at r = r0, i.e. 2 ECO-like vortices reduce to the draining bathtub when δ = 0 and R = 0. 0 Hωm(r0) = 0. Equivalently, DBCs (NBCs) can be defined by 4
R = −1 (R = 1)3. Perfectly-reflecting BCs will hereafter be Φ are dubbed QNMs [15]. The set of all eigenfrequencies is specialized to DBCs and NBCs only. often referred to as QNM spectrum. The QNM frequencies The solution in Eq. (22) may be written as a superposition ωQNM are in general complex, i.e. ωQNM = ωR + iωI, where of modes with asymptotics [18] ωR ≡ Re{ωQNM} and ωI ≡ Im{ωQNM}. The sign of ωI de- fines the stability of the QNM. According to the convention −iϖr∗ ∗ for the Fourier decomposition in Eq. (8), if: ω < 0, the mode + e , r∗ → r0 I H m(r) ∼ (24) ω − −iωr + +iωr is stable and τdam ≡ 1/|ωI| defines the damping e-folding A e ∗ + A e ∗ , r∗ → +∞ ∞ ∞ timescale; ωI > 0, the mode is unstable and τins ≡ 1/ωI de- fines the instability e-folding timescale; ωI = 0, the mode A−e−iϖr∗ + A+e+iϖr∗ , r → r∗ − h h ∗ 0 is marginally stable or stationary. When analyzing unstable H m(r) ∼ (25) ω +iωr e ∗ , r∗ → +∞ QNMs, one is commonly interested in those corresponding to the shortest instability timescales. In the present case, these + Hωm is a draining bathtub solution (R = 0) of the scattering are the fundamental m = 1 QNMs. − problem, whereas Hωm is an ECO solution of the QNM eigen- The absence of ingoing waves at infinity is equivalent to − value problem (since it is a superposition of ingoing and out- setting As = 0 in Eq. (28), i.e. to requiring going waves at the reflective surface). ∗ + − −i2(ωQNM−mB)r0 From the constancy of the Wronskians of Eq. (19), one can Ah /Ah = Re . (33) write the following useful relations between the coefficients ± ± ± One can solve Eq. (33) for ωQNM, which yields As , Ah and A∞ : ωA− = ϖA+, (26) 1 − + ∞ h ωR = mB + ∗ arg(R) + arg(Ah /Ah ] (34) + −∗ 2r0 ωA∞ = −ϖAh , (27) ∗ 1 2 − + 2 − + − −2iϖr0 ωI = − log|R| + log|A /A | . (35) ωAs = ϖ(Ah − Ah Re ), (28) ∗ h h 4r0 + − − + −2iϖr∗ ω(As A − As A ) = ϖRe 0 , (29) ∞ ∞ arg( ) dictates the difference in phase between ingoing and − 2 + 2 2 R ω(|As | − |As | ) = ϖ(1 − |R| ). (30) outgoing 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). 4. Superradiance If R is a complex number, the phase difference is a multiple of some real number between 0 and π. Without loss of gen- The amplification factor in a scattering process is defined erality (as far as QNM stability is concerned), the reflectivity by [53] R will hereafter be considered a real parameter. Note that ωR does not depend on the magnitude of . Thus, once arg( ) + 2 2 R R A mB 1 − |R| 2 Z( , ) = s − 1 = − 1 − , (31) is fixed, the introduction of dissipation (|R| < 1) does not af- ω R − − 2 As ω |As | fect the real part of the QNM frequency. On the contrary, the imaginary part changes with changing |R|2. It follows from where the last equality follows from the Wronskian relation Eq. (35) that |R|2 determines QNM stability. Such stability in Eq. (30). The amplitude of the reflected wave is greater is achieved whenever ω < 0, i.e. when than that of the incident wave at infinity (i.e. Z > 0) when I 0 < < mB. Note that the superradiance condition does not 2 2 ω A+ A− depend on . 2 h ∞ R |R| < − = + (36) Ah A∞
5. Quasi-normal modes where the last equality follows from the Wronskian relations in Eqs. (26) and (27). The last term in Eq. (36) is the inverse ∗ of the superradiant coefficient for the draining bathtub (R = Once physical BCs at r∗ → r0 and r∗ → +∞ are imposed, 0). Thus, one can write Eq. (19) defines an eigenvalue problem. If one requires purely outgoing waves at infinity, 1 |R|2 < , (37) +iωr∗ 1 + Z0(ωR) Hωm(r) ∼ e , r∗ → +∞, (32) where Z ( ) ≡ Z( ,0). In other words, the upper bound the eigenvalues, the characteristic frequencies ω , are 0 ωR ωR QNM | |2 called QNM frequencies and the corresponding perturbations on the range of values R can take to assure stability is a function of the amplification factors for the draining bathtub only. The same result can be derived from a «bounce-and- amplify» argument [26, 53]. When |R|2 = 1, the condition 3 −iϖr∗ 0 −iϖr∗ Plugging Hωm(r0) = (1 + R)e 0 and Hωm(r0) = −iϖ(1 − R)e 0 in Eq. (37) is satisfied only when Z0 < 0, i.e. when the real into Eq. (23), one can show that R = −[cos(ξ) − iϖ sin(ξ)]/[cos(ξ) + part of the QNM frequencies does not lie in the superradiant iϖ sin(ξ)]. regime (0 < ωR < mB), otherwise instabilities are triggered. 5
III. NUMERICAL RESULTS B. Quasi-normal modes
A. Numerical method Fig. 1 shows the fundamental m = 1 QNM frequencies of perfectly-reflecting ECO-like vortices with different charac- 5 { ,B} = − The numerical results to be presented in the following were teristic parameters δ for DBCs (R 1) and NBCs = obtained using a direct-integration method. The integration (R 1). Note that the bottom panels are plots of the absolute ( ) of Eq. (19) is performed using the numerically convenient value of ωI. As pointed out in Sec. II, ωR depends on arg R | |2 expansions but not on R . This means that the top panels of Fig. 1 give information about the real part of the QNM frequencies of both perfectly- and partially-reflecting ECO-like vortices. H˜ (r) = H (r, ) + H (r,− ), h h ϖ R h ϖ The results are qualitatively similar for both BCs. At first ˜ + − H∞(r) = As H∞(r,ω) + As H∞(r,−ω) order in B (i.e. for B . 0.1), ωR is a linear function of the angular velocity B, as one would expect from Eq. (34). The ∗ for the radial function Hωm(r) in Eq. (22) in the near and in initial value (corresponding to B = 0) depends on r0 or, more the far regions, respectively, where precisely, on the the inverse of logδ, in accordance with Eq. (17). Similarly, ωI grows monotonically with increasing ro- Nh tation. Both ωR and ωI change sign from negative to positive −iϖ/2 n Hh(r,ϖ) = (r − rH ) ∑ cn(r − rH ) , as B increases. Within numerical accuracy, the sign changes n=0 occur at the same critical value Bc, meaning ωR,ωI < 0 when N∞ B < Bc and ωR,ωI > 0 when B > Bc. In other words, QNMs +iωr∗ −n H∞(r,ω) = e ∑ dnr . turn from stable to unstable as the fluid spins faster and faster n=0 and, furthermore, perfectly-reflecting ECO-like vortices ad- mit zero-frequency (ω = 0) QNMs. Similar phenomenologi- ˜ +iϖ −iϖ/2 Note that Hh(r0) ≈ (1 + Rδ )δ . According to Eq. cal aspects regarding the interaction of massless bosonic fields −iϖr∗ (22), Hωm(r0) ∼ (1 + R)e 0 , meaning that one should with perfectly-reflecting Kerr-like ECOs have been reported 4 −iϖ/2 require H˜h(r0) ∼ (1 + R)δ , from which follows that in [25, 26, 54]. It was shown in particular that some ECOs −iϖ R = Rδ . Nh and N∞ are the number of terms of the par- or ECO analogues can only support static configurations of a tial sums. The coefficients cn and dn are functions of ϖ and scalar field for a discrete set of critical radii [54–57]. This also ω, respectively, and both depend on B and m. Inserting each holds true for the present case, as shown in the AppendixB. expansion into Eq. (19) and equating coefficients order by The aforementioned instability finds its origin in the pos- order, it is possible to write c1,...,cNh (d1,...,dN∞ ) in terms sible existence of negative-energy physical states inside the of c0 (d0) . The latter is usually set to 1. The choice of Nh ergoregion. In general, in BH physics, the absence of an event and/or N∞ should be a trade-off between computational time horizon turns horizonless rotating ECOs unstable [17, 25]. and accuracy. If the goal is to compute QNM frequencies, one The event horizon of Kerr BHs, which can be regarded as a assigns a guess value to ω and integrates Eq. (19) from r = r∞ perfectly-absorbing surface (R = 0), prevents the falling into ˜ − to r = r0 using the ansatz H∞ with As = 0 so that the solution lower and lower negative-energy states. The same does not satisfies the relations H˜∞ = Hωm and dH˜∞/dr = dHωm/dr at occur when considering perfectly-reflecting BCs, hence the r = r∞, where r∞ stands for the numerical value of infinity. development of instabilities. The previous step is repeated for different guess values of ω All the positive frequencies in the top panels of Fig. 1 re- until the solution satisfies the BC in Eq. (33) (one-parameter fer to exponentially growing modes which meet the super- shooting). If the algorithm is numerically stable, variations in radiance condition, that is to say that acoustic perturbations N∞ and/or r∞ do yield similar results. On the other hand, if the with such frequencies are amplified (Z > 0) when scattered aim is to estimate the amplification factors defined in Eq. (31) off perfectly-reflecting ECO-like vortices, thanks to the trans- for a given frequency ω, one integrates Eq. (19) from r = r0 fer of angular momentum and energy from the latter to the for- to r = r∞ using the ansatz H˜h so that the solution satisfies the mer. According to Eq. (37), only the absence of superradiant relations H˜h = Hωm and dH˜h/dr = dHωm/dr at r = r0 and then amplification when R = 0 (Z0 < 0) guarantees exponentially ± ˜ 2 extracts the coefficients As of the ansatz H∞ at infinity. decaying responses of perfectly-reflecting (|R| = 1) ECOs to All numerical integrations were performed using the inte- external linear perturbations. From a dynamical point of view, gration parameters Nh = 4 and N∞ = 10. When computing the instability is expected to result in the scatterer spinning QNM frequencies (amplification factors), r∞ was set to 100 slower and slower until its fundamental QNM mode, whose (400). The guess values to the QNM frequencies were chosen frequency depends on B, starts decaying rather than growing according to the numerical results reported in [25]. over time, which occurs when B = Bc (note that Z < 0 when B < Bc). The instability domain of ECO-like vortices is de- piected in Fig. 2. The threshold decreases monotonically as r0 → rH (i.e. as δ → 0).
∗ −iϖ/2 4 −iϖr0 −iϖr0 δ −iϖ/2 Using Eq. (17), one can show that e = e δ+2 ∼ δ , 5 where the last step holds as long as δ 1. Since δ 1, δ is a more suitable parameter than r0. 6
0.6 Superradiant regime 0<ω 0.2 0.1 R 0.0 R 0.0 ω ω -0.2 -0.1 -7 δ= 10 -7 δ= 10 -6 δ= 10 -6 -0.2 δ= 10 -0.4 -5 δ= 10 -5 δ= 10 -4 -4 δ= 10 δ= 10 -0.3 -0.6 10-1 10-2 10-2 10-3 10-3 Stable Unstable | | ωI <0 ωI >0 I Stable Unstable I 10-4 ωI <0 ωI >0 |ω |ω 10-4 10-5 10-5 10-6 10-6 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 B B Figure 1. Real (top) and imaginary (bottom) parts of the fundamental m = 1 QNM frequencies of ECO-like vortices with a perfectly-reflecting surface at r0 ≡ rH (1 + δ), 0 < δ 1, where rH is the would-be acoustic horizon of the corresponding draining bathtub, as a function of the 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. 0.8 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 (Z0 > 0), unless 0.6 B < Bc, it is clear that only partially-reflecting ECO-like vor- tices may be stable. However, attention must be paid to the Unstable fact that an ECO with |R|2 < 1 is not perforce dynamically c stable. In fact, for each frequency satisfying the superradi- B 0.4 ance condition, there is a range of values |R|2 can take to assure stability. Using the small-frequency approximations in Eqs. (A15) and (A16), Eq. (37) takes the explicit form 0.2 2 2m 2 ℛ=- 1 2 Γ(m) + iπψiχ(ω/2) ℛ=+ 1 |R| < 2 2m , (38) 0.0 Γ(m) − iπψiχ(ω/2) -20 -15 -10 -5 −1 m−1 2 log10(δ) where χ = β(β − m) ∏n=0 (β − m + n) . The upper bound on |R|2 represents the threshold for the manifestation of er- Figure 2. Ergoregion instability rotation parameter threshold of goregion instabilities and depends both on ω and B, as shown the fundamental m = 1 QNMs of perfectly-reflecting ECO-like in Fig. 3. vortices for both DBCs (R = −1) and NBCs (R = 1). The One is interested in setting restrictions on |R|2 which are shaded regions refer to the domain of the instability. frequency-independent, in order to avert exponentially grow- ing linear perturbations regardless of their frequency. The ab- solute upper bound on |R|2 from Eq. (37), herein dubbed 7 maximum reflectivity, is set by the maximum amplification IV. CONCLUSION factor when R = 0, i.e. to the minimum of the function −1 (1 + Z0) , and is therefore independent of δ. For suf- GW astronomy opens a new window on the universe and is ficiently slow drains (B ≤ 1), ECO-like vortices are stable expected to unveil spacetime features in the vicinity of com- against acoustic perturbations of any frequency as long as pact objects, testing both general relativity and BH physics 2 |R| . 70%. predictions. 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 been one of the strongest motivations behind ECO models. Their phenomenology 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 model for ECOs built from the draining bathtub geometry, named herein ECO-like vortices. These objects feature an ergore- gion and are endowed with a surface with reflective properties rather than an acoustic horizon. Although ECO-like vortices do 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 a low-frequency analysis and by direct-integration numerical Figure 3. Plot of (1 + Z )−1 as a function of the rotation param- 0 calculations. First, when the object’s surface is perfectly- eter B and of the frequency ω of acoustic perturbations. When 2 2 −1 reflecting (|R| = 1), an instability develops when the vor- |R| < (1 + Z0) , ECO-like vortices are dynamically stable. Note that the upper limit on |R|2 decreases as B increases. tex is spinning at a rate above some critical value of the ro- tation parameter. Despite the dependence of the instability domain on the location of the surface, it generally occurs when B > O(0.1). The instability is intimately linked to the C. Superradiant scattering ergoregion, where negative-energy physical states can form. These cannot be absorbed by the vortex’s surface and, there- Fig. 4 illustrates the amplification factors for superradi- fore, cause the exponential growth of acoustic perturbations. ant m = 1 acoustic linear perturbations scattered off ECO- The ergoregion instability of ECO-like vortices is attenuated like vortices with δ = 10−6, B = 0.6 and different reflec- or neutralized when its surface is not perfectly- but partially- tivities. The numerical results were obtained via the direct- reflecting. An absorption coefficient greater than approxi- integration method described previously, whereas the analyt- mately 30% prevents unstable QNMs to develop in ECO-like ical ones were computed using the small-frequency approxi- vortices with B below unit. The results are similar to those re- mations in Eqs. (A15) and (A16) via Eq. (A18). The close- ported in [25, 26, 49] and attests that a general way of prevent- ness between numerical and analytical results is evident. The ing such instabilities from arising is to incorporate dissipative- data for different reflectivities share some qualitatively fea- like effects into the surface. tures: the amplification factors appear to vanish at the end- Second, the stimulation of exponentially growing QNMs points of the superradiant regime (i.e. at ω = 0 and ω = mB) is optimized for more reflective surfaces and, therefore, is and have a maximum value. The graph referring to the drain- expected to generate narrower spectral lines in the emission ing bathtub (R = 0) is in agreement with numerical results cross sections, similarly to those in the absorption cross sec- previously reported in the literature [6,7]. The maximum am- tions of spherically symmetric ECOs [49]. An interesting ex- plification is about 8% when R = 0, meaning the maximum tension of the work presented herein would precisely be to reflectivity is approximately 92% when B = 0.6. When |R|2 compute the emission cross sections of ECO-like vortices, as is nonvanishing, a resonance becomes noticeable around a fre- it would be useful in probing the effects of dissipation upon quency of about 0.372 (dotted vertical lines in Fig. 4), which superradiant scattering in possible future experiments. Imple- matches the real part of a QNM frequency precisely. Like in menting a setup which reproduces the ECO-like vortex intro- classical mechanics, an acoustic linear perturbation extracts duced here appears to be a thorny issue. This would require more angular momentum and energy when its frequency co- to place a right circular cylinder at the acoustic horizon of a incides with the object’s proper frequencies of vibration. The vortex flow. Moreover, the reflectivity R was assumed to be frequency- peak’s height is determined by the imaginary part of ωQNM independent. A possible future extension may lift this as- [49], being maximum when ωI = 0. As shown in Fig. 5, the QNM which sets the peaks in Fig. 4 is marginally-stable when sumption and consider ECOs with frequency-dependent re- R ≈ 0.964 (dotted vertical line in Fig. 5). Fig. 6 confirms that flectivities. the maximum value of the amplification factor occurs indeed for a reflectivity around 0.964. 8 100 Numerical ℛ= 0.00 ℛ= 0.25 Analytical 10-1 10-2 1 Z 10-3 ℛ max(Z1) 0.00 7.7% 10-4 0.25 13.1% 0.50 24.8% 0.75 68.1% 10-5 100 ℛ= 0.50 ℛ= 0.75 10-1 10-2 1 Z 10-3 ωR = 0.372 10-4 10-5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ω/mB ω/mB Figure 4. Numerical and analytical values of the amplification factors for superradiant (0 < ω < mB) acoustic perturbations with m = 1 scattered off a ECO-like vortex with B = 0.6 and featuring a surface with reflectivity R at r0 = rH (1 + δ), where rH is the would-be acoustic horizon of the corresponding draining bathtub and δ = 10−6. The resonance matches the real part of the fundamental QNM frequency of the vortex-like ECO (dotted vertical lines). 500 0.00 ℛ= 0.95 ωR = 0.372 ℛ= 0.96 100 ℛ= 0.97 -0.02 ℛ= 0.98 50 ℛ= 0.99 I -0.04 1 Z ω 10 -0.06 5 -0.08 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.615 0.620 0.625 0.630 ℛ ω/mB Figure 5. Imaginary part of the QNM frequency which sets the Figure 6. Analytical approximation for the amplification factors plot- peaks of the amplification factors plotted in Fig. 4, as a function of ted in Fig. 4 in the neighborhood of the fundamental m = 1 QNM the reflectivity R. ωI vanishes at R ≈ 0.964 (dotted vertical line). frequency for reflectivities near R ≈ 0.964. 9 ACKNOWLEDGMENTS The amplitudes of the ingoing and outgoing waves are pro- portional to (A + iB) and (A − iB), respectively. The main This work has been supported by Fundação para a Ciência goal of the asymptotic matching is to find expressions for A e a Tecnologia (FCT) grant PTDC/FIS-OUT/28407/2017 and and B in terms of r0 (or δ), R, B, m and ω. by CENTRA (FCT) strategic project UID/FIS/00099/2013. The small-r behavior of the asymptotic solution in Eq. (A3) This work has further been supported by the European is Union’s Horizon 2020 research and innovation (RISE) pro- m (ω/2) +m Γ(m) −m grammes H2020-MSCA-RISE-2015 Grant No. StronGrHEP- Sωm(r) ∼ A r − B r . (A6) Γ(m + 1) π(ω/2)m 690904 and H2020-MSCA-RISE-2017 Grant No. FunFiCO- 777740. The authors would like to acknowledge networking In the near region, by defining support by the COST Action CA16104. −m −i ϖ Sωm(r) = r [g(r)] 2 Tωm(r), (A7) Appendix A: Analytical analysis introducing the new coordinate z = r−2 and using the condi- tions Bω 1 and ϖ 1, one can bring Eq. (13) into the A number of relevant quantities, such as QNM frequencies hypergeometric form and amplification factors, can be computed analytically in the 2 low-frequency regime (ω 1) using matching-asymptotic d d z(1 − z) + [γ − (α + β + 1)z] − αβ Tωm(z) = 0, techniques [22]. Contrarily to previous works [58], mostly dz2 dz focused on the BH case (R = 0), here the presence of out- (A8) going waves near the would-be acoustic horizon is considered where 2α = 2 + m − iϖ, 2β = m − iϖ and γ = m + 1. It is and the problem is solved for a generic reflectivity R. For that easy to check that α, β and γ satisfy the relations α − β = 1, purpose, the spacetime region outside the reflective surface at α + β = 2α − 1 = 2β + 1 and γ − α − β = iϖ. r = r0 is split into a region near the would-be acoustic hori- The hypergeometric differential equation has three singular zon, i.e. in the limit r − rH 1/ω, and a region far from it, points: z = 0,1,∞. Its most general solution in the neighbor- i.e. at infinity, where r rH . In the following, besides the hood of z = 1 reads condition Bω 1, the assumption ϖ 1 for frequencies in i γ−α−β o the superradiant regime is also considered. One starts looking Tωm(z) = C Tωm(z) + D(1 − z) Tωm(z), (A9) for asymptotic solutions to Eq. (13) in each spacetime region i where C ,D ∈ C, Tωm(z) = 2F1(α,β;α + β − γ + 1;1 − and imposing the BC in Eq. (33), and then matches them in o z) and T (z) = 2F1(γ − β,γ − α;γ − α − β + 1;1 − the overlapping region, where 1 r − rH 1/ω. ωm z). 2F1(α,β;γ;z) is the hypergeometric function. Since To solve Eq. (13) in the far region, it is convenient to rewrite i o it in the form Tωm(1) = Tωm(1) = 1, one finds that the small-r behavior of the asymptotic solution in Eq. (A9) is " 2 2 # d d mB ∆m +iϖ ∆ ∆ + ωr − − Sωm(r) = 0, (A1) T m(z) ∼ C + D(1 − z) , (A10) dr dr r r ω Using Eqs. (18) and (A7), one can show that where ∆(r) = rg(r). When r rH and Bω 1, Eq. (A1) −iϖr∗ +iϖr∗ reduces to a Bessel ODE, Hωm(r) ∼ C e + De (A11) d2 d r2 + r + (ω2r2 − m2) S (r) = 0, (A2) near the would-be acoustic horizon, where dr2 dr ωm r + 1−iϖ r + 1+iϖ whose general solution is a linear combination of Bessel func- C ≡ C , D ≡ D . (A12) rer rer tions of the first and second kinds, Therefore, the BC (33) turns into Sωm(r) = A Jm(ωr) + BYm(ωr), (A3) r2 iϖ where A ,B ∈ C. The large-r behavior of the asymptotic so- D 0 = R 2 . (A13) lution in Eq. (A3) is C r0 − 1 r 2 Given that γ ∈ Z, one must be careful when analyzing the S m(r) ∼ [A cos(ωr − ς) + B sin(ωr − ς)], (A4) ω πωr large-r (small-z) behavior of the asymptotic solution (A9)[59, 60]. One can show that [58] where ς ≡ π(m+1/2)/2, which can be written in terms of the m−1 radial function Hωm(r) and as a superposition of ingoing and i (−1) −m Tωm(z) ∼ Ti z + ψi , outgoing waves. In effect, m(α − m)m(β − m)m m−1 ( + i )e−i(ωr−ς) + ( − i )e+i(ωr−ς) o (−1) −m A B A B T (z) ∼ To z + ψo , Hωm(r) ∼ √ . (A5) ωm m(1 − α) (1 − β) 2πω m m 10 where where (−1)m+1 Γ(α + β − m) 1 1 T ≡ , W (x) = + , (B3) i m! Γ(α − m)Γ(β − m) 1 x x + 1 2 2 ψi ≡ ψ(α) + ψ(β) − ψ(m + 1) − ψ(1), 1 m B 1 2 W2(x) = + + (1 − m ) . (B4) (−1)m+1 Γ(m + 2 − α − β) 4x(x + 1) x x + 1 To ≡ , m! Γ(1 − α)Γ(1 − β) Eq. (B2) is a standard Riemann-Papparitz ODE. Defining ψo ≡ ψ(m + 1 − α) + ψ(m + 1 − β) − ψ(m + 1) − ψ(1). σ − 1 Lωm(x) = x (x + 1) 2 Gωm(x), (B5) (q)n is the rising Pochhammer symbol and ψ(·) is the digamma function [59, 60]. with 2σ = −imB, and introducing the new coordinate y = −x, Using Eq. (A7) and rearranging the terms, one gets one gets +m −m S m(r) ∼ A r + Br , (A14) ω d2 d y(1 − y) + ρ(1 − y) − κλ G (y) = 0, (B6) where dy2 dy ωm m−1 (−1) TiC ToD A ≡ + , with 2κ = −m(1 + iB), 2λ = −m(−1 + iB) and ρ = 1 − imB. m (α − m)m(β − m)m (1 − α)m(1 − β)m It is easy to check that κ, λ and ρ satisfy the relations κ + B ≡ ψiTiC + ψoToD. λ + 1 = ρ, λ − κ = m and ρ = 2 − ρ, where ρ is the complex conjugate of ρ. Eq. (A14) exhibits the same dependence on r as Eq. (A6). The general solution of Eq. (B6) reads [59] Matching the two solutions, it is straightforward to show that Gωm(y) =E12F1(κ,λ,ρ,y) Γ(m + 1) A = A , (A15) 1−ρ (ω/2)m + E2y 2F1(−λ,−κ,ρ,y), (B7) π(ω/2)m B = − B. (A16) where E1,E2 ∈ C. Equivalently, Γ(m) − 1 One can derive some relevant physical quantities from Eqs. Lωm(x) = (x + 1) 2 · σ −σ (A15), (A16) and related, namely QNM frequencies and [E1 x 2F1(κ,λ,ρ,−x) + E2 x 2F1(−λ,−κ,ρ,−x)], (B8) amplifications factors of acoustic perturbations scattered off 1−ρ ECO-like vortices. where E2 ≡ (−1) E2. The small-x behavior of Eq. (B8) The very definition of QNM requires the amplitude of the must be written in terms of the radial function H(r) for the ingoing wave at infinity to vanish. It then follows from Eq. BC in Eq. (33) to be applied. One can show that (A5) that one must impose +imBr∗ −imBr∗ Hωm(r) ∼ E1e + E2e (B9) A + iB = 0. (A17) near the would-be acoustic horizon, meaning the BC to be Eq. (A17) can be solved using a simple root-finding algo- imposed at r = r0 is rithm. On the other hand, according to Eq. (31), the amplification +2imBr∗ E2/E1 = Re 0 . (B10) factors are given by 2 The large-x behavior of Eq. (B8) can be analyzed by mak- A − iB Zm(ω,R) = − 1. (A18) ing use of the linear transformation formula 15.3.7 in [59] and A + iB rearranging the terms. In effect, one gets − 1 h + m − m i L (x) = (x + 1) 2 x 2 + x 2 . (B11) Appendix B: Static configurations ωm F1 F2 with In order to study the static configurations of Eq. (9), one defines the radial function L(r) [9] Γ(ρ) Γ(ρ) F1 = E1 2 − E2 2 Γ(λ − κ), (B12) i ( −mB)log(r2−1)+2mBlog(r) λΓ(λ) κΓ(−κ) R (r) = r e 2 [ ω ]L (r), (B1) ωm ωm Γ(ρ) Γ(ρ) F = E − E Γ(κ − λ). (B13) introduces the new coordinate x = r2 − 1 and sets ω = 0, 2 1 κΓ(κ)2 2 λΓ(−λ)2 which reduces the radial equation to It follows from the explicit expression for the energy flux d2 d across an arbitrary surface at a constant radial coordinate r + W (x) + W (x) L m(x) = 0, (B2) dx2 1 dx 2 ω that the amplitudes of the ingoing wave and of the outgoing 11 wave are proportional to (F1 −iF2) and (F1 +iF2), respec- where the dot denotes differentiation with respect to x. Like tively [9]. The absence of ingoing waves at infinity requires Eq. (B14), Eq. (B16) is a condition on the amplitudes E1 and that E2. Explicitly, 2 2 F1 − iF2 = 0, (B14) E [(3 − 4σ)r − 3]tan(ξ) + 2r0(r − 1) 1 = (r2 − 1)−2σ 0 0 . E 0 [(3 + 4σ)r2 − 3]tan(ξ) + 2r (r2 − 1) which implies that 2 0 0 0 (B17) 2 Equating the right-hand side of Eqs. (B15) and (B17), one E1 Γ(ρ) Γ(κ) Γ(λ) = · obtains a relation between r0 (or δ), B and m. 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