Acoustic Black Holes and Superressonance Mechanisms Extended Abstract

Acoustic Black Holes and Superressonance Mechanisms Extended abstract

Sofia Freitas1 1 CENTRA, Departamento de F´ısica, Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal (Dated: October 17, 2017) It has been showed that sound waves, in a moving fluid, behave analogously to scalar fields in a curved spacetime. Supersonic fluid flow can generate a ”dumb hole”, which corresponds to a region in space from which no information can be extracted. This is the analogue of a black hole in the theory of General Relativity. The formal mathematical equivalence is very powerful once it allows for the study of black holes in laboratories. Throughout this work, we study the analogue gravity description of rotating black holes, with a focus on superradiance. We analyse two distinct spacetimes: the draining bathtub and the rotating cylinder setups. Viscosity is also included on the latter.

I. INTRODUCTION Theorised in 1969 by Roger Penrose, it accounts for the loss of angular momentum in rotating BHs. Penrose ar- Black holes (BHs) are described by the theory of Gen- gued that once particles entered a special region of the eral Relativity (GR). GR was developed in 1915 by Al- BH, called the ergosphere, they could be scattered and bert Einstein and it describes gravity as the curvature of ejected with greater energy than the one they had ini- spacetime. In 1904, Einstein was troubled by the views tially. Note that this scattering process rarely occurs, as of relativity put forward by Galileo and Maxwell. On the very specific conditions must be verified. For instance, one hand, Galileo claimed that absolute motion could not one must arrange the initial trajectory such that after- be defined. Maxwell’s electromagnetism, on the other, es- wards there is a geodesic trajectory that takes the parti- tablished that light travelled at a fixed speed c. Einstein cle back outside the ergoregion. Nonetheless, the Penrose came to the conclusion these two notions could only be process was central in the establishment of its analogue compatible if c were to be kept constant across different wave process, superradiance. Superradiance is a radia- frames of reference. The subsequent view that time was tion enhancement process that occurs in dissipative sys- not absolute made way for the establishment of Special tems. In BHs, it happens at the level of the event horizon. Relativity. GR arose years later, expanding the theory Superradiance allows for energy and angular momentum to account for effects of acceleration. The latter passed to be extracted from the vacuum, resulting in a conse- its first test in 1919, when the bending of star light was quent amplification of the scattered wave packet. Super- observed during a solar eclipse [1]. Since then, the theory radiance in rotating BHs was first discovered in 1971, by has been put to test in several other occasions, the latest Zel’dovich. He showed, provided the superradiance con- being the experimental detection of gravitational waves dition is satisfied, that the scattering of incident waves in 2015 [2]. In a nutshell, GR is defined via Einstein’s in rotating dissipative bodies resulted in an amplifica- field equations, tion of such waves, producing outgoing waves of larger amplitude. The superradiance condition reads 1 G = R Rg = 8πT , (1) µν µν − 2 µν µν ω < mΩ, (2) which describe how matter and energy interact with where ω is the frequency of the incident radiation, m, spacetime, the fabric of our universe. The Ricci tensor the azimuthal number, accounts for the projection of the µν Rµν and the Ricci scalar R = g Rµν are geometrical angular momentum along the axis of rotation and Ω is entities which measure the curvature of spacetime. They the angular velocity of the body. depend solely on the metric gµν . The element on the In 1981, Unruh showed that sound waves, in a mov- right hand side of the equation, Tµν , is the stress-energy ing fluid, behave the same way as scalar fields in curved tensor and it accounts for the presence and distribution spacetimes [4]. He observed horizons could occur in day- of matter. The first solution to Einstein’s field equations to-day situations and drew analogies between fluid me- is due to Karl Schwarzschild who, in 1916, computed the chanics and GR. He called these horizons ”dumb holes”, gravitational field generated by a static point mass [3]. and proved their existence as a consequence of super- Subsequently, other solutions have been discovered. A sonic fluid flows, as illustrated in Figure 1. By finding particularly relevant one for this work is the Kerr solu- ways to reproduce behaviours of BHs and other astro- tion, which describes a rotating BH. physical bodies, one is able to carry out experiments in Despite its extremely interesting features, astrophys- laboratories, with the aim of testing the existing theories. ical BHs are of difficult observation and direct study. A leading laboratory in the field is the Quantum Grav- That is why one must rely on other, secondary, effects. A ity Laboratory in Nottingham. In a recent article [5], very important secondary effect is the Penrose process. Silke Weinfurtner’s team describes the first laboratory 2

of sound in the fluid: 2 ρ (c2 v ) vT gµν (t, x) = − − || || − , (4) c v I[3 3] − × where I[3 3] is the 3 3 identity matrix. In general, when the fluid× is non-homogeneous× and flowing, the acoustic Riemann tensor associated with the acoustic metric is non-zero. The path to arrive at this result is described promptly. FIG. 1. Propagation of information using the analogue fluid me- One starts by writing down the two fundamental equa- chanics. Beth and Alfred communicate using ”ripples” (acoustic disturbances) on the free surface of the fluid. The fluid naturally tions ruling fluid dynamics, the continuity equation and flows from Beth to Alfred, so she is always able to get her message the Euler equation. Respectively, they read: across. For Alfred to be able to reply to Beth, he must be able to guarantee the ripples he sends move faster upstream than the flow ∂ ρ + (ρv) = 0 (5) t ∇ · does downstream. But Alfred is in a supercritical region, so he can dv never send any information upstream – nothing can counterprop- ρ = ρ [∂tv + (v )v] = F, (6) agate the flow in this region. The region separating the sub and dt · ∇ supercritical regions is the analogue BH horizon. where ρ is the density of the fluid and v is the fluid velocity vector field. F in the Euler equation is defined as detection of superradiance. By working with a draining bathtub fluid flow, they observed that plane waves F = p ρ Φ, (7) −∇ − ∇ propagating on the surface of the water were amplified p being the pressure and Φ the potential of an external after being scattered off by the draining vortex. The ex- driving force. We assume the fluid to be vorticity free, perimental setup in Nottingham was first described in i.e. locally irrotational. This means the velocity can be 2002 by Unruh and Schützhold[6]. They proposed an written as v = ψ. Furthermore, the fluid is assumed to experiment in which superradiance could be detected in be barotropic, meaning∇ the fluid’s equation of state is of the laboratory by assuming a long wavelength, shallow the form ρ = ρ(p). The latter is helpful in defining the water approximation. An alternative experimental setup specific enthalpy h, worth mentioning is the one proposed by Cardoso et al., in 2016 [7]. The authors analysed the hypothesis of mim- 1 h = p. (8) icking the vortex geometry by making use of a rotating ∇ ρ∇ cylinder. They showed that, given an appropriate choice of material for the cylinder, surface and sound waves were The Euler equation (6) can thus be written as amplified. Both proposals are studied here. 1 ∂ ψ + h + ( ψ)2 + Φ = 0. (9) t 2 ∇ Next, one linearises the fundamental equations around II. ACOUSTIC ANALOGUES an average bulk motion described by the background dy- namical quantities (p0, ρ0, ψ0), plus some low amplitude A. Fluid dynamics acoustic perturbations (εp1, ερ1, εψ1). We consider only first order terms in ε. In 1981, Unruh developed a method for mapping cer- p = p + εp + (ε2), ρ = ρ + ερ + (ε2), tain aspects of BH physics into fluid mechanics. He 0 1 O 0 1 O 2 started by considering a barotropic and inviscid fluid, ψ = ψ0 + εψ1 + (ε ). (10) O whose flow is irrotational – though possibly time de- The continuity and the Euler equations can be rewrit- pendent. The equation of motion for the velocity po- ten as two versions of themselves: an unperturbed and tential describing an acoustic disturbance is identical to a perturbed one. The unperturbed equations will simply the d’Alembertian equation of motion for a minimally- be the aforementioned equations (5) and (9), while the coupled massless scalar field ψ, propagating in a (1 + 3)- perturbed equations read dimensional Lorentzian geometry, namely Continuity equation: ∂tρ1 + (ρ1v0 + ρ0v1) = 0 1 ∇ · ∆ψ = ∂ √ ggµν ∂ ψ = 0. (3) (11) √ g µ − ν − Euler equation: ∂tψ1 + h1 + v0 ψ1 = 0. (12) ∇ This is just the Klein Gordon equation, µψ = 0, ∇µ∇ The perturbation to the enthalpy can be identified as describing the dynamics of a scalar field. Under these h = p1 via a Taylor expansion. Making use of equa- 1 ρ0 conditions, the propagation of sound is governed by an tion (12), the following relation arises: acoustic metric gµν (t, x), which depends algebraically on the density and velocity of the flow and on the local speed p = ρ (∂ ψ + v0 ψ ) . (13) 1 − 0 t 1 ∇ 1 3

Together with the barotropic assumption result please refer to Reference [8]. Observe that A can be either positive or negative: A < 0 corresponds to a ∂p sink and A > 0 to a source. Respectively, they will either p = p(ρ) = p1 = ρ1 (14) ⇒ ∂ρ emulate a future acoustic horizon – acoustic black hole – or a past acoustic horizon – acoustic white hole. and equation (11), one obtains the wave equation, Dropping a position independent prefactor, the line element describing the propagation of sound waves in the ∂ (ρ (∂ ψ + v0 ψ )) t 0 t 1 · ∇ 1 (1 + 2)-dimensional draining bathtub fluid flow reads 2 + c ρ0 ψ1 + ρ0v0(∂tψ1 + v0 ψ1) = 0, (15) ∇ · − ∇ · ∇ A2 + B2 ds2 = c2 dt2 where the local speed of sound has been defined as − − r2

2A 2 2 2 2 ∂ρ dtdr 2Bdtdθ + dr + r dθ . (19) c− = . (16) − r − ∂p The coordinates for the horizon and ergosphere of the This wave equation describes the propagation of the lin- spacetime are earised scalar potential ψ . Once ψ is computed, p 1 1 1 A and ρ1 are easily calculated via equations (13) and (14). rH = | | (20) Thus, the wave equation completely determines the prop- c √ 2 2 agation of the acoustic disturbances (p1, ρ1, ψ1). The A + B re = . (21) background fields (p0, ρ0, ψ0), which appear as time- c dependent and position-dependent coefficients in the equation, are constrained to solve the equations of fluid III. DYNAMICS OF ACOUSTIC HOLES motion for an externally-driven, barotropic, inviscid, and irrotational flow. The corresponding acoustic metric is given by equation (4). A. Wave equation

In order to obtain the wave equation ruling the be- B. Vortex geometries: the draining bathtub fluid haviour of sound waves in the draining bathtub flow, we flow recover the general result of equation (15) and use the background fluid flow velocity presented in equation (17):

The first setup analysed consists of a draining bathtub 2 flow, similar to the ones we see everyday in wash basins. 2 A B c 2 ∂t ψ1 + ∂r + ∂θ ∂tψ1 (∂r(r∂rψ1)+∂θ ψ1) = 0. Without any loss of generality, one describes the vortex r r2 − r2 geometry in polar coordinates: (t, r, θ). The background (22) fluid velocity can be written as The ansatz of a plane wave,

i(ωt mθ) Aer + Beθ ψ1(t, r, θ) = R(r)e− − , (23) v0 = , (17) r allows one to separate the wave equation. We work only with the corresponding potential with the radial part. Writing R(r) = Z(r)H(r) and re- quiring Z(r) to solve ψ0(r, θ) = A log(r/a) + Bθ. (18) 1 (c2r2 A2) + 2iA(Bm r2ω) Z + − − Z = 0, (24) A and B are parameters associated to the radial and ,r 2 r(c2r2 A2) angular components of the background ﬂuid velocity, and − a is some irrelevant length scale. For a derivation of this one obtains the Schroedinger-like wave equation:

" 2 2 2 2 2 # 2 Bm c r A 2 2 1 5A H + c− ω − r− m + H = 0. (25) ,r∗r∗ − r2 − c2r2 − 4 4r4c2

Observe the introduction of the tortoise coordinate r which explicitly reads ∗

2 1 dr A − A cr A ∗ = ∆ = 1 , (26) r = r + log − . (27) dr − c2r2 ∗ 2c cr + A 4

m = 1 m = 2 1.2 1.2

1 1

0.8 0.8 B = 0.1 B = 0.1 2 2 | B = 0.3 | B = 0.3 0.6 0.6 ωm B = 0.6 ωm B = 0.6 R R | B = 0.9 | B = 0.9 0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 ω ω

m = 1 m = 2 2 2 1.8 1.6 1.5 1.4

1.2 B = 0 B = 0 2 2 | B = 1 | B = 1 1 1 ωm B = 3 ωm B = 3 R R | 0.8 B = 5 | B = 5 0.6 0.5 0.4 0.2 0 0 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 ω ω

2 FIG. 2. Reflection coefficient |Rωm| as a function of ω. Several solutions for the differential equation are produced, as one changes the value of parameter B, the angular velocity at the horizon. The left hand panels show the results for m = 1 and the right hand ones the 2 results for m = 2. |Rωm| = 1 (limit above which superradiance occurs) is plotted as a dashed line.

rA ωA ˆ Using the rescaled quantitiesr ˆ = c ,ω ˆ = c2 and B = the entire domain. However, it is possible to obtain an- B alytical solutions at the horizon, rH = 1, and at inﬁnity, A , the wave equation reads r . They read ∞ → ∞ Hrˆ∗,rˆ∗ + Q(ˆr)H = 0. (28) i(ω Bm)r∗ HH = AH e− − (32)

iωr∗ iωr∗ The potential is H = Aine− + Aoute . (33) ∞ !2 Bmˆ The Wronskian associated to H and H† is constant in Q = ωˆ V (ˆr) (29) 1 − rˆ2 − the radial coordinate . This means one can equate the horizon and infinity Wronskians, resulting in an energy 2 rˆ 1 2 2 1 5 conservation condition. Such equality, together with the V = − rˆ− m + . (30) rˆ2 − 4 4ˆr4 superradiance condition2 ω < mB, yields the result 2 From here onwards, all hats are dropped and one works Aout > 1. (34) only with the rescaled quantities. In such units, the hori- Ain zon and ergosurface occur at the following radii: Observe this ratio is, by definition, the reflection coeffi- p 2 r = 1, r = 1 + B2. (31) cient Rωm . H e | |

B. Scattering in the vortex geometry. 1 Superradiance In Schroedinger-like real-valued eﬀective potential equations, the Wronskian is conserved in the coordinate upon which the wave equation is being diﬀerentiated. The complicated form of the potential given by equa- 2 In the draining bathtub spacetime, the angular velocity at the tion (29) rules out an analytical solution that holds for event horizon is given by the quantity B. 5

Next, one works with the wave equation where ωR influences the phase of the wave and ωI mod- ifies its amplitude. Note this is the outgoing part of the 2 2 1 d H 2 1 dH wave. Moreover, the sign of ωI determines the stability 1 + 1 +Q(r)H = 0, (35) − r2 dr2 r3 − r2 dr of the solution. A negative ωI corresponds to a solution which is damped in time, hence stable. Positive ωI , on where the derivatives have all been written with respect the other hand, is an amplified and thus unstable solution to the radial coordinate r. The numerical method used to in time. solve the wave equation consists in iteratively integrating The QNMs are computed using direct integration a it for different frequencies ω, imposing the boundary con- la Chandrasekhar-Detweiler [10]. The results obtained, ditions expressed by equations (32) and (33). The out- displayed in Table I, are in accordance with the existing 2 put of each cycle consist of a pair of points (ω, Rωm ), literature. which is later plotted. A series expansion with| the aim| m ω of increasing precision near the boundaries is performed. QNM 1 0.4069 − 0.3412i The incoming wave is normalised by setting AH = 1 in equation (32). 2 0.9527 − 0.3507i 3 1.4685 − 0.3524i In Figure 2, we present some of the plots that illustrate 4 1.9765 − 0.3530i the results produced. In the plots, we have included a 2 dashed line at Rωm = 1. This line sets the frontier TABLE I. Fundamental quasinormal frequencies, for B = 0. between the cases| of| no amplification and amplification. Whenever the curves are above the dashed line, there is superradiant scattering. It is interesting to note that the intersection of each curve with the horizontal line happens when ω = mB, as predicted by the theory. Out of IV. DYNAMICS OF ACOUSTIC GEOMETRIES the parameters chosen, the maximum value obtained for 2 A. Wave equation amplification was Rωm = 1.805 for ω = 4.18, m = 1 and B = 5. Larger| amplification| is obtained for larger values of the rotation parameter B. Once rotation is the The new setup consists of fluid surrounding a rotating source of amplification in superresonant mechanisms, it is cylinder. We consider the fluid to be at rest. Taking into natural that the greater the rotational speed, the greater account a static fluid configuration, v0 = 0, the wave the amplification: more rotational energy is available. equation (15) becomes This very same reason justifies that B = 0 produces no ∂2ψ (c2 ψ ) = 0. (37) amplification. One also sees that greater amplification t 1 − ∇ · ∇ 1 occurs for m = 1 rather than for m = 2. This has to Using the plane wave equation ansatz given by equa- do with the relation between the energy carried by the ϕ(r) wave and the potential barrier produced in both cases. tion (23), where we identify R(r) = √r , it reads A complete explanation can be found in Appendix A of ω2 1 1 Reference [8]. The results obtained are consistent with ∂2ϕ + m2 ϕ = 0. (38) previous works, in particular with Figure 2 in Reference r c2 − r2 − 4 [9]. Finally, one observes the reflection factor is never larger than 2. This upper bound is also present in the Reissner–Nordstroem BH and as we will see in the cylin- B. Scattering off a rotating cylinder. drical setup evaluated later. Superradiance

C. Quasinormal modes and stability The cylinder, of radius R0, is characterised by a quantity called acoustic impedance. Its value determines the interaction wall-ﬂuid. The acoustic impedance, Z, is a If an isolated guitar string vibrates in normal modes, complex variable. The real part is called resistance and a guitar string coupled to air vibrates in QNMs. In fact, the imaginary part is the reactance. The concept of the the energy transfer occurring from the string to the sur- resistance is similar to that of a regular resistance. Think rounding air adds on an imaginary part to the frequency in terms of Ohm’s law. A given material, depending on of the (quasi)normal modes. This imaginary part directly its resistance, will dissipate more or less energy. The accounts for the losses or, in the opposite range of the reactance, on the other hand, measures the dephasing spectrum, for instabilities associated with the ampliﬁca- during the interaction. tion of the wave energy. Under the conditions described in Reference [7], Zω Generically, a complex frequency ω = ωR + iωI pro- can be written as duces solutions ∂rψ1 iρ0ω iωR(r t) ωI (r t) = . (39) Ψ e − e− − , (36) ψ1 − Zω ∼ r=R0 6

m = 1 m = 2 1.8 1.1

1.6 1 1.4 0.9 1.2 Ω = 0.1 Ω = 0.1 2 2 0.8 | Ω = 0.3 | Ω = 0.3 1 ωm Ω = 0.6 ωm Ω = 0.6 R R | Ω = 0.9 | 0.7 Ω = 0.9 0.8 0.6 0.6

0.4 0.5

0.2 0.4 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ω ω

m = 1 m = 2 45 30

40 25 35

30 20 Ω = 0 Ω = 0 2 2 | 25 Ω = 1 | Ω = 1 15 ωm Ω = 3 ωm Ω = 3 R 20 R | Ω = 5 | Ω = 5 15 10

10 5 5

0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ω ω

2 FIG. 3. Reflection coefficient |Rωm| as a function of ω. Several solutions for the differential equation are produced, as one changes the value of parameter Ω, the angular velocity of the rotating cylinder. The left hand panels show the results for m = 1 and the right hand 2 ones the results for m = 2. |Rωm| = 1 (limit above which superradiance occurs) is plotted as a dashed line. The following parameters were used: c = ρ0 = R0 = 1 and Z = 1 − i. Ω is presented in units of c/R0.

This equation is the boundary condition one imposes at than the one to m = 2, as usual. Finally, one should r = R0. For a cylinder rotating uniformly with angular mention that when Ω = 0, the curve is always below 2 velocity Ω, the frequency of the incident wave can be the dashed Rωm = 1 line. In other words, when the modified to include the angular rotation such that cylinder is not| rotating,| there is no amplification. 2 The amplification is never larger than Rωm = 2 ω ω˜ = ω mΩ. (40) | | → − when Ω < 1. Using WKB expansions, one can predict such limit. For a full discussion on the subject cf. Ref- Once again, one faces a Schroedinger-like wave equa- erence [12]. Exiting the realistic range of values for the tion and hence one can equate Wronskians at different acoustic impedance, the upper bound of 2 ceases to ex- radial coordinates. Also here the superradiance condi- ist. In fact, there is a tendency that smaller impedances tion yields the result of equation (34). return greater amplifications; and vice-versa. This result Implementing a numerical solution in all similar to the makes sense when we think of the nature of impedance, one described in Section III B, one produces the plots of which is that of a resistance. This result is illustrated in Figure 3. A very obvious distinction between the curves Figure 4, where we plot different values of Z for a fixed in the two spacetimes has to do with the fact that here the 2 rotational speed Ω = 0.7 c/R0. curve never reaches Rωm = 0. This is a consequence of the definition of the| acoustic| impedance, which has been set to Z = 1 i. This value is standard for several known materials [11].− Out of the plots displayed, the C. Quasinormal modes and stability maximum amplification for m = 1 occurs when Ω = 3 2 and it corresponds to Rωm = 43.677, at ω = 1.18. In the spirit of studying superradiance-triggered insta- For m = 2, the maximum| | amplification is at Ω = 5, bilities, one looks at the QNMs of the system. The modes 2 with Rωm = 28.314, at ω = 2.20. The location of the are computed in a slightly different fashion than before. maxima| is| related to the QNMs – see Figure 5. Note the By confining the cylinder, we look at solutions whose maximum amplification associated to m = 1 is greater frequencies ω are invariant under changes of such con- 7

m = 1, Ω = 0.7 m = 1 3.5 14 Z = 0.3(1 i) Ω = 4.5 Z = 0.7(1 − i) Ω = 5.0 3 Z = 2(1 − i) 12 Ω = 5.5 Z = 5(1 − i) Ω = 6.0 Z = 8(1 − i) 10 2.5 −

2 2 8 | | 2 ωm ωm R R | | 6 1.5 4

1 2

0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.6 1.8 2 2.2 2.4 2.6 2.8 ω ω

2 2 FIG. 4. Reflection coefficient |Rωm| as a function of ω. By FIG. 5. Reflection coefficient |Rωm| as a function of ω. We keeping the rotation speed of the cylinder constant (Ω = 0.7 c/R0), focus our analysis in the relation between the maximum amplifi- we see how different acoustic impedances influence the reflection cation factor and ωQNM by drawing vertical lines that correspond coefficient. One can clearly observe that the smaller the impedance, to Re(ωQNM). The QNMs are drawn in the same colour as the the larger the amplification. Finally, it is interesting to note the respective reflection curves. Ω is in units of c/R0. superradiant regime ceases to exist as soon as ω reaches 0.7, as predicted by the superradiance condition (mΩ = 0.7). D. Confined geometry finement radius. The confinement is achieved by adding a cylinder to our setup. It has radius R1 > R0 and is con- Rather than using the confinement as a means of com- centric with the first one. We impose the outer cylinder puting QNMs, one examines how it can impact the sta- to behave as a perfect mirror, reflecting back all incident bility of the solutions. We turn now our attention to the radiation. The boundary condition in terms of the radial values of ωI in subcritical rotational speeds: Ω < c/R0. field reads As one approaches subsound rotational speeds, ωI be- ϕ comes smaller and smaller. At a certain point, it may ∂ ϕ = 0. (41) r − 2r even become negative. In fact, for some rotational speeds Ω and at given ratios R1/R0, there are stable solutions In this configuration, whenever both ω and ω are con- R I (ωI < 0). Working with other values of Ω and studying stant, we have found a QNM. The requirement for con- where the crossing ωI = 0 occurs, we obtain the results stancy of ω ensures the frequency at infinity to be the a shown in Figure 6, which fit to R1/R0 = Ω . a is a free same as the one found in the confined setup. Ergo, the parameter. frequency found is that characteristic of an open system, a QNM frequency by definition. The results are in Ta- 250 ble II. The QNMs are unstable (ωI > 0), which is not

200 Ω ωQNM (m = 1) ωQNM (m = 2) 3.0 1.1744 + 0.1772i – 150

3.5 1.3669 + 0.2928i – 0 /R 4.0 1.5610 + 0.4050i – 1 R 4.5 1.7562 + 0.5148i 2.0124 + 0.1912i 100 5.0 1.9522 + 0.6228i 2.1845 + 0.3262i 5.5 2.1487 + 0.7295i 2.3618 + 0.4564i 50 6.0 2.3457 + 0.8351i 2.5428 + 0.5825i

0 TABLE II. Frequencies of QNMs, for m = 1 and m = 2. The 0 0.5 1 1.5 2 2.5 frequencies presented correspond to the fundamental mode. The Ω entries of the table marked with – are those for which we could not find the corresponding ωQNM. We were able to find QNM down to FIG. 6. Coordinates R1/R0 and Ω for which ωI = 0. For in- Ω = 2.8 for m = 1 and Ω = 4.2 for m = 2. creasingly lower rotational speeds, the point where the curve for ωI (R1/R0) crosses the ω = 0 axis is increasingly higher. The fit surprising, as the frequencies of Table II correspond to equation is R1/R0 = (21.1154 ± 0.0175)/Ω. domains where superradiance is present – cf. Figure 3. Comparing the QNM frequencies displayed in Table II To decode the nature of the fit, we can think of the with the plots of Figure 3, one identifies a correspondence physical consequences of setting ωI = 0. In the limit of between the maximum amplification and the location of vanishing ωI , superradiant instabilities are non-existing. the QNMs. See Figure 5, where we have plotted the More precisely, we are in the regime where ω = mΩ, ω R 2 curves, signalling the QNM frequencies as well. being a purely real frequency. Our knowledge of normal | ωm| 8 modes allows us to recognise that, given finite boundary that there is a relative motion between various parts of conditions, ω is inversely proportional to the distance the fluid. Including viscosity in the treatment done so between such finite boundaries. In our specific case, we far means that a modification of the fluid equations is in can thus write order. At this point, we substitute the Euler equation by the Navier-Stokes equation, 1 mΩ, (42) µ R1/R0 ∼ ρ ∂ v + ρ v v = p + µ 2v + ξ + ( v) . t · ∇ −∇ ∇ 3 ∇ ∇ · which is in accordance with the numerical results. (43) This formulation of the Navier-Stokes equation holds for compressible flows, where we require µ and ξ to be constant throughout the flow. µ and ξ are, respectively, the V. DYNAMICS OF ACOUSTIC GEOMETRIES IN VISCOUS FLOWS dynamic and second (bulk) viscosities of the fluid. Linearising and following the procedure described in Section II A, we obtain the perturbed version of the A. Governing equations and wave equation Navier-Stokes equation, Viscosity is a phenomenon of internal friction, which p1 α 2 ρ1 ∂tψ1 = v0 ψ1 + ψ1 v0 , (44) causes an irreversible transfer of momentum from points − · ∇ − ρ0 ρ0 ∇ − ρ0 ∇ where the velocity is large to those where it is small. 4 Processes of internal friction occur in a fluid only when where α, the viscosity parameter, is defined as α = ξ+ 3 µ. different fluid particles move with different velocities, so The corresponding wave equation reads

α 2 2 α 2 ∂t ∂tψ1 + v0 ψ1 ψ1 + c ψ1 + v0 ∂tψ1 + v0 ψ1 ψ1 = 0. (45) · ∇ − ρ0 ∇ ∇ · − ∇ · ∇ − ρ0 ∇

In this equation we have used the fact that the back- where M is a mass unit, L stands for length and T for ground flow is irrotational, v = 0, and incompress- time. In the new units, viscosities of realistic fluids are ∇ × ible, v0 = 0. in the range Making∇ · use of the plane wave ansatz (23) and working 10 5 αˆ [10− , 10− ] cm. (51) once more in a static fluid configuration, one obtains the ∈ wave equation in terms of the radial field, The choice of cm as length unit has to do with the ex- 2 2 ω 1 2 1 perimental realisation of the setup; in particular with ∂r ϕ + 2 2 m ϕ = 0. (46) R = 1 cm. For the remainder of the article, all hats are c iωα/ρ0 − r − 4 0 − dropped and we work with the rescaled quantities. It can be noticed viscosity is only relevant when dealing with high frequencies ω. The viscous term can be thought B. Scattering in the presence of viscosity of as a correction to the local speed of sound. We define an effective local speed of sound, as Working with the viscous wave equation (45), one 2 2 ωα starts by realising that the effective potential in the wave ceff = c i . (47) − ρ0 equation is complex. Consequently, the Wronskian is not In view of analysing the role of viscosity in the analogue constant in the radial coordinate [13]. Instead, it obeys treatment, we first derive its units. Secondly, we define dW αω3 the range of realistic values in such units. Numerically, = 2i ϕ(r) 2. (52) dr 4 ω2α2 | | we work with c = ρ = 1. This choice is equivalent to ρc + ρ rescaling the quantities This leads to a modified condition for the reflection co- ω α efficient R 2 in viscous superradiant systems: ωˆ = , αˆ = . (48) | ωm| c cρ0 2 2 ω < mΩ = Rωm > 1 αω . (53) Dimensional analysis yields ⇒ | | − Identically to the previous sections, we plot the reflec- M 2 [α] = (49) tion coefficient Rωm as a function of the frequency ω LT – Figure 7. We| work| with Ω = 0.9 and Ω = 3 for both [ˆα] = L, (50) m = 1 and m = 2, while varying the values of α in 9

Ω = 0.9 m = 1 Ω = 0.9 m = 2 1.8 1.2

1.6 1 1.4

1.2 0.8 7 7 α = 10− α = 10− 2 2 | 1 α = 10 6 | α = 10 6 − 0.6 − ωm 5 ωm 5 α = 10− α = 10− R 0.8 R | 4 | 4 α = 10− α = 10− 0.6 0.4

0.4 0.2 0.2

0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ω ω

Ω = 3 m = 1 Ω = 3 m = 2 45 16

40 14 35 12 30 7 10 7 α = 10− α = 10− 2 2 | 25 α = 10 6 | α = 10 6 − 8 − ωm 5 ωm 5 α = 10− α = 10− R 20 R | α = 10 4 | α = 10 4 − 6 − 15 4 10

5 2

0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ω ω

2 FIG. 7. Reflection coefficient |Rωm| as a function of ω. The plots presented here provide the modified solutions to Figure 3. For each pair of parameters (Ω, m), several orders of magnitude ofα ˆ are studied. We observe increasing viscosity yields less amplification. In the range of viscosities of real fluids and the frequencies ω considered, the superradiant limit is, in good approximation, |Rωm| = 1. As usual, we plot the limit above which superradiance occurs as a dashed line.

accordance with the real fluids range presented in equa- 1 tion (51). One observes greater viscosity returns less am- 0.9 plification. This was to be expected, as a viscous fluid behaves as a dissipative means for energy. In order to 0.8 0.7 quantitatively assess how viscosity impacts the reflection Ω = 0.9, m = 1 Ω = 0.9, m = 2 coefficient, one can compute the areas associated with 0.6 Ω = 3.0, m = 1 the various curves of Figure 7. Posterior normalisation Ω = 3.0, m = 2 0.5 with respect to the non-viscous case produces the results Normalised Areas displayed in Figure 8. 0.4 0.3

0.2 7.5 7 6.5 6 5.5 5 4.5 4 − − − − − − − − C. Quasinormal modes. Superradiant instabilities log10 α in the presence of viscosity FIG. 8. Area normalisation. For the same Ω, viscosity’s impact is higher for m = 2. One should pay special attention to the case Starting with the already known results for the non- where α = 10−7, as the ratio area with viscosity to area with no viscous case, displayed in Table II, and increasing α up viscosity is virtually 1. This validates our previous approach of to several orders of magnitude, one observes ωI decreases assuming water to have null viscosity. Note water has a viscosity as viscosity increases. This behaviour is in agreement of order of magnitude 10−8. with the results of Figure 7, as smaller values of ωI mean less amplification. However, these modifications to the QNMs frequencies due to viscosity occur only for orders 3 of magnitude greater than α 10− . Hence, the QNMs frequencies of real fluids suffer∼ no relevant changes due to viscosity. 10

D. Confined geometry in the presence of viscosity lenging predictions, the experimental tests must be done under very specific conditions. Tests of strong-field gravity, for instance, can only be performed near astrophysi- Running the same parameters as in Section IV D, with cal bodies such as BHs and neutron stars. Probing this α = 0, one observes that, in the range of realistic viscosi- type of results is not an easy task and it often deems ties,6 no modifications arise. Due to the constancy of the itself impossible due to technological lags. plots, we assume the fit of Figure 6 for the case of no vis- Acoustic analogues bridge the gap between theory and cosity to hold here as well. One concludes the behaviour experiment, as they allow for the testing of phenomena – of the frequencies for subcritical velocities is independent as Hawking radiation and superradiance, to name but a of viscosity, in the domain of real fluids. few – one would not be able to otherwise. Such effects are virtually impossible to study in astrophysical BHs with the existing technology. Studying two distinct setups, we introduced the reader VI. FINAL REMARKS to the analogue framework whilst producing new results on the topic. In particular, we saw horizons are not a Albert Einstein’s GR is a colossal achievement of 20th- necessary condition for superradiance. On top of that, century physics. The claim that what we perceive as we successfully concluded on how viscosity affects the the force of gravity actually arises from the curvature of superradiance results in the cylinder geometry. spacetime is not a straightforward one. The scientific paradigm Einstein introduced is still present nowadays, with GR being our best candidate for a theory of grav- ACKNOWLEDGMENTS ity. Since its initial publishing, GR has been subjected to many experimental tests. The most recent triumph S. Freitas would like to thank Professor Dr. V´ıtorCar- was in 2015 with the detection of gravitational waves, a doso for all the good advice, geniality, perfectionism and century after their initial prediction. GR is a very hard knowledge. The completion of this work would have not theory to test at will. Besides its mathematically chal- been possible without his guidance.

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