On the Superradiance-Tidal Friction Correspondence

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Shasvath Dtd ue2,2021) 22, June (Dated: a- y hyaepyial h aeeet ahrta mere whether a ask than rather to effect, interest same of the rel- also physically full is phe- are in explicitly it they same shown Moreover the been are not ativity. has friction this tidal nomenon, Hartle-like and tering electromagnetic [5–8]. cylinder of non-relativistic dis- context a first the off in was scattering in waves phenomenon ergoregion, Zel’dovich The an by and neither. cussed horizon requires event it an both fact with of asso- therefore presence and today holes, the is black superradiance rotating with Although ciated energy more emitted. receiving the it thus by than body back emitting scattered the hole, superradiantly be- black are phe- implication waves This the that amount. superradiance, ing same as known interaction, the hole is gains this black nomenon body by the small momentum Thus the angular while and negative. momentum mass is angular loses the and horizon fre- as energy angular the of sense greater reaching flux a same prograde the with the that (for again in quency) but found orbits rotates, generally hole body is black the It the where of infinity orbits calculated. that perturbation reach be a which can as spacetime) emitted in hole’s waves (treated black gravitational infinity body central of internal orbiting flux an the the by as and treated system be the The gravita- can system. the horizon the calculate by event infinity to towards instead emitted waves used sim- tional a is (or black formalism) formalism ratio Teukolsky ilar extreme-mass the of Rather be- case binaries. the transfer hole in momentum bodies angular the the tween calculating in body. cists central the from away further per- draw involves satellite, to the this it accelerating hole, mitting and black hole black the the of braking frequency rotational the ecneto lrcmatrltvsi bodies. relativistic ultracompact of context he hl ti oeie bevdta uerdatscat- superradiant that observed sometimes is it While physi- by used rarely however, is, approach Hartle’s eoeo f“ia rcin (well-known friction” “tidal of henomenon sehne ihicesn anisotropy increasing with enhanced is t 3, mtepito iwo h yaisof dynamics the of view of point the om faiorpcmte,wihi required is which matter, anisotropic of mrsn lc oewt t horizon its with hole black a omprising oa epnet ia n centrifugal and tidal to response ional gravitational the to close very radius a u aeilbde.I hspprwe paper this In bodies. material ous † t hysol neg ia friction tidal undergo should they ity etdta uerdatscattering superradiant that cepted n ailKennefick Daniel and c aeilbde)addiscuss and bodies) material act n -27,Germany D-72076, en, omn fteteMembrane the the of lopment e rass72701 Arkansas le, 10 Spain 0100, ,4, 3, ‡ 2 analogy. Since it is difficult to compare black holes to In Section III we show that the anisotropy can am- normal material bodies, it is convenient to consider a hy- plify the conventional tidal friction result, so that hyper- pothetical material body with a field like that of a black compact objects, if they exist and are anisotropic, may hole, a body even more compact than a neutron star. have augmented tidal coupling with an orbiting satel- Since we know little about the structure of such objects, lite. An interesting result is that bodies with significant if they exist, a common stand-in is the so-called mirror anisotropic pressure, when they are exposed to a tidal Kerr spacetime, which is simply a Kerr black hole with a force, form a tidal bulge which is nearly orthogonal (in perfectly reflecting mirror placed just in front of the event the direction of its major axis) with respect to the bulge horizon. From the point of view of radial motion there is formed in isotropic systems. Moreover, these systems are little difference between a perfect mirror and a perfectly characterized by a negative tidal Love number and a pro- transparent central body, so the mirror Kerr system is a late (rather than oblate) shape caused by rotation. convenient surrogate for an ultracompact star. In Section IV we study the relation between superra- Investigating this mirror-Kerr spacetime Richartz et diance and tidal friction in more detail, and identify an al. [9] have shown recently that superradiance disappears effective viscosity and an effective tidal Love number for with the introduction of the mirror. This is not very black holes. surprising, since by definition a perfect mirror will pro- In Section V we carefully study the extent to which su- duce a reflected wave equal in amplitude to the incident perradiance can be said to occur in bodies with no event wave, rather than one with greater amplitude. Richartz horizon. Moreover, we speculate about the viscosity of et al. demonstrated their result for a scalar field, so in ultracompact objects and about the possibility of having this paper we perform their calculation for a tensor field floating orbits around them. to demonstrate that the result does also work for grav- In Section VI we study the mirror-Kerr spacetime. itational waves. A natural interpretation of the result There are two key issues discussed. One is the intro- would be that hypercompact stars whose material was duction of an imperfect mirror, designed to model a vis- transparent (or nearly so) to gravitational waves should cous central body which can absorb gravitational wave not exhibit superradiance in a binary system, and there- energy, though not perfectly as in the black hole case. fore, if superradiance and tidal friction are equivalent, The other is to take account of the potential barrier sur- should not undergo tidal friction either. But we would rounding the black hole-like body, which can set up a normally expect to observe tidal friction in any binary cavity within which waves reflect back and forth between system consisting of material bodies. We are interested the mirror and the potential barrier as discussed in [12]. in finding whether both superradiance and tidal friction We show that superradiance and the so-called ergoregion can be found in the case of purely material objects (with- instability are both possible in this system, depending on out an event horizon) and whether they are still clearly the reflectivity of the mirror (which is probably related two ways of describing the same physical effect. to the viscosity of the material system). We conclude in Tidal friction should occur in the ultracompact mate- Section VII with a discussion of the interpretation of our rial system if the central body, whatever its nature, has various results. viscosity. So we address the role of viscosity in the binary system, whether viewed within the superradiance or the tidal friction framework. It has been known since the II. SUPERRADIANCE AND TIDAL TORQUE 1970s that viscosity plays a central role in the absorption ON BLACK HOLES of gravitational wave energy by a fluid body [10, 11]. The introduction of viscosity will permit us to maintain the In the 1970s Hartle [3, 4] showed that black holes could identification of tidal friction and superradiance in ultra- transfer their spin angular momentum to an orbiting relativistic objects. If viscosity goes to zero (as in the body by a process remarkably similar to tidal friction in mirror-Kerr example, which presumes the central body material systems such as the Earth-Moon. In this Sec- is transparent to gravitational waves) then both tidal friction we provide an outline of Hartle’s tidal theory and tion and superradiance goes to zero. If viscosity remains, its connections to the phenomenon of black hole super- then both effects remain operative. radiance. As such, our discussion here adds little new In Section II we will review the standard Newtonian information on the subject but, nevertheless, it serves to and post-Newtonian theories of both Hartle-like tidal set the scene for the main analysis of this paper. friction and superradiance and show that, to lower post- Hartle derived a formula for the torque exerted on a Newtonian order, they produce identical results. Kerr black hole, of mass M and spin angular momentum Since we are interested in hypercompact objects we J = aM 2, by a test body of mass µ M in a circular take note of results which suggest that such objects can- orbit around the black hole (for simplicity≪ we assume an not support themselves without some anisotropy in their equatorial orbit with angular frequency Ωorb). The orbit internal pressure. There has been, to our knowledge, no is assumed to lie in the weak-field region with a radius study of how pressure anisotropy would affect tidal in- b M and as a consequence the orbital velocity is itself teractions, even in the Newtonian case, so we begin by small,≫ v2 = M/b 1. calculating tidal friction where such anisotropy exists. In addition, Hartle’s≪ analysis assumed a slowly spin- 3 ning black hole in the sense that only leading order terms where with respect to the Kerr parameter a are retained. Then 32 µ2 the angular frequency of the hole’s horizon can be ap- ˙ 10 EN = 2 v (6) proximated as, 5 M The analysis leading to this result only assumes v 1, a a ≪ ΩH = (1) without placing any restriction on the Kerr parameter, 2r ≈ 4M + 0 a< 1. The negative sign of the leading (and higher) ≤ 2 order spin-dependent terms in (5) is associated with su- where r+ = M + M√1 a 2M is the event horizon radius in standard Boyer-Lindquist− ≈ coordinates. The two perradiance since it represents an energy loss for the black hole. characteristic frequencies of the system, Ωorb and ΩH, are assumed to obey The situation is quite different for a Schwarzschild black hole. According to (5), for a = 0 the horizon flux MΩorb MΩH 1 (2) first appears at a higher post-Newtonian order and it is ≪ ≪ positive, that is, it represents energy removed from the This ordering implies that within Hartle’s model we are orbiting body and absorbed by the black hole. not allowed to take the Schwarzschild limit a 0. We We shall now derive a spindown formula based on the → are allowed to make Ωorb arbitrarily small so that we have horizon flux (5). For a circular orbit the angular momen- a quasi-stationary test-body with respect to the hole (as, tum flux down the black hole horizon is given by (see for in fact, Hartle did). instance [16]) Hartle’s formula for the tidal torque exerted on the black hole is given by [3, 4]: 1 J˙H = E˙ H (7) Ωorb 2 3 ˙ 8 µ M J = 6 J (3) Expanding the angular frequency in a post-Newtonian −5 b series (see Ref. [15]) and combining it with the above where an overdot represents a time derivative. As we flux formula we obtain: have mentioned, this expression is accurate to leading 32 µ2M 6 33 order with respect to both J and v. An equivalent form J˙ = Ω 1+3a2 4+ a2 v2 H − 5 b6 H − 4 of (3) is, " 2 6 4 ˙ 32 µ M + (a2) v3 + (v4) (8) J = 6 ΩH (4) − a O O − 5 b # This result implies that a slowly-rotating Kerr hole is A similar result of even higher precision has been derived always spun down under the action of the torque caused in Refs. [17, 18]. If we wish to truncate this series at by a distant orbiting body (provided (2) holds). Global (v2) order (or lower) then we need to make sure that conservation of angular momentum then requires that the Oin the slow-rotation limit a 1 we also have a v = rotational energy removed from the hole is gained by the (M/b)1/2; it is easy to see that≪ this condition implies≫ (2). orbiting body. The overall negative sign in the J˙H flux (=spindown) This gain of energy by the small body at the expense is a direct consequence of superradiance in the gravita- of the black hole bears an obvious resemblance to the tional wave energy flux E˙ H. At the same time, and at situation where energy is removed by a black hole via leading order with respect to a and v, J˙H is identical to superradiant scattering of gravitational waves [13]. It can the Hartle torque (4). Therefore the spindown described be shown that this is more than a mere similarity and we by (4) can be fully attributed to superradiance. In other will investigate the extent to which Hartle’s spin-down words Hartle’s “tidal friction” is simply superradiance. effect is physically indistinguishable from superradiance. It is also clear that (8) can be viewed as an extension of This equivalence can be demonstrated by considering Hartle’s formula for an arbitrary spin a (note that similar the gravitational wave energy flux at the horizon as calcu- formulae have been derived in the past, see Refs. [19, 20]). lated by a post-Newtonian approximation of the Teukol- In particular, the angular momentum flux at the horizon sky black hole formalism [14]. This arduous calculation of a Schwarzschild black hole (more precisely of a hole is described in Ref. [15]; the final result for the energy with ΩH Ωorb) is flux at the horizon produced by a test body in a circular ≪ 32 µ2M 6 equatorial orbit is: J˙ Ω (9) H ≈ 5 b6 orb 1 33 E˙ = E˙ v5 a 1+3a2 a 1+ a2 v2 and represents a spin-up torque. It can be seen that this H N − 4 − 16 " expression displays the same dependence with respect to M and b, as well as the same numerical prefactor 32/5 + 1+ (a2) v3 + (v4) (5) (albeit with opposite sign), as that of the slow rotation O O # result (4) (or (8)). 4

The two torque results we have discussed so far, that the outside Universe with certain properties determined is, the torque exerted on the black hole in systems with by the mass and spin of the black hole. In this viewpoint MΩ 1 and MΩ 1 and with Ω /Ω 1 the surface has an effective viscosity which explains the orb ≪ H ≪ H orb ≫ (eqn. (4)) or ΩH/Ωorb 1 (eqn. (9)), can be combined dissipation connected with the exchange of angular mo- into a ‘global’ torque formula:≪ mentum between the black hole and the orbiting body. It is worth noting, however, that the analogy is not per- 2 6 32 µ M fect where the actual shape of the tidal distortion of the J˙H (Ωorb ΩH) (10) ≈ 5 b6 − event horizon is concerned. Several studies have shown not This is valid for any ratio Ω /Ω and at leading order that the tidal bulge raised in an event horizon is in H orb the position one would expect from the case of material with respect to the small parameter M(Ωorb ΩH) 1. A key property of (10) is that it predicts| a− zero torque| ≪ bodies [4, 21, 22]. on the black hole in the special case of a black hole- Fang & Lovelace [21] and Vega, Poisson & Massey [22] argue that this is the result of the teleological nature synchronous orbit, Ωorb = ΩH. In fact, this is generally true for all circular equatorial orbits in Kerr spacetime of the horizon, based on Hartle’s [4] original point that – this can be verified by a simple inspection of the fully the boundary condition which defines the location of the horizon is imposed at future null infinity and thus has an relativistic formula for J˙H as calculated with the help of the Teukolsky formalism (see Ref. [16] for details). inherently different causality to that which applies to the surface of a material body. Although the actual position of the tidal bulge is different in the black hole case, it A. The nature of tidal bulges on the event horizon produces the same effects, in terms of energy and angular momentum transfer, as does a bulge in a material body. In a binary system, if the body upon which tidal bulges are raised rotates with respect to the source of the tidal III. TIDAL TORQUE ON NEWTONIAN gravitational field then the viscosity of the body causes ANISOTROPIC STARS the bulges to attempt to rotate with it, even though they “ought” to remain directly under the body which is the source of the tidal field. The upshot is a tidal A. Why anisotropic stars? lag, in which the bulges take up a position some few degrees away from their “natural” position. This intro- The second stage of our analysis of the superradiance- duces a torque acting on the body because the pull of tidal friction correspondence requires a discussion of the the source is no longer purely radial, but also tangen- tidal torque exerted on material fluid bodies. The classic tial. This process transfers angular momentum from the Newtonian tidal theory was first formulated in the late rotating body to the orbiting body (in the most fami- 19th century by Darwin [23] and is still in use today. lar cases), with some additional dissipation of energy, for One of the modern discussions of Darwin’s theory is pro- instance by heating. In cases where the orbiting body vided in a recent paper by Poisson [24], in the context of actually has a larger angular frequency than that of the tidally deformed black holes and their similarities with rotating body (as in the case of Phobos, the moon of Newtonian tides. Mars), then a tidal lead will occur and angular momen- Our own discussion of tidal friction in this Section re- tum will be transferred the other way, from the satellite, lies heavily on Poisson’s analysis, differing only in that operating to speed up rather than retard the rotation we consider a Newtonian star made of anisotropic mat- of the planet. This process is speculated to explain the ter (this property is encoded in the presence of two scalar moonlessness of the planet Venus, whose extremely slow pressures, radial and tangential, as described below). rotation would cause any satellite to lose orbital angular The motivation for choosing to study anisotropic stars momentum and eventually crash into the planet below. is this: from the known solutions of compact relativis- In a similar process, we may expect nearly Schwarzschild tic stars it is clear that these objects cannot be made black holes to more quickly draw and swallow orbiting arbitrarily compact (i.e. the stellar radius R⋆ approach- objects than would rapidly rotating black holes. ing the Schwarzschild radius 2M⋆) unless their matter is At first glance it is not at all apparent that there should highly anisotropic. For instance, this is the case for uni- be an analogous process operating in black holes. The form density spheres where only the anisotropic Lemaitre mass of the black hole is concentrated at a point (at solution can approach the ‘black hole’ limit R⋆ 2M⋆ least classically) and it seems counter-intuitive that the [25, 26]. Another type of ultracompact objects,→ the so- event horizon should have a viscosity. Nevertheless Har- called gravastars, are also known to require a layer of tle showed that an interaction analogous to tidal friction anisotropic pressure, see [27]. does take place between the event horizon of a black hole Since only relativistic hypercompact stars can closely and an orbiting body. This insight led to the develop- mimic black holes, it makes sense, if these bodies are ment of a different view of the black hole, known as the likely to exhibit pressure anisotropies, to understand how membrane paradigm [1], in which the event horizon is such anisotropies affect tidal deformation, albeit in New- conceived of as a surface which interacts physically with tonian theory. As we shall see, the presence of anisotropy 5 in matter leads to some rather interesting behaviour. C. The perturbed star

We now consider the previous anisotropic star, with B. The unperturbed star rotation and under the action of an imposed quadrupolar tidal field (the tidal field source will be later specified as that of an orbiting test-body). Working in the stellar rest To begin our analysis we first need to consider the un- frame and using Cartesian coordinates xi we can express perturbed spherically symmetric star. The key stellar the tidal field in terms of the time-dependent symmetric parameters are the mass M , radius R , and pressures ⋆ ⋆ and trace-free (STF) tensor (t). The induced tidal p (radial) and p (tangential). The star is assumed to ij r t potential is E have a uniform density

1 i j 3M Φtidal = ij x x (17) ⋆ 2E ρ = 3 (11) 4πR⋆ Under the action of this ﬁeld a stellar scalar parame- It is also convenient to deﬁne the anisotropy parameter ter X can be decomposed into a background spherically symmetric piece and a quadrupolar perturbation, i.e.

σ = pr pt (12) i j − X = X0(r)+ δX = X0(r)+ Xij x x (18) The hydrostatic equilibrium of the system is described by (we use the index 0 to identify background parameters) with Xij (t) being itself a STF tensor. Stellar rotation, with angular frequency vector Ωˆzi (a ‘hat’ labels a unit vector), leads to a centrifugal potential 2σ0 p′ = ρΦ′ (13) which can also be decomposed in a similar way: r0 − 0 − r 1 1 1 where a prime stands for a radial derivative. The grav- V = Ω2(x2 + y2)=Ω2 r2 + C xixj (19) cf 2 3 2 ij itational potential Φ0 can be calculated in a standard way: where C = δ /3 zˆ zˆ . The star is assumed to be ij ij − i j 2 slowly rotating, in the sense that Ω ΩK, where ΩK M⋆ r 3 1/2 ≪ 1 ∼ Φ (r)= 3 (14) (GM⋆/R ) is the mass-shedding spin limit . 0 −2R − R2 ⋆ ⋆ ⋆ The tidal field deforms the shape of the star and produces a velocity field δvi which is superimposed on the The stellar model would be complete if equations of bulk rigid-body rotation. This velocity can be decom- state (that is, functional relations between the pressures posed as [24] and the density) were available but no such physically- motivated relations are known for anisotropic matter. 2 2 j k l δvi = (r + γR⋆)Vij x + βVklx x xi (20) The only viable alternative is to use a convenient para- metric relation between pt0,pr0 and ρ that can smoothly where γ, β are dimensionless constants and Vij (t) is an- vary from isotropy to extreme anisotropy. One such other STF tensor. parametrization is given in Ref. [26]. By taking the New- The perturbed stellar surface can be parametrised with tonian limit of the general relativistic solution derived in the help of the deformation tensor eij (t) as: that paper we have the following relation: i j R = R⋆(1 + eij nˆ nˆ ) (21) 4πr2 1 p (r)= p (r)+ ρ2 Q (15) t0 r0 3 2 − where nî = xi/r is the unit radial vector. The departure from sphericity is accompanied by an additional ‘body’ The parameter Q controls the degree of anisotropy, al- contribution Φbody to the total gravitational potential. lowing a smooth transition from isotropic stars (Q =1/2, We thus have the relativistic Schwarzschild solution [28]) to the most 1 extreme case of anisotropy, the so-called ‘Lemaitre vault’ δΦ=Φ +Φ = xixj +Φ xixj (22) tidal body 2Eij ij solution, pr0 = 0 and Q = 0 [25, 26]. The minimum allowed stellar radius for a given Q is [26]:

−1 1 min −1/Q Strictly speaking, the spherically symmetric portion of Vcf should R⋆ =2M⋆ 1 3 (16) − have been included in the calculation of the structure of the unperturbed star. However, it turns out that the rotational correc- which shows that Q = 0 (Q = 1/2) represents the most tion to the equations of Section III B can be ignored at the level (the least) compact object. of precision of the tidal deformation calculation. 6

The body potential can be calculated in the standard sion: way [24], 1 2 j k δpt ∂i Ω Cjkx x δΦ 3 M⋆ 2 − ρ − Φij = 3 eij (23) −5 R⋆ δσ 2 δσ nˆ nˆj ∂ + + ν 2δv =0 (27) − i j ρ r ρ ∇ i The velocity perturbation at the surface can be written as As a final equation we need to formulate a boundary condition of a vanishing force at the stellar surface. This j nˆj δv (R⋆)= ∂tR (24) can be done by defining the traction vector from the fluid i ij stress-energy tensor, t = T nˆj . The desired surface Moreover we can assume an incompressible flow, δρ = boundary condition is ti =0 which leads to j 0 ∂j δv = 0. From these equations we can obtain β =→ 2/5 and p nˆ ρν(∂ δv + ∂ δv )ˆnj =0 (28) − r i − j i i j 2 ∂teij = (1+ β + γ)R⋆Vij (25) D. Calculating the tidal deformation Up to this point the perturbation analysis has been identical to that of isotropic fluid stars hence it is not The torque exerted on the star by the tidal field can surprising that the previous equations match those of be computed once the deformation tensor eij is available. Ref. [24]. However, the remaining equations to be dis- The calculation of this parameter is the subject of this cussed, the Euler equation and the surface boundary con- subsection. dition, do depend on the presence of anisotropic matter. The total surface pressure can be approximated as (us- The linearised Euler equation describing the hydrody- ing the fact that pr0(R⋆)=0), namics of an incompressible viscous fluid with anisotropic ′ pressure can be written as (see the Appendix for a deriva- p (R⋆) p (R) R2nˆ nˆ pij + r0 eij (29) tion) r ≈ ⋆ i j r R ⋆ Using this and eqn. (20) in the surface condition (28), j k 1 2 j k ∂tδvi + 2Ωǫijkzˆ δv = ∂i Ω Cjkx x δΦ 2 − k k l 2ρν (2 + β + γ)Vjknˆ +(1+2β)Vklnˆ nˆ nˆj 1 1 j 2 δσ 2 ′ ∂iδpt nî nˆ ∂j δσ + + ν δvi (26) kl pr0(R⋆) kl − ρ − ρ r ρ ∇ =n ˆj nˆknˆl p + e (30) r R ⋆ This equation features the (perturbed) Coriolis and cen- The various terms in this equation can be collected in two trifugal forces, pressure and gravitational forces, and a independent groups, leading to γ = (2+β)= 8/5 and viscous force with (uniform) shear viscosity coefficient ν. − − ′ As far as the tidal interaction problem is concerned ij pr0(R⋆) ij 2ρν ij pr = e 2 (1+2β)∂te (31) the full Euler equation (26) can be greatly simplified if − R⋆ − R⋆ the star is slowly rotating (Ω ΩK) and the tidal field is slowly-varying (more precisely,≪ for the particular case In the final step of this calculation we need to use the of tides raised by an orbiting body of angular frequency Euler equation (27); this should eventually become a dif- Ωorb we assume Ω Ωorb ΩK). Under this double ferential equation for eij . Using the previous results, assumption we can| easily− show| ≪ that the first two inertial jl terms in (26) are negligible with respect to the tidal force 1 2 jl pt jl 1 jl (whereas the centrifugal force is not), see also [24]. The xl Ω C Φ 2 − ρ − − 2E ! viscosity force is also assumed to be small; the mathe- k σkl jl matical requirement is τν Ω Ωorb 1 (τν is the viscous 2xlnˆj nˆ = ν(5+2β)V xl (32) delay timescale, see eqn. (39)| − below| ≪ for its definition). In − ρ − essence this is a statement of a small phase-lag between After collecting terms of similar structure we find the instantaneous position of the orbiting body and the ij ij direction of the raised tidal bulge on the star. It should σij =0 pr = pt (33) be also pointed out that the viscosity term can even be → much smaller than the other terms in the Euler equation and (indeed it can be exactly zero) but, nevertheless, it must ν p′ (R ) 3 M be retained as the leading dissipative term. After all, this (3 2β)∂ e r0 ⋆ + ⋆ e R2 − t ij − ρR 5 R3 ij is the term responsible for the system’s ‘tidal friction’. ⋆ ⋆ ⋆ 1 Adopting the above assumptions the Euler equation = Ω2C (34) can be approximated by the simpler, amputated expres- 2 ij −Eij 7

The pressure gradient can be eliminated with the help of which subsequently combine to give the hydrostatic equilibrium equation (13). Then, after −1 ˙ −1 using (15) and some trivial algebra, we arrive at: eij (t) fij (t) τν fij (t) fij (t τν ) (43) ≈ A − ≈ A − 3 τ0 5 R⋆ 2 This is clearly consistent with (40). Restoring the explicit ∂teij + eij = Ω Cij ij (35) 4 M −E form of fij , A A ⋆ 3 where we have introduced the ‘viscous time delay’ ap- 5 R⋆ 2 eij (t) Ω Cij ij (t)+ τν ˙ij (t) (44) pearing in the tidal theory of isotropic stars [24]: ≈ 4 M⋆ −E E A 19 νR This solution looks similar to the classic tidal deforma- ⋆ −1 τ0 = (36) tion result: modulo the scaling factor , the deforma- 2 M⋆ A tion is simply equal to the time-retarded driving term fij . The only difference between our result (35) and the cor- In a binary system consisting of an orbiting moon and a responding formula for isotropic stars is encoded in the fluid planet the solution (43) describes a tidal bulge on anisotropy factor, the planet’s surface that follows the position of the moon with a time lag τν . 3 This picture is accurate for isotropic and weakly 5Q = 5(Q Qcrit) (37) A≡ − 2 − anisotropic stars; increasing the degree of anisotropy (Q For Q = 1/2 (isotropy) we have = 1, while with de- moving away from 1/2 towards Qcrit = 3/10) results in A an overall amplification of the tidal deformation and a creasing Q (increasing anisotropy) changes sign and + A longer time-lag τν . However, the limit Q Q cannot becomes negative; this happens at Q = Qcrit = 3/10. → crit The parameter is minimised at Q =0, where = 3/2. be studied within our formalism because of the diver- Equation (35) describes the time-evolutionA of the− stel- gence in both τν and eij . lar deformation for a given external tidal field . Ina Eij more compact notation, E. The case of strong anisotropy −1 τν e˙ij + eij = fij (t) (38) A Once the anisotropy parameter Q becomes less than where, as before, an overdot stands for a time-derivative Qcrit, making < 0, the system shows an unexpected and a new effective viscous timescale has been defined: response to tidalA perturbations. The exact solution of the τ0 deformation equation (38) contains an exponential term τν (39) exp(t/ τ ) which becomes dominant at late times t ≡ ∼ | ν | ≫ A τν , signaling the presence of an instability. However, We also note that the source term fij (t) is identical to the| | fact that the characteristic timescale of this solution that of the isotropic problem. Already at this point, and is τν casts serious doubts on the physical relevance of this before solving (38), the possibility of < 0 is an indi- A instability: the limit ν 0 implies an arbitrarily high cation that strongly anisotropic stars would behave in a growth rate for the tidal→ deformation, a prediction that qualitatively different way under the action of a given makes little sense. tidal field. This indeed will turn out to be true as we It would not be too surprising if the reader finds the shall shortly see. nature of this solution reminiscent of the ‘runaway so- The exact solution of (38) can be easily found (e.g. see lutions’ of the famous Lorentz-Dirac equation of motion Ref. [24]) but it is not particularly useful or physically for an accelerating charged particle (see [29] for a detailed transparent. A more useful approximate solution can be discussion of this equation). After all, the < 0 version obtained by assuming that the deformation eij (t) varies of (38) is mathematically equivalent to the Lorentz-DiracA on a timescale tf , equation. And as it was the case for the unphysical solutions of that equation, the runaway solution of (38) can eij fij tf (40) be dismissed on the basis of its timescale. It can be eas- e˙ij ∼ ≡ f˙ | | | ij | ily verified that a solution eij /e˙ij τν would generically violate the conditions underpinning∼ the validity of (32) Treating τ /t as a small parameter (in agreement with ν f (for instance the inertial acceleration term may outgrow our earlier assumption τ Ω Ω 1 which led to the ν orb the tidal force) and therefore (38) itself. approximate Euler equation| − (27)) we| ≪ adopt the following The upshot of this discussion is that the tidal deforma- ansatz tion of strongly anisotropic stars (0

This solution represents a rather counter-intuitive be- parameters, hence suggesting a "no-hair" property for haviour for such systems. Firstly, we can see that ro- neutron stars [31]. tation drives the body to a prolate rather than the usual Our findings suggest that taking the "black hole limit" oblate shape. Secondly, in the presence of a tidal field of ultracompact stars (i.e. R⋆ 2M⋆ ) may reveal sourced by an orbiting satellite we can see that, in the a discontinuity in the "I-Love-Q"→ relations. Beyond limit of zero viscosity, the induced tidal bulge is at right some compactness (that is never reached by neutron angles to the line pointing towards the satellite. In other or quark stars described by any known realistic equa- words the bulge is rotated 90 degrees with respect to the tion of state), the stellar matter is likely to experi- bulge of isotropic or weakly anisotropic stars. Viscos- ence a "phase-transition" and forcibly become strongly ity causes an additional slight rotation to the bulge; the anisotropic, thus producing a system with k2 < 0, > 0 bulge’s major axis now is orthogonal with respect to the that deviates from a black hole. Of course, in orderQ to time-advanced position of the orbiting body. strictly establish this discontinuity in k2, one would have to repeat our analysis in full General RelativityQ and obtain the same sign for these parameters. F. Strongly anisotropic systems: ‘anomalous’ quadrupole moment and Love number G. The tidal torque The counter-intuitive tidal dynamics of < 0 stars is A reflected in the sign of the so-called tidal Love number As discussed in Ref. [24], the tidal torque exerted on k2, that is, the proportionality factor appearing in the the star is to be calculated in the global inertial frame linear relation between the tidal field and the induced (centered at the stellar center of mass). Using an overbar quadrupolar moment, Qij = (2/3)k2 ij where to label quantities in that frame we have the following − E expression for the tidal torque: 3 1 2 Qij = d xρ xixj r δij (46) − 3 3 j k ¯ 3 j ¯kl Z Ni = d xρǫ¯ ijkx¯ ∂ Φtidal = d xρǫ¯ ijkx¯ x¯l − − E is the STF quadrupole moment tensor. This is related to Z Z (50) the deformation eij as [24] Carrying out the integration we can obtain the rate of change of the stellar spin, 2 2 Qij = M⋆R⋆eij (47) 5 J˙ = ǫ Q¯j ¯lk (51) i − ijk lE Combing these expressions with the previous solutions where the quadrupole moment tensor is given in for e , we find, ij eqns. (46) & (47) above. Ignoring the possibility of any

3 −1 precessional motion (that is, we assume a perturbation k = (48) i i 2 4A that does not change the orientation of J = Jzˆ is space) we have Normally the Love number is a positive quantity, however, according to our result strongly anisotropic stars J˙ = ǫ Q¯i ¯lj zˆk (52) − ijk lE ( < 0) have k2 < 0. AThe response of rotating anisotropic stars to the ac- The coordinate transformation from the rotating to tion of the centrifugal force is equally ‘anomalous’ with the inertial frame is of course independent of the stel- respect to the induced quadrupole moment . This is de- lar model. Hence the entire analysis of Ref. [24] applies fined in the usual way, Q = diag( /3, Q /3, 2 /3). here too without any change. Using the solution (44) it ij −Q −Q Q Isolating the centrifugal term in the solution for eij (i.e. is found that only the viscous term τ ˙ contributes ∼ ν Eij the Cij term) we obtain the quadrupole moment to the tidal torque. It is then a matter of simple inspection to conclude that the final result for J˙ is that of the R5Ω2 −2 = ⋆ (49) isotropic star rescaled with a factor : Q − 2 A A ˙ 1 ˙ Then, clearly, strongly anisotropic systems have > 0 J = 2 Jiso (53) which is of opposite sign with respect to the quadrupoleQ A moment of isotropic stars (this basically means that ro- It is important to emphasize that a factor −1 comes tation makes the body prolate rather than oblate). from the rescaled viscous timescale (39) whileA another These results are of some importance when viewed similar factor originates from the overall prefactor in the from the perspective of the "I-Love-Q" universality rela- right hand side of (44). This second contribution is also tions, very recently shown to exist in neutron and quark present in the inviscid system and represents a rescaling stars for a variety of realistic equations of state [30, 31]. in the tidal Love number k2 k2/ . With increasing stellar compactness, the "I-Love-Q" re- The occurrence of the 2 →factorA means that the tidal lations appear to approach the corresponding black hole torque formula (53) is commonA for both strongly and 9 weakly anisotropic stars and, moreover, that and The torque (57) displays the well-known result that a systems are tidally torqued at the same rate (assumingA −A all small body orbiting rapidly (slowly) with respect to the other parameters identical). The fact that 3/2 1 central body’s rotation will cause the big body to spin- means that anisotropy can enhance the magnitude− ≤A≤ of the up (spindown) provided viscosity is present. An equilib- tidal torque, but as the extreme limit Q 0, 3/2 rium is reached when the system becomes synchronised, → A→− is approached the effect is reversed. Ωorb =Ω. For the particular case of a tidal field sourced by a test If Ωorb Ω⋆ (as in Hartle’s calculation) the torque body of mass µ in a circular equatorial orbit of radius b (57) reduces≪ to the spin-down torque: and angular frequency Ωorb we have κ ν µ2R6 J˙ ⋆ Ω (59) 3µ 1 N 2 6 = mˆ mˆ δ (54) ≈− M⋆ b Eij − b3 i j − 3 ij A In the opposite limit Ωorb Ω (which includes the pos- where sibility of a non-rotating star)≫ we have a spin-up torque:

i 2 6 mˆ = [cos χ, sin χ, 0], χ = (Ωorb Ω)t (55) ˙ κ ν µ R⋆ − JN 2 6 Ωorb (60) ≈ M⋆ b is the unit vector pointing at the instantaneous position A of the orbiting body. After repeating the calculation de- These last three Newtonian expressions look very similar scribed in Ref. [24] the torque (8) becomes to the previous formulae for the torque exerted on a black hole, see eqns. (4), (9) and (10). 2 5 9 τ0 µ R The match between the torque for Newtonian bodies J˙ = ⋆ (Ω Ω) (56) 2 2 b6 orb − and black holes becomes exact if we make the obvious A identifications R r , Ω Ω and subsequently This is our final result for the tidal friction on an ⋆ ←→ + ←→ H −2 define an effective black hole Love number, viscosity and anisotropic star. Due to the factor it displays an en- anisotropy: hanced torque with respectA to isotropic stars, but apart from that it is identical to the classic tidal torque for- k ν M H H = (61) mula. 570 AH Then the torque exerted on a black hole, eqn. (10), be- IV. THE SUPERRADIANCE -NEWTONIAN comes TIDAL FRICTION CORRESPONDENCE 2 6 ˙ 57kH νH µ r+ JH = 6 (Ωorb ΩH) (62) After discussing two incarnations of tidal friction, in H M b − A the context of black holes (where it was identified with the more exotic notion of superradiance) and in that of It is clear that the identification (61) implies a high Newtonian fluid stars, we have prepared the ground for degree of degeneracy between the three effective parame- making a formal connection between them. ters. Part of the degeneracy can be lifted if we make use That there is a connection between these two seem- of the effective horizon viscosity appearing in the mem- ingly alien notions had been known for some time [1, 3]. brane model of black hole event horizons [1]. The horizon Apart from repeating this early analysis we make a new viscosity is given by the remarkably simple formula contribution to the subject by including anisotropic stars ν = M (63) which, as we have discussed, might be more ‘realistic’ H models for ultracompact relativistic bodies. 2 and it allows us to fix the hole’s Love number , kH = We have seen that the Newtonian tidal friction torque H/570. is given by (eqns. (56) & (36) ) A Which would be the most natural choice for a black hole? AlthoughA = 1 could be a reasonable initial κ ν µ2R6 AH J˙ = ⋆ (Ω Ω) (57) guess (thus making (61) identical to the identification of N 2 M b6 orb − A ⋆ The numerical prefactor κ depends on the internal structure of the star, more precisely the combination κ/ is a A 2 This Love number result may seem at odds with Refs. [32, 33] surrogate for the tidal Love number. For our uniform star which show that the Love number of a Schwarzschild black hole is we have κ = 171/4; given that the Love number for the exactly zero. We do not believe there is a contradiction here: our −1 same system is k2 = (3/4) we can write an equivalent Love number is an effective quantity emerging as part of of the expression for the torqueA by making the substitution tidal torque identification between a black hole and a material body whereas the vanishing Love number calculated in [32, 33] is a result of the computation of the actual event horizon defor- κ 57k2 (58) mation. 2 → A A 10

Ref. [24]), a more intuitive choice would be to exploit the V. SUPERRADIANCE FROM HORIZONLESS R⋆ 2M⋆ property of the Lemaitre ‘star’ and identify COMPACT OBJECTS? →with the minimum value of , that is, = 3/2. AH A AH − In that case the effective Love number becomes negative, The established close connection between relativistic k = 1/380, a property also shared by the Lemaitre H − superradiance and Newtonian tidal friction can be ex- star. ploited if we wish to understand the gravitational physics A relation like (61) or (63) is not that absurd; after all, of an extreme mass ratio system in which the Kerr hole is replaced by a putative ultracompact and horizonless the event horizon is “viscous” in the sense that it absorbs everything that falls in it. Note that (63) is always used massive body. This is not an entirely unreasonable set 2 up: a class of compact horizonless objects – where gravas- at a level of precision at which r+ =2M + (a ). Hence a relation ν M is the only possibility thatO makes sense tars and boson stars are among the most popular mem- H bers – is sometimes invoked as a theoretical alternative on dimensional∼ grounds (in non-geometric units M GM/c in (63)). Expressed in human units, the horizon→ to Kerr black holes for explaining the true nature of the supermassive “dark” objects commonly found in nuclei of viscosity is: massive galaxies (including our own Milky Way). Leaving aside the question of how realistic such objects could be (surely none of them has the astrophysical sta- GM M ν = 8.6 1014 cm2s−1 (64) tus enjoyed by black holes and neutron stars) we instead H c ≈ × M ⊙ prefer to focus our attention to the notion of “superradiance” in these systems. This is partially motivated by the fact that the simplest compact and horizonless object This is an enormous value; it dwarfs by many orders of one could build (albeit in an entirely unphysical manner) magnitude the predicted viscosity in neutron star matter is a black hole with its horizon enveloped by a perfect – the most dense state of matter known (see next Sec- mirror. tion). So for stellar mass compact objects a black hole is The resulting “mirror-Kerr” object is supposed to far more “viscous,” for the purposes of tidal friction, than mimic a horizonless material body in the sense that it any other compact or hypercompact object is likely to be. is designed to reflect, rather than absorb, any incoming As we shall note, however, for the purposes of comparing gravitational waves. The role of the mirror is to provide to tidal friction in the Earth that an Earth mass black the total reflection boundary condition, with respect to hole does not have such an enormous viscosity, since the radial motion, that the geometric center does in mate- event horizon effective viscosity scales with mass. rial bodies. We ignore the energy removed by the wave The identification between (10) and (57) via (61) and dissipated as heat as a result of its coupling with the is of key importance for the discussion of this paper. body’s matter. Note, however, that doing so is probably These formulae (and their limits for large or small ra- tantamount to presuming that the body is inviscid! The presence of the mirror has a striking effect on tios Ωorb/Ω and Ωorb/ΩH) describe completely different physical mechanisms for producing tidal friction: su- superradiance. In fact, superradiance disappears alto- perradiance as opposed to normal fluid viscosity. Nev- gether. This was recently pointed out in Ref. [9] in the ertheless the resulting torques are described by identi- context of scalar wave propagation in Kerr spacetime. cal formulae, at least within the specified approximation The same conclusion remains true in the case of gravitational waves – this is discussed in detail in a later Section. M(Ωorb ΩH) , M(Ωorb Ω) 1. | − | | − | ≪ Extrapolation of this result to the case of more realis- It therefore makes perfect sense to consider superra- tic horizonless bodies would imply that these objects do diance and viscous friction as equivalent notions, in the not display any superradiance of the General Relativistic sense that they can produce the same torque between the type. This certainly can be a correct statement, however, members of a binary system. it does not account for the likely presence of Newtonian- type “superradiance” in the form of tidal friction. As In particular, it can be said that a Newtonian system we have discussed, at leading order the two effects are can exhibit “superradiance”, J˙ < 0, in the same way a N dynamically equivalent. What is really required of the tidally distorted black hole does. And it is equally legal horizonless object is to consist of viscous matter. to consider a black hole as a “viscous” body, experiencing tidal friction as Newtonian fluid bodies do. To make the discussion more concrete let us consider the particular example of the anisotropic ‘star’ of Sec- To conclude this discussion it should be pointed out tion III. Some degree of anisotropy in the fluid pressure that the tidal friction - superradiance equivalence is un- seems to be a key ingredient in relativistic stellar mod- likely to survive beyond (a) precision. At higher or- els with high compactness, such as gravastars (these ob- ders with respect to theO spin, the black hole torque is jects consist of a de Sitter spacetime vacuum spherical expected to be sensitive to General Relativistic physics interior which is cloaked by a thin shell of high-density such as frame dragging and ergoregions – notions com- anisotropic fluid matter, see [27]). pletely alien to Newtonian physics. Our earlier result for the tidal torque exerted on an 11 anisotropic star by an orbiting test-body, eqn. (57), can For neutron star matter we have cs 0.1c and λ R⋆ be translated to an effective “horizon” energy flux (this and as a result ν ν . The effective∼ black hole≪ hori- ns ≪ max is identical to the tidal work W˙ of Ref. [24]): zon viscosity νH does not really have to do anything with particle collisions but nevertheless we can still compare κ ν µ2R5 it against (69) with R replaced by r =2GM/c2. Using E˙ =Ω J˙ = ⋆ Ω (Ω Ω) (65) ⋆ + ⋆ orb N 2 M b6 orb orb − our previous result we find A ⋆ νns νH For orbital motion with Ωorb < Ω the central object ex- 1 (70) hibits “superradiance”, E˙ ⋆ < 0, and this flux is gained by νmax(R⋆) ≪ νmax(r+) ∼ the orbiting body itself as a result of angular momentum Most notably, the effective horizon viscosity is compara- conservation. ble to the maximum viscosity ν . At the same time the small body radiates gravitational max The upshot of this discussion is that a ‘realistic’ ultra- waves at infinity with a flux (given here at a quadrupole- compact star could be, at least in principle, as viscous as formula precision) a black hole or thereabout. Combined with the enhance- 2 ment due to the anisotropy factor < 1 this would im- ˙ 32 µ 10 ˙ A E∞ 2 v (66) ply that the flux E⋆ might be an important factor in the ≈ 5 M⋆ gravitational wave-driven inspiral of small bodies around such stars. If nothing else, these systems can be “super- For obtaining a numerical value for the flux (65) we ob- radiant” even if negligibly so. viously need to know the coefficient of kinematic viscosity Continuing along the same line of reasoning we could, ν. Unfortunately this is not possible: the detailed com- for example, ask if “floating” orbits can exist. By def- position – let alone properties – of matter at such high inition these are test-body orbits where the orbital en- densities as the ones required to build a supermassive- ergy lost to gravitational radiation at infinity is bal- ultracompact star with R 2M is completely un- ⋆ ⋆ anced by the horizon flux of the central object. It is known (assuming that such objects≈ are realised in Na- known that such orbits are not possible around black ture). Nonetheless we can get some idea of how viscous holes (see Ref. [36] for a very recent discussion) but this dense matter is by making contact with neutron stars – may not be necessarily true for other compact systems. the most dense astrophysical objects known to us3. A For the present case such orbits would obey E˙ = E˙ . typical viscosity for neutron star matter is [35], ∞ ⋆ The resulting orbital radius is − 5/4 8 2 4 ρ 10 K 2 −1 2/5 2/5 9/5 ν 10 cm s (67) bf 855 Ω −4/5 ν R⋆ ns ≈ 1014 grcm−3 T (71) M ≈ 128 ΩK A M⋆ M where T is the neutron star’s core temperature and ρ its Note that this expression was derived using a weak-field density (both quantities have been normalised to values b M approximation but it could also be indicative of typical for mature neutron stars). f strong≫ field orbits. As implied by our previous discussion, The viscosity (67) could be, perhaps, viewed as a lower a prerequisite for an orbit to float is the presence of highly limit for the viscosity in supermassive ultracompact stars. viscous matter ν νmax, assisted by an anisotropy factor We can also come up with an upper limit using the gen- < 1. In an earlier∼ paper [36] we showed that for a eral kinetic theory formula for the coefficient of dynamic |A|floating orbit to exist in a compact binary system, one and kinematic shear viscosity, η and ν, needs the central spinning body to have a prolate shape. η 1 It is worth recalling that anisotropic pressure naturally ν = = csλ (68) produces a prolate shape in a rotating body (see the dis- ρ 5 cussion after eqn. (45)).

Here cs is the speed of sound, λ is the mean free path As we have argued, there appears to be strong reasons length associated with the particle collisions responsible to believe that ultracompact relativistic stars with vis- for viscosity and ρ is the density of these particles. cous interiors do experience a phenomenon which can be From this relation we can obtain a maximum viscosity equivalently described as tidal friction or superradiance, rather as is the case for black holes. In the next sec- by setting cs = c and λ = 2R⋆, that is (we temporarily restore c) tion we will show that eliminating viscosity, which would clearly reduce tidal friction to zero, also eliminates su- 2 perradiance. ν = cR (69) max 5 ⋆ The question naturally arises whether ordinary material systems (such as the Earth-Moon) also can be thought of as superradiant, when they are observed to undergo tidal coupling. It is not the purpose of our pa- 3 The next densest astrophysical objects, white dwarfs, have a per to consider this issue, but it is worth noting that if much smaller shear viscosity, see for instance [34]. one replaced the Earth with a black hole of the same 12 mass, the effective viscosity of that black hole would be and Y¯ ( s), where s is the spin-weight of the field in not too dissimilar to that of the present Earth. Thus, an question− (s = 2 for gravitational perturbations) and Earth-sized black hole would not be appreciably better at an overbar denotes± a complex conjugate. Equating the absorbing gravitational waves than the Earth is. While of Wronskian evaluated at infinity and at the horizon leads course the Earth-sized black hole would have far smaller to a relation between the wave amplitudes at these two tidal friction than the Earth does, we would still expect boundaries. This relation is the one predicting wave su- the tidal friction it did exhibit to be describable in terms perradiance when the frequency obeys ω˜ = ω mΩH < 0 of gravitational wave superradiance. (here m is the usual spherical harmonic integer− mode associated with the azimuthal coordinate ϕ). For our own discussion of gravitational wave scattering VI. SUPERRADIANCE IN THE PRESENCE OF we will follow a slightly different path, namely, one that A MIRROR makes more contact with the scalar wave analysis of [9] and that can be phrased in terms of reflection and trans- The comparison between black holes and ultracompact mission amplitudes. For this purpose, the wave equation relativistic stars can also take a different form, one in (72) is not a suitable equation because its solutions are which the ‘star’ is modelled as a black hole enveloped not pure plane waves (for example, at infinity we have solutions of the form Y (r ) r±se∓iωr∗ ). This by a ‘mirror’ surface. Clearly this ‘mirror-Kerr’ body is → ∞ ∼ a purely theoretical construct which, nevertheless, pos- unwelcome property is a direct consequence of the long- range character, 1/r, of the V potential. sesses some of the properties of more realistic relativistic ∼ stellar models. A more suitable equation can be build with the help From a mathematical point of view the role of the mir- of the Sasaki-Nakamura formalism (see Ref. [15] for a ror is to mimic the total reflection imposed on the radial review), one which by design features a short-ranged po- motion of waves at the stellar center. Moreover, the effec- tential. Written in a Schrödinger form and for a fixed spin s = 2 this equation is (see [39] for details): tive potential governing wave propagation is qualitatively − similar to that of compact stars (e.g. the system is en- d2X dowed with spacetime w-modes [37]). As far as superra- 2 + VSNX =0 (73) diance is concerned it is known that the addition of the dr∗ mirror eliminates it altogether. This has been recently where X is the new wavefunction. The Sasaki-Nakamura pointed out, for scalar waves, by Richartz et al. [9]. potential VSN has the asymptotic behaviour Our discussion in this Section considers the interaction V (r )= ω2 + (1/r2) (74) of gravitational waves with a mirror-Kerr object. To this SN → ∞ O end we first generalise the analysis of [9] and reach the 2 (r+−r−)/2M VSN(r r+)=˜ω + e (75) same conclusion as they did, namely, the absence of su- → O perradiance. However, we also take few more steps and and indeed it decays faster at radial infinity than the make contact with the so-called ergoregion instability. Teukolsky potential V . This phenomenon is characteristic of ultracompact stars Eqn. (73) admits plane wave solutions at infinity and and is also a property of the mirror-Kerr system [12]. In at the hole horizon. In particular the solution X+, de- the end, we provide a unified description of the transition scribing incoming waves (amplitude Ain) which are par- from superradiance to the ergoregion instability. tially reflected (amplitude Aout) and partially transmitted down the hole (amplitude Atr), has the asymptotic form A. Superradiant scattering of gravitational waves X (r )= A e−iωr⋆ + A eiωr⋆ (76) + → ∞ in out X (r r )= A e−iωr˜ ⋆ (77) Studying the scattering of waves (gravitational or not) + → + tr by a Kerr black hole is greatly facilitated by the well- We can then define reflection and transmission coeffi- known Teukolsky formalism [14],[38]. The required wave cients in a standard way, equation is not the original Teukolsky equation but its A A Schrödinger-type variant. In the notation of [38] (their = out , = tr (78) eqn. (5.2)) this equation is R Ain T Ain

2 The A amplitudes are related to the amplitudes of the d Y solutions of the Teukolsky equation (72) in a simple 2 + V Y =0 (72) dr∗ way [39] (these are the B amplitudes in Ref. [38]), thus permitting the connection of and using the Wron- where Y is the radial wavefunction, r∗ is the usual tor- R T skian equality W (r ) = W (r = r+) of that latter toise coordinate and V plays the role of the scattering equation. The result→ of ∞ this exercise is potential (its specific form will not be needed here). 2 As detailed in [38] the constant Wroskian property of c0 ω˜ 2 = | | 1 2 (79) (72) can be applied to the two independent solutions Y (s) |R| 2 − ω D|T | |C| 13 where c0, (the so-called Starobinski constant) are com- B. Wave and Particle Superradiance plex functionsC of the wave-frequency ω and the black hole parameters M,a. Although their detailed forms are not Having argued that black hole tidal friction and su- needed, we note that they can be found in Ref. [15]. The perradiant scattering are just two ways of looking at the parameter is real and positive and is given by: D same unique wave-amplification mechanism, it is worth- while to ask whether another mechanism which extracts 16(2Mr )5(˜ω2 +4ǫ2)(˜ω2 + 16ǫ2) c (ω, ω˜)= + | 0| (80) energy and angular momentum from the black holes, the D η d 2 | || | Penrose process, is in some way related. Descriptions of superradiant scattering as simply the Penrose process with for waves (instead of particles) are fairly common in the d = 2Mr 8M 2 4a2m2 12iamM (81) literature and even more so in classroom accounts. How- + − − 2 2 ever there is a difficulty with this comparison, which is +p (8 24iωM 16ω M ) that the Penrose process can take place only with the − − 2 + ( 16M + 16amωM + 12iam + 24iωM )r+ ergoregion of a black hole (or other sufficiently hyper- − compact body) but superradiance does not depend upon The functions η and ǫ can also be found in the Sasaki- the ergoregion for its existence. This has recently been Nakamura formalism section of Ref. [15]. pointed out by Richartz & Saa [8], who observe that the At first glance the result (79) seems counterintuitive: first discussion of superradiance, by Zel’dovich [5, 6], in- in some part of the parameter space c0/ > 1, implying volved a rotating cylinder with no ergoregion. While it is a reflection > 1 and thus suggesting| C| the existence of |R| true that superradiance requires negative energy modes superradiance even when ω˜ > 0, contrary to established in the system with which the scattered wave interacts, results [38]. However, superradiant amplification must this can be achieved without an ergosphere. be formulated in terms of the ratio of the incoming en- The Penrose process relies crucially on the existence of ergy flux to that of the outgoing radiation. Unlike the the ergosphere, wherein geodesics with negative energies case of scalar waves where this ratio is identical to the exist. This allows for particle decay processes occurring amplitude ratio, i.e. = E˙ /E˙ = A 2/ A 2, the R out in | out| | in| inside the ergosphere to transfer a portion of the black gravitational wave fluxes are related to the amplitudes hole’s energy and angular momentum to decay products in a more subtle way. Expressing the gravitational wave that exit the ergosphere. fluxes from [38] in terms of the Sasaki-Nakamura ampli- For slowly rotating black-holes, the radius of the ergo- tudes, sphere, when expanded to first order in a, coincides with 2 the event horizon radius: ˙ 8ω 2 Ein = 2 Ain (82) 2 c0 | | r = r =2M + (a ) (88) |C| | | ergo + O 8ω2 E˙ = A 2 (83) There is therefore no ergoregion at (a), making it im- out c 3 | out| O | 0| possible for the Penrose process to occur at this order. 8ωω˜ On the other hand, expanding the energy flux reflec- E˙ = D A 2 (84) tr 2 c | tr| tion coefficient (cf. (87)) to first order in a, we get: |C| | 0| 2 mq0 2 2 With these we can define a new pair of reflection and flux =1 q0 + q a + (a ) (89) transmission coefficients: R − |T | 4Mω − |T | O ˙ 2 where the constant in (87) is expanded as = q0+qa+ Eout 2 2 D D flux= = |C| 2 (85) (a ) (q0 and q are constants that are independent of a R E˙ c0 |R| O in | | but, in general, are complicated functions of the orbital ˙ Etr ω˜ 2 radius r and harmonic integer m). flux = = (86) For sufficiently high values of m and a range of radii T E˙ in ω D|T | r, it can be shown that superradiance ( flux > 1) could Then using (79) persist at (a). Contrast this with theR Penrose process which is completelyO dependent on the ergoregion, a sec- ω˜ 2 flux =1 =1 flux (87) ond order effect in a. Therefore, even though both mech- R − ω D|T | − T anisms are forms of scattering that steal the black hole’s This expression re-establishes the standard condition for energy and angular momentum, they are fundamentally superradiance: ω˜ < 0 flux < 1 and flux > 1. distinct processes. At this point, if we⇔ replace T the horizonR with a perfect mirror we should have Atr = 0 = flux = 0. In → T T C. From superradiance to the ergoregion instability other words, the presence of the mirror fixes flux = 1 and eliminates superradiance altogether regardlessR of ω˜. This is in agreement with the findings of [9] for the black The disappearance of superradiance is not the only hole-scalar waves system. change caused by the replacement of the event horizon 14 with a mirror. A perhaps more dramatic consequence is sibility of an imperfect mirror, meaning that the mirror the appearance of the ergoregion instability. This is a dy- causes only partial reflection while absorbing the rest of namical type of instability, already known in the context the impinging wave. Thus, the imperfect mirror allows of compact relativistic stars [40, 41]. If sufficiently com- a smooth transition from an event horizon (total absorp- pact and rapidly rotating, such stars have an ergoregion tion) to a perfect mirror (total reflection) type of bound- and a potential ‘cavity’ between the stellar center and the ary condition. Technically this would mean that an in- peak of the potential in the vacuum exterior. The insta- coming wave of amplitude Dinc just before interacting bility sets in through the system’s trapped w-modes [42]; with the mirror is reflected back with an amplitude these are the spacetime modes associated with the cavity D = λD , 0 λ 1 (90) of the potential. ref inc ≤ ≤ Our system consisting of waves propagating in a Then λ =1 would correspond to Vilenkin’s perfect mir- mirror-Kerr field is naturally prone to the ergoregion in- ror system while λ =0 can be thought as the event hori- stability. In fact, this is the same system considered by zon limit. Vilenkin [12] (albeit for scalar waves) in one of the first The ergoregion instability calculation is based on the discussions of the ergoregion instability, and much before use of two linearly independent solutions of (73). The it was associated with unstable w-modes. The required first solution is the one given earlier, X+, with the in- cavity in the potential (V or V ) is formed between the SN coming amplitude set to unity, Ain = 1. The second mirror and the peak of the potential (the peak is located solution, denoted as X−, has an asymptotic behaviour near the Kerr spacetime’s unstable circular photon or- iωr⋆ 4 X−(r )= A e bit). This is also where the hole’s ergoregion is located . → ∞ up −iωr˜ ⋆ iωr˜ ⋆ Impinging radiation with frequency ω can become un- X−(r r+)= Adowne + e (91) stable through the ergoregion mechanism provided ω < → This solution represents a plane wave of unit amplitude mΩH, i.e. the same condition that would lead to superradiance if the mirror was not there. In this sense moving radially outwards from the vicinity of the hori- superradiance and the ergoregion instability seem to be zon/mirror and being partially transmitted to infinity two sides of the same coin. However, and unlike the puny (amplitude Aup) and partially reflected back (amplitude amplification caused by superradiance, the ergoregion in- Adown). stability leads to an exponentially growing wave reflected The first relation between wave amplitudes corre- back to infinity. In the language of normal modes, the sponding to the X+ solutions (cf. eqn. (76)) is in fact instability is associated with a complex-valued eigenfre- nothing but eqn. (79) with unit incident-wave amplitude quency. This is also the reason why this effect was not Ain =1 : present in the previous real-frequency analysis. c 2 ω˜ A 2 = | 0| 1 A 2 (92) How the exponential growth comes about can be very | out| 2 − ω D| tr| intuitively understood by the scattering experiment de- |C| scribed in the Vilenkin paper [12]. This consists of an The second set of relations corresponding to the X− solu- initially incoming pulse, partially reflected and partially tions may be trivially found by comparing eqns. (76) and transmitted through the black hole potential. The trans- (91). A variable transformation of the form r⋆ r⋆ → − mitted part gets reflected at the mirror and becomes ra- leaves the Sasaki-Nakamura equation (73) functionally dially outgoing. Then this secondary wavepacket gets unchanged. Therefore, the X+ and X− solutions are partially reflected and partially transmitted. If the con- symmetric under this transformation, provided we swap dition ω˜ < 0 is satisfied the transmitted part escapes to the variables ω ω˜. Interchanging these variables in ↔ radial infinity after having been amplified in the ergore- eqn. (92) (and appropriately labeling the amplitudes), gion. The same process is repeated everytime a sub-pulse the amplitude relation corresponding to the X− solutions is reflected towards the black hole. As a result of this an is found to be: observer far away registers an exponentially growing wave 2 2 c˜0 ω 2 signal. Adown = | | 1 ˜ Aup (93) | | ˜ 2 − ω˜ D| | This procedure is essentially the same for scalar and |C| gravitational waves and therefore it would be sufficient where c˜0 = c0(˜ω), ˜ = (˜ω) and ˜ = (˜ω,ω). to sketch the calculation here and refer the reader to The total gravitationalC C wave energyD DE registered at in- Ref. [12] for full details. However, apart from consider- finity with respect to the energy E0 of the initial pulse ing gravitational perturbations, we expand the original can be calculated with the help of these amplitude re- calculation in one more way: we accommodate the pos- lations, the reflection condition (90) and the energy formula (82). Essentially, this is the same quantity as the reflection coefficient, i.e. = E/E . The result is: Rflux 0 4 ∞ The set up described here bears a strong similarity with the so- ω˜ λ 2n called ‘black hole bomb’ mechanism devised by Press & Teukol- =1 A 2 + λ2 ˜ A A 2 Rflux − ω D| tr| DD| tr up| λ sky [43] where a cavity is formed between the event horizon and n=0 0 a mirror placed outside the potential peak. X (94) 15 with VII. CONCLUDING DISCUSSION

c˜0 1 λ0 | | (95) When comparing the superradiant scattering which oc- ≡ C˜ Adown | | | | curs in a binary black hole system with the tidal friction in a material system like our Earth and its Moon, one can This result generalises Vilenkin’s formula (his eqn. (5)) to initially be struck by the indirectness of the analogy pro- the case of gravitational waves and an imperfect mirror. posed by Hartle [3, 4]. This analogy, that superradiant Consider first the case ω˜ > 0. From (93) this implies scattering of gravitational waves in the black hole binary 2 2 2 2 C˜ λ Adown / c˜0 < 1 for the range of allowed values be treated just as if it was a case of tidal friction between |of |λ.| Hence| the| infinite| | summation in (82) converges, two orbiting bodies, was subsequently elaborated further and we have: in the membrane paradigm [1]. Tidal friction, as its name implies, relies on some form 2 ω˜ 2 2 AtrAup of dissipation to facilitate the angular momentum trans- flux =1 | | Atr + λ ˜ | | (96) R − ω D| | DD 1 λ2/λ2 fer between the two bodies in the system. Since it is im- − 0 possible, in the unequal mass case, to satisfy both energy It is easy to verify that for λ = 1 (the perfect mirror and angular momentum conservation, some dissipation limit) this expression reduces to flux =1, in other words must occur. In the case of material bodies, this dissipa- R the entire energy of the initial pulse is reflected back to tion is due to tidal heating. In the black hole case the infinity. When λ< 1 part of the initial energy is absorbed dissipation is due to the event horizon’s ability to per- by the hole, naturally resulting in flux < 1 (note that fectly absorb gravitational radiation. Clearly material R ω˜ > 0 implies λ0 > 1, see eqn. (93)). bodies are not good absorbers of gravitational energy. The phenomenology is much more interesting when They are virtually transparent to gravitational waves. ω˜ < 0. This entails λ0 < 1 and then we need to con- For this reason, when replacing the black hole with a sider separately the subcases λ0 < λ 1 and λ < λ0. To material body, a typical model is to think of it as a black ≤ begin with, we first note that λ = 0 (the event horizon hole with a mirror placed just in front of the horizon. limit) leads to The notion is that the material body is virtually transparent to gravitational waves, which pass right through ω˜ and (speaking radially), come right back out again, as if =1+ | | A 2 > 1 (97) Rflux ω D| tr| reflected. It has been shown by Richartz et al. [9], and confirmed earlier in this paper, that such a mirror de- This is the standard superradiance result. stroys superradiance. Yet tidal friction is certainly well For λ < λ0 we again obtain flux > 1 but this time the known in material bodies, so why do we persist in think- R superradiance is enhanced with respect to the previous ing that superradiance and tidal friction are equivalent, result: if destroying one leaves the other intact?

2 To answer this question it is instructive to consider 2 AtrAup replacing like with like. The effective viscosity of the flux = flux λ=0 + λ ˜ | | (98) R R | DD 1 λ2/λ2 black hole event horizon (for black holes of stellar mass − 0 or greater) is a huge quantity, many orders of magnitude Finally, the ergoregion instability emerges when ω˜ < 0 higher than even the viscosity of bodies like the Earth and for a mirror reflectivity λ0 < λ 1. Then the sum (which is high by the standards of most materials). What ≤ in (94) is divergent, making flux + (note that this would happen if we replaced the black hole with a sim- R → ∞ divergence is a prediction of linear perturbation theory ilarly sized (in mass and radius) physical body with a – once the wave amplitude has grown sufficiently, non- viscosity close to that of the event horizon? It seems linear effects would become important and should be ac- clear that the mirror model would not be an appropriate counted for). Hence, the critical value λ = λ0(ω) marks one in that case. Most materials are nearly transparent to the transition from a monotonically increasing superra- gravitational waves, but hypothetical materials with such diance to the ergoregion instability. large viscosity would not be. The absorption of gravita- The fact that an imperfect mirror restores superradi- tional waves by a viscous fluid has been studied in the ance, enhances it and eventually promotes it to the er- weak field case by Esposito [10] and in the strong field goregion instability can now be viewed from the perspec- case (of a fluid ball close to a Schwarzschild black hole) by tive of a viscous material body. The imperfect mirror, Papadopoulos & Esposito [11]. They show clearly that with λ < 1 is a dissipative surface because we associate an arbitrarily viscous fluid will be arbitrarily efficient at it with a material body with the property that it can ab- absorbing incident gravitational radiation. Since this is sorb a portion of the gravitational radiation which passes the case, it strongly implies that a hyper-compact highly through it. Where a material body has a very high vis- viscous fluid star with an orbiting satellite would exhibit cosity, and therefore can absorb a significant proportion tidal friction resulting in a transfer of the star’s angular of gravitational wave energy, then the mirror becomes momentum to the satellite, since the dissipation would significantly imperfect and superradiance is restored. consist of the near-complete absorption of the incident 16 gravitational energy from the orbiting satellite. report more progress on this subject in the near future. The parallel between tidal friction and superradiance has rarely been mentioned in print in recent years, though an important exception is provided by the interesting pa- ACKNOWLEDGEMENTS per of Cardoso & Pani [44]. One question which naturally arises, which is also discussed by them, is whether the KG is supported by the Ramón y Cajal Programme of case of dissipation in material bodies, mediated through the Spanish Ministerio de Ciencia e Innovación and by viscosity, is characteristically different from tidal friction the German Science Foundation (DFG) via SFB/TR7. in the black hole case, in which the dissipation is pro- The authors would like to thank Stephen O’Sullivan and vided by the absorptive capacity of the event horizon. Eric Poisson for fruitful discussions. It might be tempting to interpret the viscosity mediated kind of tidal friction in material bodies as an electromagnetic mediated kind of interaction, on the grounds that Appendix A: The linearised Euler Equation viscosity is an electromagnetic process. It seems to us, however, that the electromagnetic tidal friction is of the In this Appendix we give an outline of the derivation type discussed in Zel’dovich [5, 6], and Richartz & Saa [8], of the linearised Euler equation (26) describing the per- in which the electromagnetic field actually mediates be- turbed state of an anisotropic fluid in Newtonian gravity. tween the two bodies, and in which one can think of the This is done by first deriving the relevant hydrodynami- effect as electromagnetic superradiance. Accordingly one cal equations in General Relativity and then taking their should think of the black hole and the hypercompact ul- Newtonian limit. traviscous star as fully equivalent systems. The stress-energy tensor for a fluid in an arbitrary One other possible source of absorption, without imag- spacetime with metric gµν with distinct radial and tan- ining hyperviscous materials, is the possibility of a hyper- gential pressures, pr and pt, is given by (e.g. [45]): compact object trapping gravitational waves inside its T = (ρ+p )u u +p g +p g +(p p )k k (A1) own potential barrier, producing what are known as w- µν t µ ν t µν t µν r − t µ ν modes, gravitational waves trapped close to a hypercom- where kµ is a unit four-vector orthogonal to the fluid’s pact object (more compact than a neutron star, for in- four-velocity uµ. The equations of motion are determined µ stance). An orbiting particle around such a body would by µTν = 0; with the help of the projection tensor ν ∇ ν ν µ send down gravitational waves which would temporar- µ= δρ + u uρ (δν is the usual Kronecker delta) these ily disappear behind the potential barrier, creating w- lead⊥ to the relativistic Euler equation: modes, and transferring angular momentum to the orbit- (ρ + p )a = ν p ν (σkρk ) (A2) ing body. If the w-modes are long lived, then the particle t µ −⊥µ ∇ν t−⊥µ ∇ρ ν could spiral in rapidly and plunge down to merger before where σ = pr pt is the pressure anisotropy and aµ = it ever gets back the angular momentum. ν − u ν uµ is the four-acceleration. In our investigation of the superradiance-tidal friction Considering∇ the case of a static and spherically sym- analogy we have taken a closer look to the tidal inter- metric fluid star, the vectors defined earlier assume the action between a fluid star made of anisotropic and vis- following forms: cous matter and an orbiting small body. The choice of µ t anisotropic matter was dictated by the need of modelling u = (u , 0, 0, 0) (A3) a ‘star’ with such a high degree of compactness that could kµ = (0, kr, 0, 0) (A4) be “mistaken” for a (supermassive) black hole by observa- aµ = (at,ar, 0, 0) (A5) tions. Although our analysis was Newtonian, and therefore not strictly applicable to ultracompact relativistic The metric for this background system takes the familiar form g = diag( eν(r),eµ(r), r2, r2 sin2 θ). The relativis- objects, we have found an interesting set of results. Sys- µν − tems with weakly anisotropic matter show a response to tic hydrostatic equilibrium is then governed by tidal fields similar to that of isotropic stars. On the 2σ ∂ p = (ρ + p )Γt (A6) other hand, strong anisotropy causes a rather counter- r r − r rt − r intuitive response, with respect to both rotational and t tidal forces: rotation makes the shape of the body pro- where Γrt = (1/2) dν/dr is a Christoffel symbol. Perturbing the system away from this equilibrium in- late rather than oblate, the tidal Love number becomes µ t i negative and the tidal bulge is rotated by roughly 90 de- duces a fluid flow δu = (δu , δv ) and a change in the grees with respect to its orientation in isotropic of weakly density/pressure δρ,δpr,δpt. In addition, we adopt the anisotropic stars. At the same time, and regardless of the Cowling approximation where the perturbation in the metric is ignored. At the same time, kµ retains its unit degree of anisotropy, the resulting tidal torque is given µ by a single universal formula. These results are clearly magnitude and orthogonality to u , which implies the interesting and deserve further and more detailed work following: using the full machinery of relativistic gravity and/or the k δkr = δkθ = δkφ =0, δk = r δur (A7) use of the full Euler equations. We hope to be able to t − ut 17

With the help of these relations the perturbed Euler To obtain the Newtonian limit of the linearised Euler equation becomes, equation, we assume ρ p ,p , u 1, g (1 + 2Φ) ≫ r t t ≈ tt ≈− µ ν ν where Φ is the Newtonian potential. We then have (i = aµ (δρ + δpt) + (ρ + pt)[ u ν δuµ + δu ν uµ uµδu aν ] r, θ, φ ) ν ν ∇ ρ ∇ − { } = µ ν δpt µ ρ[ δ(σk kν ] (A8) −⊥ ∇ −⊥ ∇ 2 ρ∂ δv = ∂ δv δi δρ ∂ Φ+ ∂ δσ + δσ (A11) Decomposing this equation into its tangential and ra- t i − i i − r r r r dial components, and simplifying each separately, we obtain: In this Newtonian expression we can easily add ‘by hand’ the force due to the perturbed gravitational potential and t 2 (ρ + p )u ∂ δv = δp , i = θ, φ (A9) the viscous force ν δui: t t i −∇i t { } ∇ and ρ∂ δv = ∂ δv ρ∂ δΦ t i − i i − i tt i 2 2 t g r δr δρ ∂rΦ+ ∂rδσ + δσ + ν δvi (A12) (ρ + pt)u ∂tδvr σgrr t ∂tδv = − r ∇ − u 2 = ∂ δp Γt (δρ + δp ) δσ (A10) Writing this expression in Cartesian coordinates leads to − r r − rt r − r eqn. (26) which is the desired result.

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