PHYSICAL REVIEW D 98, 064036 (2018)

Black hole binaries: Ergoregions, surfaces, wave scattering, and quasinormal modes

Thiago Assumpção,1 Vitor Cardoso,2,3,4 Akihiro Ishibashi,5 Maurício Richartz,1 and Miguel Zilhão2 1Centro de Matemática, Computação e Cognição, Universidade Federal do ABC (UFABC), 09210-170 Santo Andr´e, São Paulo, Brazil 2CENTRA, Departamento de Física, Instituto Superior T´ecnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal 3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 4Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland 5Department of Physics and Research Institute for Science and Technology, Kindai University, Higashi-Osaka, 577-8502, Japan

(Received 25 June 2018; published 18 September 2018)

Closed photon orbits around isolated black holes are related to important aspects of physics, such as strong lensing, absorption cross section of null particles and the way that black holes relax through quasinormal ringing. When two black holes are present—such as during the inspiral and merger events of interest for gravitational-wave detectors—the concept of closed photon orbits still exist, but its properties are basically unknown. With these applications in mind, we study here the closed photon orbits of two different static black hole binaries. The first one is the Majumdar-Papapetrou geometry describing two extremal, charged black holes in equilibrium, while the second one is the double sink solution of fluid dynamics, which describes (in a curved-spacetime language) two “dumb” holes. For the latter solution, we also characterize its dynamical response to external perturbations and study how it relates to the photon orbits. In addition, we compute the ergoregion of such spacetime and show that it does not coincide with the .

DOI: 10.1103/PhysRevD.98.064036

I. INTRODUCTION 14,15]. In the eikonal limit, the oscillation frequency of such modes is a multiple of the orbital frequency while their High-frequency electromagnetic or gravitational waves — characteristic damping time is related to the instability time propagating around extremely compact objects such as – black holes (BHs)—can travel in closed orbits. These can scale of such orbits [2 5]. Second, the absorption cross be either stable or unstable, according to whether or not section of a highly energetic scalar field scattered by the small deviations from the orbit grow away from it as time gravitational field of a spacetime endowed with a photon evolves. In the case of a Schwarzschild BH, the union of all sphere is simply described by the orbital frequency and such orbits forms a sphere, the so-called Schwarzschild instability time scale of the associated null closed orbits [6]. photon sphere. Any inward-directed lightray emitted inside Third, the extreme bending of light near the photon sphere the photon sphere will eventually end in the singularity, also allows the formation of multiple (possibly infinite) — while any outward-directed lightray emitted outside even- images on the optical axis [7,8] in addition to the primary tually reaches future null infinity. Hence the photon sphere image typically formed by any weak . In is not a horizon (for which no lightray would ever be able to particular, strong gravitational lenses produce an infinite exit) but has a similar fundamental role. number of Einstein rings, and the astrophysical observa- Several BH phenomena are directly related to the tion of a sequence of such rings would be another presence of a photon sphere [1–13]. First, the characteristic successful test of general relativity in the strong field modes of oscillation of a spherically symmetric BH can be regime [8]. Last but not least, the presence of stable related to the parameters of the null closed orbits [1–4, photon spheres implies the existence of a trapping region in spacetime where matter can accumulate, potentially leading to nonlinear instabilities [9–11,16]. The concept of a photon sphere, useful for Published by the American Physical Society under the terms of Schwarzschild BHs, can be generalized to the concept the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to of a photon surface in an arbitrary spacetime [17]. The the author(s) and the published article’s title, journal citation, definition of a photon surface, in simple terms, requires that and DOI. any null geodesic tangent to the surface will remain

2470-0010=2018=98(6)=064036(13) 064036-1 Published by the American Physical Society THIAGO ASSUMPÇÃO et al. PHYS. REV. D 98, 064036 (2018) everywhere tangent to it. More specifically, in this work our We note that partial results concerning the structure of main interest is in closed null hypersurfaces, i.e., non- lightray or scalar-field propagation in the background of spacelike, chronal hypersurfaces S that admit a null BH binaries were reported earlier. The evolution of scalar geodesic (i) which is tangent to S and (ii) whose projection fields in the background of an inspiraling BH binary was onto a given spatial cross section of S forms a “closed orbit” investigated in Ref. [26]. The shadow of a static BH binary 1 on the cross section. The notion of a photon surface is (unphysical, since it contains a strut holding the two BHs independent of any symmetry that the spacetime may apart) was also recently reported [27,28]. However, unlike possess and is applicable to the situation of two very our analysis, these references do not address the question of compact objects that inspiral into each other and eventually the structure and time scales associated to the different merge, forming a single compact object. These events, possible closed photon surfaces. Therefore, through our notably the merger phase, being important sources of work, we hope to pave the way for future investigations of gravitational-waves, are of interest for detectors like closed null orbits in more realistic situations of interest for LIGO and LISA. Photon surfaces and closed photon orbits LIGO and LISA. around such binaries, although never studied before, could provide important astrophysical information about BH-BH II. THE MAJUMDAR-PAPAPETROU SOLUTION (and BH-) collisions, specially now that the gravitational-wave astronomy era has begun [18]. An exact solution in general relativity which describes The main objective of this work is to provide the first two or more static BHs is known as the Majumdar- step towards understanding closed photon orbits around Papapetrou (MP) solution [20–22]. In a cylindrical coor- binary systems of astrophysical compact objects. Due to the dinate system the two-BH version of the MP solution is complexity of simulating and studying collisions of BHs written as and other compact objects, we focus on simpler geometries of BH binaries that are static and, therefore, never merge. dt2 ds2 ¼ − þ U2ðdρ2 þ ρ2dϕ2 þ dz2Þ; ð1Þ We expect that our results are useful for low-frequency U2 binaries whose velocities are much smaller than the speed of light. This work is also a first step towards understanding with how other peculiarities—such as ergoregions—extend to M M dynamical spacetimes. There is a clear motivation for that, Uðρ;zÞ¼1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð2Þ since it could lead to Penrose-like phenomena or interesting ρ2 þðz − aÞ2 ρ2 þðz þ aÞ2 superradiant effects [19]. The first toy model we consider is that of two electrically This solution represents two maximally charged BHs in charged BHs in equilibrium [20–22] (the Majumdar- equilibrium, each with mass M and charge Q ¼ M. Here, Papapetrou solution of Einstein’s equations), for which and throughout this work, we use geometric units. In these we study null geodesics both on the symmetry plane coordinates, their horizons are shrunk to two points at between the BHs and on the plane containing both BHs. z ¼a (hence, the parameter a measures the distance Our second model is based on analogue , the fact between them). The spacetime ADM mass is 2M. that sound propagation in moving fluids is formally The Lagrangian for geodesic motion in the Majumdar- equivalent to a scalar field evolving in curved spacetime Papapetrou spacetime is given by [23,24]. Points where the fluid velocity exceeds the local _2 sound speed behave as an ergoregion; if this happens on a t 2 2 2 2 2 2L ¼ − þ U ðρ_ þ ρ ϕ_ þ z_ Þ¼−δ; ð3Þ normal to a certain surface, such surface is an analogue U2 horizon. By considering a double sink hydrodynamical fluid flow (an exact solution of the fluid-dynamic equa- where dots refer to derivatives with respect to an affine δ tions) we are able to study the analogue of a binary BH parameter. The constant on the right-hand side is zero δ 0 δ 1 system. We should note that extending the notion of ( ¼ ) for null geodesics and one ( ¼ ) for timelike _ 2 ergoregions to dynamical curved spacetimes is nontrivial geodesics. The two conserved quantities are E ¼ t=U and _ 2 2 in general, but quite straightforward in acoustic setups. Our L ¼ ϕU ρ , which we identify as energy and angular results may thus help in understanding whether phenomena momentum per unit rest mass, respectively. such as superradiance exist in BH-binary spacetimes [19].2 A. Geodesics in the meridian ϕ = 0 plane 1 Note that the “closed orbit” here defined is not the same as a We can fix the azimuth angle by setting ϕ ¼ 0 (this is closed geodesic. an arbitrary choice as any fixed angle yields the same 2The existence of ergoregions in static spacetimes describing two Kerr BHs held apart by a strut was established in Ref. [25]. behavior) and study geodesic motion confined to the A generalization to dynamical regular spacetimes has not been meridian plane. For a generic separation between the done. BHs, the geodesic equations are not separable, as noted

064036-2 BLACK HOLE BINARIES: ERGOREGIONS, PHOTON … PHYS. REV. D 98, 064036 (2018)

4 20

2

10

/M 0 ρ

–2

/M 0 ρ

–4 –10 –5 0 5 10 z/M –10 FIG. 1. There are different closed orbits for light in a static binary. We find, generically, three different closed trajectories: one global outer geodesic that encircles both holes, an “8-shaped” –20 trajectory and two “smaller” ones encircling only each of the –20 –10 0 10 20 BHs. These plots show the different “light rings” for a MP z/M solution with a ¼ 10M. The “8-shaped” orbit (black curve) has a period T ¼ 98.79M in coordinate time. The global outer orbit FIG. 2. Same as Fig. 1, but for massive free particles instead (blue curve) has a period T ¼ 95.47M. The period of the orbits of light (i.e., timelike instead of null geodesics). The “8-shaped” around a single hole (red curves) is T ¼ 26.39M. orbit (black curve) has energy E ¼ 0.9159 and period T ¼ 291.64M in coordinate time. The global outer orbit (blue curve), has energy E ¼ 0.9644 and period T ¼ 474.01M. The orbits in Ref. [29]. However, when the two BHs coalesce, around a single hole (red curves) have energy E ¼ 0.9159 and 0 a ¼ , and it is easy to solve the corresponding geodesic period T ¼ 38.84M. equationspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi[2]. We find that there is a null geodesic at r ¼ z2 þ ρ2 ¼ 2M. highly relativistic particles, timelike geodesics approach With the aid of standard numerical integration methods, null geodesics. It is impossible to find closed timelike orbits three types of closed null and closed timelike orbits were for arbitrary energies. identified, as shown in Figs. 1 and 2: (i) An orbit which encloses both horizons, shown in blue in both figures. For small separations, i.e., B. Geodesics in the symmetry z = 0 plane a ≪ M, the coordinate radius of such an orbit is 2M, Halfway between the two BHs, there is a symmetry as expected from the analysis of the a ¼ 0 case. plane (z ¼ 0) in which the geodesic equations are sepa- The corresponding coordinate period is ∼16πM.At rable. By setting a ¼ 1, which simply amounts to rescaling large separations, we find that the period of null units as ρ → ρ=a and M → M=a, Eq. (2) becomes outer orbits is T ∼ 20πM þ 4a, for values of M in 1 ≲ ≲ 10 2M the range M . U ¼ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð4Þ (ii) Orbits which enclose only one horizon, shown in red ρ2 þ 1 in both figures. At large separations, such geodesics only “see” the gravitational field of one BH. Indeed, Substituting E and L and setting δ ¼ 0 in the normalization we find that then the orbit is nearly circular with condition, the equation for null geodesics becomes radius ∼M, and a coordinate period ∼8πM,as ρ_2 ρ 2 expected from the above. þ Veffð Þ¼E ; ð5Þ (iii) An “8-shaped” orbit which encloses each of the horizons once, shown in black in both figures. L2 V ðρÞ¼ : ð6Þ For the two-BH MP geometry these three types of orbits eff ρ2U4 seem to exist for any separation a. These results are ’ 2 27 8 in complete agreement with both Chandrasekhar s and Provided that M > = , the effective potential Veff has Shipley and Dolan’s recent results [29,30]. A full descrip- two critical points, which correspond to a stable (inner) and tion of the equations and normalization conditions is given an unstable (outer) closed orbit.3 The analytic expressions in the Appendix. We found, numerically, that all these null for these two radii are [31] geodesics are unstable: the numerical integration requires fine tuning to follow them. On the other hand, timelike 3These statements refer to stability on the z ¼ 0 plane only. In geodesics are more stable and complete a larger number reality, both orbits are expected to be globally unstable: any of full orbits before collapsing to the singularity (some perturbation on the z direction would give rise to motion towards “8-shaped” trajectories are actually long-term stable). For one of the holes.

064036-3 THIAGO ASSUMPÇÃO et al. PHYS. REV. D 98, 064036 (2018)

6 6 6 For timelike geodesics, the number of roots of the 4 4 4 equation dV =dρ ¼ 0 (and consequently the number of 2 2 2 eff

0 0 0 circular orbits) depends not only on the mass of the BHs but –2 –2 –2 also on the angular momentum of the particle due to an –4 –4 –4 extra term in the effective potential [32]. The Lyapunov –6 –6 –6 exponent λ associated with the unstable circular orbits can –6 –4 –2 0 2 4 6 –6 –4 –2 0 2 4 6 –6 –4 –2 0 2 4 6 be computed in a straightforward way [2,5]. A detailed FIG. 3. Null geodesics on the symmetry plane of a two-BH MP calculation for null geodesics is given in the Appendix. In spacetime, characterized by a ¼ M=5. The radii of the circular our context, the Lyapunov exponent λ describes the time orbits do not depend on L and are expressed in units of a. scale in which a deviation δρ from a circular orbit grows 0 0366882 From left to right, the energy of the orbits are E ¼ . , exponentially, i.e., λ is characterized by δρ ∼ eλt. As seen in 0.0366939, 0.0367020. The red curve shows the portion of the Fig. 4, the Lyapunov exponent attains its maximum value motion between two consecutive apastrons, for which the Δϕ 2π 0.03158 at M ¼ 2.197, and approaches 0 for very large M. accumulated angle is given by ¼ q. When q is a rational 6 −8 number, the motion is periodic, and the geometric properties of For instance, at M ¼ 10 , the exponent is 8.38 × 10 . the plot can be extracted from q. From left to right, q is given by More general closed timelike orbits—other than circular— (1 þ 7=10), (1 þ 33=47), and (1 þ 45=64). Notice that q in- also exist, as shown in Fig. 5. creases monotonically with E. Geometrically this means that the Closed orbits in the symmetry plane exhibit a “zoom- motion is divided into more “leaves”, which creates more vertices whirl” behavior, as seen in Figs. 3 and 5. A brief remark ρ 1 167 in the figures. The stable circular orbit has a radius in ¼ . a, concerning these orbits is in order. Applying the formalism ρ 9 740 and the unstable orbit is at out ¼ . a. presented in Ref. [33], we associate geometric properties of 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi symmetric orbits in the plane z ¼ with the accumulated    Δϕ 4 2 1 2 angle between two consecutive apastrons given by ρ M 1 2 −1γ − 1 out ¼ þ cos cos ; Z 9 3 ρ a dϕ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Δϕ ¼ 2 dρ; ð7Þ 2 ρ ρ 4 π 1 2 p d ρ M 1 − 2 − −1γ − 1 in ¼ 9 sin 6 3 cos ; where ρp and ρa are the radial coordinates of the periastron pffiffiffi and apastron respectively. We may look for orbits for γ ≡ 1–27 4 2 ρ2 1 2 Δϕ 2π where =ð M Þ. Notice that in ¼ þ =M þ which ¼ q, where q is a rational number. We can O 1 2 ρ2 −5 4 2 O 1 2 ð =M Þ, and out ¼ þ M þ ð =M Þ. When always write any noninteger rational number uniquely as ≫ ρ → 2 ∈ Z M a we have out M, so that the single hole solution q ¼ w þ v=z, where w ¼bqc , and v

0.15 10 0.030 0.10 5 5 0.05 0.025 0.00 0 0

–0.05 0.020 –5

/a –0.10 –5 λ –10 0.015 –0.15 –0.15 –0.10 –0.05 0.00 0.05 0.10 0.15 –5 0 5 –10 –5 0 5 10 0.010 FIG. 5. Timelike geodesics on the symmetry plane of a two- 0.005 BH MP spacetime, characterized by a ¼ M=5. From left to right, the energy and angular momentum of the orbits are: 0.000 0 5 10 15 20 (E ¼ 0.0920047, L ¼ 0.1a), (E ¼ 0.456023, L ¼ a), and M/a (E ¼ 0.585419, L ¼ 10a). From left to right, q is given by 56=111, 29=53, and 27=28. The radii of both circular orbits FIG. 4. Lyapunov exponent as a function of the BHs masses for (stable and unstable) depend on M and L=a for timelike geo- 0 null geodesics in the z ¼ plane of the MP spacetime. The black desics. From left to right, the inner circular orbit radii ρin=a are dots are the numerical results, while the red dashed line given by 0.0977,0.3093,0.8892. There are no unstable outer corresponds to the analytic expression. orbits for these combinations of M and L=a.

064036-4 BLACK HOLE BINARIES: ERGOREGIONS, PHOTON … PHYS. REV. D 98, 064036 (2018) consecutive “leaves” throughout the motion, so the number in a rotating vortex representing the draining bathtub v tells how many vertices the orbit will skip before drawing geometry [37]. the next “leaf”. These properties can all be recovered using the effective metric experienced by sound waves (for simplicity we set the background density to unity), III. THE DOUBLE SINK SOLUTION 2 2 2 2 2 2 Our second toy model for a BH binary is inspired in ds ¼ −ðcs − v Þdt − 2dtðvxdx þ vydyÞþdx þ dy ; analogue gravity, a formal equivalence between the propa- 2 2 ≡ −ðα − βiβ Þdt þ 2dtβ dxi þ γ dxidxj: ð10Þ gation of sound waves in fluids and a scalar field in a i i ij (analogue) curved spacetime [24,34,35]. In particular, a 0 Φ v ∇Φ When there is no separation between the sinks (a ¼ ), the sound wave associated with a velocity field ¼ is metric above simplifies to governed by the equation 4 □Φ 0 ds2 ¼ −ðc2 − 4=r2Þdt2 þ dtdr þ dr2 þ r2dϕ2: ð11Þ ¼ : ð8Þ s r The curved spacetime geometry associated with the Transforming t and r into new variables t˜ and r˜, defined ’ ˜ 2 2 2 D Alembertian is fixed by the background fluid flow by r ¼ r=c˜ s and dt ¼ dt=cs þ 2rd˜ r=˜ ðcsðr˜ − 4ÞÞ, brings and contains horizons if and when the fluid flow exceeds, the acoustic metric above to the canonical form [38] at some point, the local sound speed [24,34,35].We   consider a generalization of the 2 þ 1 “draining bathtub” 4 1 c2ds2 − 1 − dt˜2 dr˜2 r˜2dϕ2: 12 geometry [35], which here describes two acoustic “dumb” s ¼ ˜2 þ 1 − 4 þ ð Þ r ˜2 holes. Our background flow is two dimensional, consists r on two sinks of unit strength at x ¼a, and can be written as [36], 1. Soundcurves 2xðr2 − a2Þ When the two holes are sufficiently close to one another, v ¼ −A ; ð9aÞ the metric reduces to (12), and the soundcurves are located x r4 þ 2a2ð−x2 þ y2Þþa4 pffiffiffi at r ¼ 2 2 [2] (here and in all numerical calculations we   1 1 1 assume cs ¼ ). The coordinate time that sound takes to − “ ” 8π vy ¼ Ay 2 − 2 þ 2 2 ; ð9bÞ travel across this single merged sink is T ¼ (corre- y þðx aÞ y þðx þ aÞ sponding to a frequency Ω 1=2), and the associated ¼ pffiffiffi λ 1 2 2 4A2r2 Lyapunov exponent is ¼ =ð Þ. v2 ; For double sinks, we find that there are different sound ¼ 4 2 2 − 2 2 4 ð9cÞ r þ a ð x þ y Þþa curves, shown in Fig. 6. Global outer orbits (shown in 2 2 2 0 blue) surrounding both BHs exist for any value of sepa- with r ¼ x þ y . The constant A> is arbitrary and ration a. Inner orbits around each sink (shown in red) and fixes the fluid speed at some radius. For simplicity, we “8-shaped” orbits (shown in black) exist only for values of assume A ¼ 1 throughout the paper.

A. Horizons, ergoregions and soundcurves 2 Some notions of curved spacetime geometry can be used to formally describe the propagation of sound waves, with 1 velocity cs, in irrotational fluids like the one determined by Eq. (9). Among these, we find useful the following y 0 2 2 definitions. Ergocurves are curves at which v ¼ cs and play the role of spacetime ergosurfaces. Acoustic horizons, –1 on the other hand, are curves at which n · v ¼ cs, where n is the normal vector at each point in the curve. Acoustic horizons play the role of event horizons when the acoustic –2 geometry is stationary. Finally, soundcurves (which are –3 –2 –1 0 1 2 3 analog light rings or analog photon surfaces) are closed x ≡ curves for sound waves; a sound wave of velocity vs FIG. 6. Different “soundcurves” when the separation between ðdx=ds; dy=dsÞ (with s an affine parameter) therefore the sinks is a ¼ 2. These curves were computed by numerically satisfies, on a soundcurve, kvs − vk¼cs. These analog solving the null geodesic equations for the analogue metric (10) light rings were recently investigated experimentally with the velocity field (9).

064036-5 THIAGO ASSUMPÇÃO et al. PHYS. REV. D 98, 064036 (2018) separation a>1. It is worth noticing that these orbits 1.5 are not symmetric, as opposed to the ones found in the 1.0 Majumdar-Papapetrou spacetime. Furthermore, the global outer orbits can cross the other two types of orbits for small 0.5 values of a, in contrast to what happens in MP spacetimes. For double sinks separated by a large distance a ≫ 1, the y 0.0 coordinate period of the global geodesic (encircling both –0.5 holes) is T ∼ 15.6 0.3 þ 4a. This is in rough agreement with expectations: the total crossing time is of order 4a plus –1.0 two semicircles, each of which takes 2π to complete. In the –1.5 same regime, the coordinate period of the inner closed lines –2 –1 0 1 2 is ∼12.5 ∼ 4π, as expected from the single-hole analysis. x For separations a ≲ 5 the travel time does not strongly depend on the separation itself and is ∼8π. Thus, depending FIG. 7. Acoustic horizon (solid line) surrounding both sinks for on the dynamical aspect under investigation, the two sinks separation a ¼ 1. The horizon exists only if a ≤ 1. The ergocurve can be considered to be merged at these separations. (dashed line) clearly does not coincide with the horizon, except at a few points. 2. Acoustic horizon Defining ϕ ¼ y0ðxÞ, the normal to the curve yðxÞ can be The function yðxÞ informs on the location of the ergocurve. written as In general, the acoustic horizon and the ergocurve do   not coincide. To show this, assume that they do and that ϕ 1 2 2 − pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ cs everywhere along the horizon surface. At any n ¼ 2 ; 2 : 1 þ ϕ 1 þ ϕ given point x where yðxÞ is a smooth function, and vx and vy are nonzero, the horizon equation (13) implies that It is straightforward to show that the equation n · v ¼ cs determining the horizon becomes a first order differential dy v v v þ x ¼ 0 ⇒ y þ x ¼ 0 ⇒ v2 ¼ 0; ð15Þ equation for yðxÞ, dx v v v pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y x y v v c2ðv2 − c2Þ 0 x y s s 0 ≠ 0 y ðxÞþ 2 2 ¼ ; ð13Þ which clearly contradicts the assumption that vx and cs − vx vy ≠ 0. Consequently, the ergocurve and the acoustic where the background velocities are given by (9). Since any horizon can only coincide at points for which either acoustic horizon engulfing both sinks must necessarily vx ¼ 0 or vy ¼ 0 or yðxÞ is not smooth. This fact is intercept the symmetry line (y axis), we look for solutions illustrated in Fig. 7, which exhibits four points where they of (13) that pass through x ¼ 0. We find that a common coincide. horizon engulfs both sinks only if a ≤ 1. When a>1, the In other words, for acoustic binaries the horizon lies sinks are sufficiently apart and we recover a single horizon inside the ergocurve, raising the interesting prospect of around each sink. phenomena such as superradiance [19,39] to occur. These For a ≤ 1, by searching the parameter space of the results suggest that similar phenomena may be present in initial conditions of Eq. (13), we have found that the gravitational BH binaries which may lead to new effects acoustic horizon is always a smooth curve with y 0 during the inspiral. pffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ¼ ð1 þ 1 − a2Þ and y0ð0Þ¼0. In Fig. 7 we plot the shape of the horizon at the critical separation a ¼ 1. B. Wave scattering and quasinormal modes There is an established connection between photon 3. Ergoregion surfaces and the dynamical response of perturbed, sin- Although a notion and definition of ergoregion for gle-BH spacetimes [2,13,40,41]. In a nutshell, the photon dynamical spacetimes does not exist, the acoustic version surface works as a trapping region where scalar, electro- described above is readily extended to any geometry. magnetic or gravitational perturbations can linger. The 2 2 Solving the equation v ¼ cs for the double sink flow instability time scale of null geodesics is then related given by Eq. (9) yields directly to the lifetime of massless perturbations. We wish to understand if such connection carries over to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 binaries. The analysis of wave dynamics in BH binaries y ¼ 2 − a2c2 − x2c2 þ 2 1 þ a2c2ðx2c2 − 1Þ: c s s s s is notoriously difficult. For single BH spacetimes with s certain symmetries, a linearized study where the relevant ð14Þ partial differential equations are transformed into ordinary

064036-6 BLACK HOLE BINARIES: ERGOREGIONS, PHOTON … PHYS. REV. D 98, 064036 (2018) differential equations is possible. Sound waves Φðt; r; ϕÞ ∼ 1 KΦ ¼ − ð∂ − LβÞΦ; ð19Þ Φðt; rÞeimϕ in single acoustic dumb-hole spacetimes were 2α t studied years ago, and it was concluded that [38,42]: L (i) The early-time response is strongly dependent on the where denotes the Lie derivative, to write our evolution initial conditions, but is followed by quasinormal system in the form ringdown [4]. In the eikonal regime, such ringdown ∂ Φ ¼ −2αKΦ þ LβΦ is described by sound waves circling the soundcurve t   and decaying away. For the geometry (12), the γij γij ∂tKΦ ¼ α KKΦ − Di∂jΦ − ∂iα∂jΦ þ LβKΦ; fundamental (longest-lived) frequencies ωm for the 2 2 first few azimuthal numbers m are [38,42]4 (see [43]

for a ringdown analysis in presence of vorticity), where Di is the covariant derivative with respect to the 2-metric γij and K is the trace of the extrinsic curva- ω1 ¼ 0.20342 − 0.17062i; ð16aÞ 1 ture Kij ¼ 2α Lβγij. To numerically evolve this system, we employ the ω 0 47636 − 0 17537 2 ¼ . . i; ð16bÞ method-of-lines, where spatial derivatives are approxi- mated by fourth-order finite difference stencils, and we ω3 ¼ 0.73427 − 0.17621i; ð16cÞ use the fourth-order Runge-Kutta scheme for the time integration. Kreiss-Oliger dissipation is applied to evolved ω4 ¼ 0.98823 − 0.17648i: ð16dÞ quantities in order to damp high-frequency noise. A complication arising from metric (10) is the presence (ii) The ringdown stage eventually gives way to a late- of curvature singularities at x ¼a. To deal with these we time power law tail, where the field decays as use a simple excision procedure, whereby our evolved quantities are multiplied by a mask function Mðx; yÞ Φ ∼ t−ð2mþ1Þ: ð17Þ ∂tu → Mðx; yÞ∂tu; Appealing to the correspondence between quasinormal Φ modes and geodesic properties [2,44], one expects a where u ¼ , KΦ and Mðx; yÞ is a smooth transition relation of the type function that evaluates to 1 outside the horizon and 0 inside.6 2 1 We evolve the system with initial conditions given by ω Ω − 1 2 λ m − ð n þffiffiffi Þ m ¼ m iðn þ = Þ ¼ i p ; ð18Þ   2 4 2 2 2 2 X X − Y −ðR−R0Þ Φ 2σ2 ðx; yÞ¼ c0 þ c1 þ c2 2 e ; ð20aÞ to hold at large m. This relation indeed describes extremely R R well the results above for the quasinormal modes of acoustic holes.5 Nevertheless, as shown in Ref. [44] for KΦ ¼ 0; ð20bÞ the draining bathtub geometry, expansions of ωm can be ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 2 obtained to arbitrary order in powers of m. where X ¼ x − x0, Y ¼ y − y0 and R ¼ X þ Y . The constants cm characterize the multipole of order m. The function Φðx; yÞ represents a Gaussian pulse of width σ 1. Numerical approach centered at ðx0;y0Þ and peaked at R ¼ R0. For the numerical evolutions of (8), we used the EINSTEINTOOLKIT infrastructure [45–47], which is based 2. Results on the CACTUS Computational Toolkit [48], a software We extract the scalar field Φ at x 30, y 0 and at framework for high-performance computing. The space- ¼ ¼ x 0, y 30. The first point lies along the symmetry line; time metric is fixed according to Eq. (10), and we evolve ¼ ¼ the second extraction point lies along the line joining the the Klein-Gordon equation on top of this background using two sinks and is closer to one than to the other. The general the same numerical code as the one employed in [49], features of our results remain the same at other extraction herein accordingly adapted to 2 þ 1 evolutions. To reduce points. the system to a first order form, we introduce the “canonical Figure 8 shows the result of evolving a Gaussian, momentum” of the scalar field Φ quadrupolar initial data, in the background of a single sink (a ¼ 0). The signal is exponentially damped at 4 These numbers were obtained using a continued fraction intermediate times, and we find excellent agreement (to representation [4]. 5In fact, one can immediately see that the imaginary part is captured to better than 4% already for the m ¼ 1 mode. 6We thank D. Hilditch for discussions about this point.

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FIG. 8. Evolution of a quadrupole (c0 ¼ c1 ¼ 0, c2 ¼ 1) Gaussian (with x0 ¼ y0 ¼ 0, R0 ¼ 5, σ ¼ 1=2) profile around a single (a ¼ 0) acoustic hole. The scalar was extracted at x ¼ 30, y ¼ 0. A ringdown, exponentially decaying, phase is visible, and followed by a late-time power-law tail. This and subsequent plots show the absolute value of the scalar.

within ∼2%) with prediction (16b), and hence also with the soundcurve analysis. The late-time tail, also clear in Fig. 8, is well described by Φ ∼ t−4.9, in good agreement with prediction (17). Figure 9 shows the result of evolving monopolar and quadrupolar waves in the background of a binary with 1 FIG. 9. Scattering of a sound wave off a binary dumb-hole with finite but small separation, a ¼ . We find that the signal a ¼ 1. The Gaussian, centered at the origin, has a width σ ¼ 1=2 still has all the features of the single-BH spacetime: a and is localized at R0 ¼ 5. The field Φ is extracted along the single ringdown stage, followed by a late-time power law x-axis (black solid line) and along the y-axis (red, dashed line) at tail. Both the ringdown and the tail are well-described by a distance of 30 from the origin. Top: Monopolar wave, with single BH geometries. In particular, due to mode-mixing c0 ¼ 1. For monopolar waves the prompt signal quickly gives −1 1 caused by the lack of axisymmetry, an initial pure-m way to a power-law tail Φ ∼ t . at late times, in agreement with 1 data will evolve to a superposition of different m’s. prediction (17). Bottom: Quadrupolar wave, with c2 ¼ . We find ω 0 472 − 0 170 ∼ −1.2 In particular, the m ¼ 0 mode seems to be present at a ringdown ¼ . . i and a power law t . late times in the waveform and causes the latetime power- law tail to be well described by Φ ∼ 1=t [cf. Eq. (17)]. In other words, binaries separated by a distance a ≲ 1 of this BH. After a time ∼25 the ringdown mode of the behave very similarly to a single BH. This is not perturbed rightmost BH reaches the extraction point x ¼ 30 completely surprising and is most likely connected to on the x-axis (black solid curve). Later, starting at t ∼ 55 such binaries having a common horizon (cf. Fig. 7). This the ringdown of the leftmost BH can be seen, followed feature is also present in nonlinear evolutions of Einstein by the decaying power-law Φ ∼ t−1. The global mode is equations for the BH-binary problem, where ringdown is encoded in the envelope of this signal, and the only reason seen to be triggered as soon as merger occurs, and in a it is not more clearly visible is because the power-law tail very linear way [50–54]. sets in very quickly. Such behavior is similar to the setting When the separation is made to increase, the different in of global modes of anti–de Sitter spacetime, or of scales in the problem become noticeable, as seen in Fig. 10. ultracompact exotic objects [12,13,55,56]. For a ¼ 5, the coordinate period of the global geodesic is of The two lower panels in Fig. 10 show the result of order ∼35, larger by about a factor of 3 than the period of evolving a monopolar and quadrupolar data, centered at the the geodesics encircling each BH. origin and localized away from the two holes. The signal is Focus on the top panel of Fig. 10. The initial data is peaked at some intermediate time, most likely the result of localized around only one of the BHs (located at x ¼ 5). interference between different modes. We do not have a Therefore the initial data excites this BH first and to a simple and elegant explanation for such feature. For any higher extent than the second BH, farther apart. On the initial data we looked at, late-time tail is always dominated x-axis, the initial ringdown is well described by the modes by t−1, that of the circularly symmetric mode.

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IV. ON THE STRUCTURE OF CIRCULAR NULL GEODESICS FOR ETERNAL AND FOR COALESCING BINARIES One may think that even though our geometries of BH binaries described by the MP metric are stationary, they can be interpreted as a “snapshot” of a dynamically coalescing process of two BHs, and therefore the sequences of the circular null geodesics obtained in this paper could mimic a photon surface around a dynamical geometry of colliding BHs in a more astrophysically realistic situation. However, there is a sharp contrast between the stationary MP binary and a dynamically colliding BH binary. One should first note that the BH binaries in the MP solution never merge to form a single BH; they have two mutually disconnected components of the event horizon and therefore allow for closed null/timelike orbits outside the event horizon (see left panel of Fig. 11). In contrast, when two BHs dynami- cally coalesce, the event horizon as a three-dimensional null hypersurface has only a single connected component from the beginning. Let us recall that the future event horizon of a BH is formed by null geodesic generators which have no end points in the future but can in general admit past end points. For stationary BHs, the event horizon is a Killing horizon and the horizon null geodesic generators have no past end points. As for dynamical BH system, such as those formed by or by the merger of two BHs, (some of) the horizon null geodesic generators have past end points, which form a subset in the spacetime. An important feature is that such a past-end- point set—called crease set—must be spacelike and achronal rather than timelike or null, since the event horizon itself must be an achronal null hypersurface. Now when we say, “a single” BH or “two” BHs, we refer to as the number of the cross section components of the event horizon and a Cauchy surface. It has been shown in [57] that this notion of the “number of cross sections” is in fact time slice dependent when the event horizon admits a crease set as a part, since in that case one can choose a Cauchy slice such that it crosses the crease-set as many times as one wishes (middle panel of Fig. 11). Furthermore, depending upon the choice of Cauchy FIG. 10. Scattering of a sound wave off a binary dumb-hole slicing, even the topology of the horizon cross section with a ¼ 5. The Gaussian width is σ ¼ 1=2. The field Φ is changes (for instance, when the crease set is two dimen- extracted along the x-axis (black solid line) and along the y-axis sional, it can even be temporarily toroidal [58–60]). For (red, dashed line) at a distance of 30 from the origin. Top: 1 5 0 3 the mathematical structure of the crease set and the Quadrupolar wave, with c2 ¼ , x0 ¼ , y0 ¼ , R0 ¼ . First classification of the possible (temporal) horizon cross ringdown stage for x matches well that of a one single, individual BH. Second ringdown is of lower frequency. Ratio is 1.2. Middle: section topologies, see Ref. [57]. For the dynamically Monopolar wave, with c0 ¼ 1, x0 ¼ y0 ¼ 0, R0 ¼ 10. Bottom: coalescing “two” BH binary, the temporarily “two BHs” Quadrupolar wave, with c2 ¼ 1, x0 ¼ y0 ¼ 0, R0 ¼ 10. are connected by a crease set, and the achronal nature of this crease set prevents any causal geodesic curve (either null or timelike) from forming a closed orbit around either attempting to make a physical interpretation of closed null “each single” BH or the “8-shaped” closed orbit that orbits found in static MP solution. would encompass the “two BHs.” An exception is a global The observation above would be manifest if, for exam- outer closed orbit. Therefore one has to be careful when ple, one examines closed null geodesics in Kastor-Traschen

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FIG. 11. Left: Null geodesic curves in two MP BHs, whose event horizon consists of two disconnected components. Depicted are two circular orbits around each of the disconnected components (blue lines) and a “8-shaped” orbit (green line) around the event horizon. Middle: Two BHs merge to form a single BH. The event horizon has only a single connected component and admits a “crease set” (depicted by thick red-line) consisting of the past end points of some of the null geodesic generators of the event horizon. Here “two black holes” is meant to be the number of cross sections of the event horizon on a particularly chosen Cauchy time slice as illustrated. Note however that since the event horizon has, as a part, the crease set, which is spacelike, the notion of the number of cross sections (as well as the topology of the cross section) is dependent on the choice of Cauchy time slice. Also because of the spacelike nature of the crease-set, null/timelike orbits cannot be of “8-shaped.” Inside the event horizon there are apparent horizons, for which null geodesic curves can possibly form circular orbits (blue lines). Right: Two ultracompact stars collide to form a single BH. The system can admit a “8-shaped” null orbit (green line) as well as circular null orbits (blue lines) around each of the ultracompact stars if they are sufficiently compact. Further study is needed to see whether these “8-shaped” and circular null orbits eventually can go around outside the event horizon (as illustrated here) or must go inside the event horizon, after the BH formation.

(KT) solution [61] with two BHs. The KT solution also geodesics that we describe should capture basically the describes configuration of multiple BHs with maximal physics at play in gravitational-wave detectors. charge, i.e., M ¼jQj, but it includes a positive cosmo- logical constant. Therefore those multiple BHs are on a de V. CONCLUSION Sitter expanding universe. The time-reverse of the solution We have given the first steps in our understanding of can describe coalescing BHs, and their event horizon must wave dynamics around BH binaries. We found three types have crease sets. Since each BH has its own apparent of null geodesics, which should play a role in the decay horizon inside the event horizon, one can presumably find of perturbed binaries. Our results for sound waves in an e.g., “8-shaped” closed null orbits that go around two acoustic geometry indeed indicate that the different time “apparent horizons” but the orbits should entirely be inside scales can be associated to such geodesics. There are many the event horizon. open issues, concerning the specific features of the wave If we consider, instead of a coalescing BH binary, a dynamics, and how it generalizes when the binary itself has coalescing binary, each of the stars is compact dynamics. In particular, defining and exploring the notion enough to admit its own photon sphere, then we can find of closed null orbits and ergoregions for dynamical space- “8-shaped” closed null/timelike orbits, besides the global times is challenging, but potentially important, as it raises null closed orbits. See rightmost panel of Fig. 11. the possibility of gravitational lensing effects, superradiant The above notes of caution refer to the purely formal effects and Penrose energy extraction. description of such geodesics and event horizons. Our detectors are placed some finite distance away from the ACKNOWLEDGMENTS black hole binary and are active for a short amount of time: they have no access to a teleological quantity such as T. A. is grateful to Instituto Superior T´ecnico for hos- eternal horizons. Instead, they measure properties related to pitality where part of this work was carried out. T. A. local quantities, such as apparent horizons. Therefore, the acknowledges support from the São Paulo Research

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2 2 2 2 Foundation (FAPESP), Grants No. 16/00384-1 and No. 17/ L ðU þ ρ∂ρUÞ 2ρ_ z_ ∂ U − ðE þ z_ − ρ_ Þ∂ρU ρ̈− þ z ¼ 0; 10641-4. V. C. is indebted to Kindai University in Osaka ρ3U5 U for hospitality where this work was started. V. C. acknowl- edges financial support provided under the European ðA4Þ ’ “ Union s H2020 ERC Consolidator Grant Matter and 2 2 2 2 ’ ” L ∂ U 2ρ_ z_ ∂ρU − ðE − z_ þ ρ_ Þ∂ U strong-field gravity: New frontiers in Einstein s theory, ̈z − z þ z ¼ 0: ðA5Þ Grant No. MaGRaTh–646597. M. R. acknowledges partial ρ2U5 U financial support from the São Paulo Research Foundation (FAPESP), Grant No. 2013/09357-9, and from Conselho To find symmetric orbits we set initial conditions as Nacional de Desenvolvimento Científico e Tecnológico ρð0Þ¼0, zð0Þ¼z0, and z_ð0Þ¼0. By virtue of Eqs. (A2) (CNPq). M. Z. is funded through Program No. IF/00729/ and (A3) we find 2015 from FCT (Portugal) and would also like to thank the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hospitality of Kindai University in Osaka where part of this ρ_ 0 2 − 0 ð Þ¼ E Veffð ;z0Þ: ðA6Þ work was done. This work has received funding from the ’ European Union s Horizon 2020 research and innovation With the above initial conditions, we are able to find programme under the Marie Skł odowska-Curie Grant various orbits by choosing values of E, L, and z0. Setting No. 690904. This work was supported in part (A. I.) by L ¼ 0 means constraining the motion to the plane that JSPS KAKENHI Grants No. 15K05092, No. 26400280. contains both holes, as explained in the main text. On the The authors would like to acknowledge networking support other hand, to look at the symmetry plane z ¼ 0 we need to by the COST Action CA16104. choose nonzero values for ρð0Þ and L.

APPENDIX: GEODESICS IN THE 2. Oscillating particle in the MP metric MAJUMDAR-PAPAPETROU GEOMETRY: We start with a special case of the three-body problem in THE SYMMETRY PLANE which one the bodies is much lighter than the other two. This appendix provides further details on the geodesics The small body is a test particle following a timelike 0 along the symmetry plane of the MP geometry. geodesic motion at z ¼ , while the two massive bodies are the two extremely charged BHs in the Majumdar- Papapetrou metric. Setting δ ¼ 1 in Eq. (3) yields 1. Integration of geodesic equations in the MP metric 1 ρ_2 þ ¼ E2; ðA7Þ The geodesic equations are found by varying the action U2 with respect to the Lagrangian in Eq. (3). The equations of 2 motion are the well-known Euler-Lagrange equations, where U ¼ Uðρ;z¼ 0Þ. The effective potential 1=U tends to 1 as ρ → ∞ and z ¼ 0. Thus, the solution is bounded ρ 0 1 1 2 2 2 1 d ∂L ∂L and oscillatory around ¼ for =ð þ MÞ

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2 with V0 ¼ 1=ð1 þ 2MÞ . For large separations (in our Combining Eqs. (A12) and (A16), the Lyapunov exponent units, when M → 0) we recover the Newtonian result. is given by rffiffiffiffiffiffiffiffiffiffi 3. Lyapunov exponent pffiffiffiffiffiffiffiffiffiffiffiffi 00 λ − V One can compute the Lyapunov exponent associated ¼ K1K2 ¼ 2: ðA17Þ 2t_ with the unstable circular orbits by considering perturba- tions in the radial coordinate ρ and in the canonical momentum pρ ¼ ∂L=∂ρ_ [2,5]. Let pρ ¼ δpρ and We recall that all quantities are evaluated at the unperturbed ρ ¼ ρ þ δρ, then the equation for the perturbation reads (circular) solution. With that in mind, we can arrive at an out λ ρ ρ expression for computed for ¼ out, the outermost and      2 d δpρ 0 K1 δpρ unstable circular null geodesic. The conditions V ¼ E and ¼ ; ðA11Þ V0 ¼ 0 yield dt δρ K2 0 δρ   where L 2 U ¼ ρ2U4 and U0 ¼ − : ðA18Þ   E 2ρ −1 d ∂L 2 −1 K1 ¼ t_ ;K2 ¼ðtU_ Þ : ðA12Þ dρ ∂ρ By substituting these expressions in Eq. (A17) and using The Lyapunov exponent λ is the eigenvalue of the matrix in the definition of E, we find pffiffiffiffiffiffiffiffiffiffiffiffi Eq. (A11), namely K1K2. It is worth noting that the

Lyapunov exponent depends on the choice of parametriza- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tion. For that reason the perturbed equation is written as an −3U þ 4ρ U00 λ ¼ : ðA19Þ ordinary differential equation with respect to the coordinate 2ρ2U5 time t, so that associated time scale is the one measured by ρ¼ρout an observer at infinity. By virtue of the Euler-Lagrange equations, It is possible to solve Eq. (A11) by applying the standard ∂L d d procedure to solve coupled differential equations with ¼ ðgρρρ_Þ¼ρ_ ðgρρρ_Þ: ðA13Þ constant coefficients. The solution is ∂r dλ dρ sffiffiffiffiffiffi Taking the derivative of Eq. (6) with respect to ρ yields pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi K1 δpρðtÞ¼δp0 coshð K1K2tÞþ δρ0 sinhð K1K2tÞ; dρ_ 1 K2 ρ_ ¼ − V0; ðA14Þ dρ 2 ðA20Þ where V0 ¼ dV=dρ. From Eqs. (6) and (A14), sffiffiffiffiffiffi ρ_ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi d d 2 0 K2 ρ_ ðgρρρ_Þ¼gρρρ_ þ ρ_ gρρ δρðtÞ¼δρ0 coshð K1K2tÞþ δp0 sinhð K1K2tÞ; dρ dρ K1 1 − 0 2 − 0 ðA21Þ ¼ 2 gρρV þðE VÞgρρ 1 2 2 − 0 ¼ ½gρρðE VÞ : ðA15Þ where δp0 ¼ δpρð0Þ and δρ0 ¼ δρð0Þ. We can compute λ 2gρρ numerically from the previous expressions. To simplify the δ 0 For circular geodesics ρ_ ¼ 0, which implies that V ¼ E2 procedure, we set p0 ¼ and find that and that the radius is a critical point of the potential.     Equations (A13) and (A15) imply 1 δρ 1 ρ −ρ λ −1 ðtÞ −1 numðtÞ out     ¼ cosh δρ ¼ cosh δρ ; ðA22Þ ∂L 1 1 t 0 t 0 d d 2 2 0 00 ¼ ½gρρðE − VÞ ¼ − gρρV : dρ ∂ρ dρ 2gρρ 2 ρ where numðtÞ is the solution of the radial equation found ðA16Þ with a standard numerical integration technique.

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[1] V. Ferrari and B. Mashhoon, Phys. Rev. D 30, 295 [32] A. Wünsch, T. Müller, D. Weiskopf, and G. Wunner, Phys. (1984). Rev. D 87, 024007 (2013). [2] V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. [33] J. Levin and G. Perez-Giz, Phys. Rev. D 77, 103005 (2008). Zanchin, Phys. Rev. D 79, 064016 (2009). [34] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). [3] H. Yang, D. A. Nichols, F. Zhang, A. Zimmerman, Z. [35] M. Visser, Classical Quantum Gravity 15, 1767 (1998). Zhang, and Y. Chen, Phys. Rev. D 86, 104006 (2012). [36] L. M. Milne-Thomson, Theoretical Hydrodynamics, Dover [4] E. Berti, V. Cardoso, and A. O. Starinets, Classical Quantum Books on Physics (Dover, New York, 2002). Gravity 26, 163001 (2009). [37] T. Torres, A. Coutant, S. Dolan, and S. Weinfurtner, arXiv: [5] N. J. Cornish and J. Levin, Classical Quantum Gravity 20, 1712.04675. 1649 (2003). [38] E. Berti, V. Cardoso, and J. P. S. Lemos, Phys. Rev. D 70, [6] Y. Decanini, G. Esposito-Farese, and A. Folacci, Phys. Rev. 124006 (2004). D 83, 044032 (2011). [39] T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W. [7] R. J. Nemiroff, Am. J. Phys. 61, 619 (1993). Tedford, and S. Weinfurtner, Nat. Phys. 13, 833 (2017). [8] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 [40] V. Cardoso and P. Pani, arXiv:1707.03021. (2000). [41] E. Chaverra, M. D. Morales, and O. Sarbach, Phys. Rev. D [9] J. Keir, Classical Quantum Gravity 33, 135009 (2016). 91, 104012 (2015). [10] V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, [42] V. Cardoso, J. P. S. Lemos, and S. Yoshida, Phys. Rev. D 70, and P. Pani, Phys. Rev. D 90, 044069 (2014). 124032 (2004). [11] P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Phys. Rev. [43] S. Patrick, A. Coutant, M. Richartz, and S. Weinfurtner, Lett. 119, 251102 (2017). Phys. Rev. Lett. 121, 061101 (2018). [12] V. Cardoso, E. Franzin, and P. Pani, Phys. Rev. Lett. 116, [44] S. R. Dolan, L. A. Oliveira, and L. C. B. Crispino, Phys. 171101 (2016); 117, 089902(E) (2016). Rev. D 85, 044031 (2012). [13] V. Cardoso and P. Pani, Nat. Astron. 1, 586 (2017). [45] F. Löffler, J. Faber, E. Bentivegna, T. Bode, P. Diener, R. [14] K. Glampedakis, G. Pappas, H. O. Silva, and E. Berti, Phys. Haas, I. Hinder, B. C. Mundim, C. D. Ott, E. Schnetter, G. Rev. D 96, 064054 (2017). Allen, M. Campanelli, and P. Laguna, Classical Quantum [15] G. Pappas and K. Glampedakis, arXiv:1806.04091. Gravity 29, 115001 (2012). [16] P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. D [46] EINSTEINTOOLKIT, Einstein Toolkit: Open software for 96, 024039 (2017). relativistic astrophysics. [17] C.-M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. [47] M. Zilhão and F. Löffler, Int. J. Mod. Phys. A 28, 1340014 Phys. (N.Y.) 42, 818 (2001). (2013). [18] L. Barack et al., arXiv:1806.05195. [48] Cactus developers, CACTUS COMPUTATIONAL TOOLKIT. [19] R. Brito, V. Cardoso, and P. Pani, Lect. Notes Phys. 906,1 [49] P. V. P. Cunha, J. A. Font, C. Herdeiro, E. Radu, N. Sanchis- (2015). Gual, and M. Zilhão, Phys. Rev. D 96, 104040 (2017). [20] S. D. Majumdar, Phys. Rev. 72, 390 (1947). [50] A. Buonanno, G. B. Cook, and F. Pretorius, Phys. Rev. D [21] A. Papapetrou, Proc. R. Irish Acad., Sect. A 51, 191 (1947). 75, 124018 (2007). [22] J. B. Hartle and S. W. Hawking, Commun. Math. Phys. 26, [51] E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, M. 87 (1972). Hannam, S. Husa, and B. Bruegmann, Phys. Rev. D 76, [23] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). 064034 (2007). [24] C. Barcelo, S. Liberati, and M. Visser, Living Rev. Rela- [52] L. T. London, arXiv:1801.08208. tivity 8, 12 (2005); 14, 3 (2011). [53] V. Baibhav, E. Berti, V. Cardoso, and G. Khanna, Phys. Rev. [25] M. S. Costa, C. A. R. Herdeiro, and C. Rebelo, Phys. Rev. D D 97, 044048 (2018). 79, 123508 (2009). [54] R. Brito, A. Buonanno, and V. Raymond, arXiv:1805.00293. [26] M. Okounkova, L. C. Stein, M. A. Scheel, and D. A. [55] V. Cardoso and G. Khanna, Phys. Rev. D 91, 024031 (2015). Hemberger, Phys. Rev. D 96, 044020 (2017). [56] M. R. Correia and V. Cardoso, Phys. Rev. D 97, 084030 [27] P. V. P. Cunha, C. A. R. Herdeiro, and M. J. Rodriguez, (2018). Phys. Rev. D 97, 084020 (2018). [57] M. Siino, Phys. Rev. D 58, 104016 (1998). [28] P. V. P. Cunha, C. A. R. Herdeiro, and M. J. Rodriguez, [58] S. A. Hughes, C. R. Keeton, P. Walker, K. T. Walsh, S. L. arXiv:1805.03798 [Phys. Rev. D (to be published)]. Shapiro, and S. A. Teukolsky, Phys. Rev. D 49, 4004 (1994). [29] S. Chandrasekhar, Proc. R. Soc. A 421, 227 (1989). [59] S. L. Shapiro, S. A. Teukolsky, and J. Winicour, Phys. Rev. [30] J. Shipley and S. R. Dolan, Classical Quantum Gravity 33, D 52, 6982 (1995). 175001 (2016). [60] R. Emparan, M. Martinez, and M. Zilhao, Phys. Rev. D 97, [31] M. Patil, P. Mishra, and D. Narasimha, Phys. Rev. D 95, 044004 (2018). 024026 (2017). [61] D. Kastor and J. H. Traschen, Phys. Rev. D 47, 5370 (1993).

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