arXiv:hep-th/0404096v3 30 Aug 2004 uharttn bet n ocue hti h frequency the if that concludes ω one object, rotating a such oe[,5 ] edn akteapie ctee wave, black scattered the amplified the of back velocity Feeding angular 6]. the 5, now [4, hole being Ω with ing, aigojcsfrsprain hnmn,weethe where phenomena, superradiant condition for objects tating the make could one mirror a unstable. by surround- system by cylinder is that rotating wave [3] the scattered in ing anticipated the also was then It body, amplified. the of velocity angular osdrn aeo h form object. the absorbing of wave cylindrical a rotating Considering a when by hit happens what paper waves examined a scalar was superradiant in it systems later where [3] much classical Zel’dovich only for considered However, was para- scattering Klein’s by 2]. raised [1, problems the dox after time, long a for ¶ § ‡ † Japan 169-8555, Tokyo ∗ Shinjuku, Okubo, lcrncades [email protected] address: Electronic [email protected] address: Electronic [email protected] address: Electronic lcrncades vcardoso@teor.fis.uc.pt address: Electronic rsn drs:SineadEgneig aeaUniversi Waseda Engineering, and Science address: Present erbakhl soeo h otitrsigro- interesting most the of one is hole black Kerr A uerdatsatrn skoni unu systems quantum in known is scattering Superradiant fteicdn aesatisfies wave incident the of m < ω st utf h ag erASbakhlsaesal n sm 04.70.-s and numbers: stable PACS are holes mi built black the be Kerr-AdS can which large bomb unstable. at why a order justify distance such in to that minimum showing that us a example found is explicit is an there It b give unstable hole location. t black become growing mirror’s the the to the is compute of and This function process a unstable. this system detail the in investigate make scatte superradiant can to one rise hole giving satisfied are conditions aeipnigo erbakhl a eapie si scat it as amplified be can hole black Kerr a on impinging wave A etod ´sc optcoa,Uiesdd eCoimbra, de Universidade F´ısica Computacional, de Centro .INTRODUCTION I. lolast uerdatscatter- superradiant to leads also Ω ... nvriaed lav,Cmu eGmea,8005- Gambelas, de Campus Algarve, do Universidade F.C.T., etoMliicpia eAto´sc ETA Departa CENTRA, Astrof´ısica - de Multidisciplinar Centro Departa CENTRA, Astrof´ısica - de Multidisciplinar Centro nttt ueirTenc,A.RvsoPi ,14-0 L 1049-001 1, Pais Rovisco T´ecnico, Av. Superior Instituto h lc oebm n uerdatinstabilities superradiant and bomb hole black The m < ω e − eatmnod ´sc,IsiuoSpro T´ecnico, Superior F´ısica, Instituto de Departamento etoMliicpia eAto´sc CENTRA, Astrof´ısica - de Multidisciplinar Centro v oic as1 0901Lso,Portugal Lisboa, 1049-001 1, Pais Rovisco Av. iωt + imφ ,weeΩi the is Ω where Ω, nietupon incident Dtd eray1 2008) 1, February (Dated: o´ .S Lemos S. P. Jos´e sa .C Dias C. J. Oscar ´ hjnYoshida Shijun io Cardoso Vitor ty, obcnit faKr lc oesrone yamir- hole a constant by black a surrounded the hole at black Specifically, placed Kerr ror a model. of consists field bomb scalar a using nfc ihylkl htsalKr-d lc oe are holes black Kerr-AdS small that but unstable. likely possible only highly be- not fact basically is black in it is these suggest for we this Furthermore, excited show, not holes. black shall are we modes Kerr-AdS superradiant As large cause stable. least are at holes simple that, a given showing unstable. have be be argument [10] therefore would Reall could and holes Hawking It black Fortunately, attracted Kerr-AdS recently. has that attention which of expected deal spacetime, great (AdS) it a in Sitter incorporated anti-de naturally “mirror” is rotation. hole a black with the spacetime of mir- function A the a of as and function a location, ror’s as timescales fre- growing oscillation and the study quencies We coordinate. radial Lindquist for irr o xml,i n osdr asv scalar massive a considers one mass own if with its field example, provides Teukolsky’s For sometimes and Nature Press mirror. [7]. is bomb This hole energy black mirror. extracted the pres- total destroys radiation sure the the ampli- finally Then until hole back exponentially black grow bounce time. should the each will and itself wave mirror black fying the the the between mirror, surrounds forth likes reflecting one and one if a as Indeed, by energy hole hole. rotational black much the as from extract can one eew netgt ndti h lc oebm by bomb hole black the detail in investigate we Here µ < ω ig ypaigamro rudteblack the around mirror a placing By ring. ∗ § † ‡ msae n silto rqece as frequencies oscillation and imescales o h ytmbakhl lsmirror plus hole black system the for h mass the nadto,oragmnsenable arguments our addition, In . l erASbakhlssol be should holes black Kerr-AdS all -0456Ciba Portugal Coimbra, P-3004-516 m fPesadTuosy We Teukolsky. and Press of omb rrms elctd ealso We located. be must rror µ et eF´ısica, de mento F´ısica, de mento cteigo erbakhl,then hole, black Kerr a off scattering 3 ao Portugal Faro, 139 eso h oei certain if hole the off ters µ so,Portugal isboa, ¶ ffcieywrsa irr[,9]. [8, mirror a as works effectively r , r = r 0 where , r steBoyer- the is 2

II. THE BLACK HOLE BOMB given by (4) behaves as

A. Formulation of the problem and basic equations e−iωt Φ e±iωr∗ , r (8) We shall consider a massless scalar field in the vicinity ∼ r → ∞ Φ e−iωte±i(ω−mΩ)r∗ , r r , (9) of a Kerr black hole, with an exterior geometry described ∼ → + by the line element:

2 2 where the tortoise r∗ coordinate is defined implicitly by 2 2Mr 2 2Mra sin θ ρ 2 2 2 ds = 1 dt 2dtdφ + dr dr∗ r +a − − ρ2 − ρ2 ∆ dr = ∆ . Requiring ingoing waves at the horizon,   which is the physically acceptable solution, one must im- 2Mra2 sin2 θ +ρ2dθ2 + r2 + a2 + sin2 θ dφ2 , pose a negative group velocity vgr for the wave packet. ρ2 −iωt −i(ω−mΩ)r∗   Thus, we choose Φ e e . However, notice (1) that if ∼ with ωmomentum J = Ma, superradiance regime, waves appear as outgoing to an and has an event horizon at r = r = M +√M 2 a2. A + − inertial observer at spatial infinity, and energy is in fact characteristic and important parameter of a Kerr black being extracted. Notice that, since we are working with hole is the angular velocity of its event horizon Ω given positive ω, superradiance will occur only for positive m, by i.e., for waves that are co-rotating with the black hole. a This follows from the time and angular dependence of Ω= . (3) the wave function, Φ ei(−ωt+mφ). The phase velocity 2Mr+ ∼ ω along the angle φ is then vφ = m , which for ω > 0 and In absence of sources, which we consider to be our case, m> 0 is positive, i.e., is in the same sense as the angular the evolution of the scalar field is dictated by the Klein- velocity of the black hole. Gordon equation in curved spacetime, µΦ=0. To ∇µ∇ Here we shall consider a Kerr black hole surrounded make the whole problem more tractable, it is convenient by a mirror placed at a constant Boyer-Lindquist radial to separate the field as [11] r coordinate with a radius r0, so that the scalar field −iωt+imφ m will be required to vanish at the mirror’s location, i.e., Φ(t,r,θ,φ)= e Sl (θ)R(r) , (4) Φ(r = r0) = 0. With these two boundary conditions, m ingoing waves at the horizon and a vanishing field at the where Sl (θ) are spheroidal angular functions, and the azimuthal number m takes on integer (positive or nega- mirror, the problem is transformed into an eigenvalue tive) values. For our purposes, it is enough to consider equation for ω. positive ω’s in (4) [4]. Inserting this in Klein-Gordon The frequencies satisfying both boundary conditions equation, we get the following angular and radial wave will be called Boxed Quasi-Normal frequencies (BQN fre- m equations for R(r) and Sl (θ): quencies, ωBQN ) and the associated modes will accord- ingly be termed Boxed Quasi-Normal Modes (BQNMs). 1 m ∂θ (sin θ∂θSl ) The quasi stems from the fact that they are not station- sin θ ary modes, and that BQN frequencies are not real num- 2 2 2 2 m m bers. Instead they are complex quantities, describing the + a ω cos θ + Alm S =0 , (5) − sin2 θ l decaying or amplification of the field. One expects that   2 2 2 2 2 2 ∆∂r (∆∂rR)+ ω (r + a ) 2Mamωr + a m for a mirror located at large distances, or for small black − holes, the imaginary part of the BQNs will be negligibly ∆(a2ω2 + A ) R =0 , (6)  − lm small and thus the modes will be stationary, correspond- ing to the pure normal modes of the mirror in the absence where Alm is the separation constant that allows the split of the wave equation, and is found as an eigenvalue of (5). of the black hole. The BQNMs are of course different For small aω, the regime we shall be interested on in the from the usual quasinormal modes (QNMs) in asymptot- next subsection, one has [5, 12] ically flat spacetimes, because the latter have no mirror and satisfy outgoing wave boundary conditions near spa- 2 2 Alm = l(l +1)+ (a ω ) . (7) tial infinity, they describe the free oscillations of the black O hole spacetime. In the following we shall compute these Near the boundaries of interest, which are the horizon, modes analytically in a certain limit, and numerically by r = r , and spatial infinity, r = , the scalar field as directly integrating the radial equation (6). + ∞ 3

B. The black hole bomb: analytical calculation of This wave equation is a standard hypergeometric equa- 2 the unstable modes tion [14], z(1 z)∂z F + [c (a + b + 1)z]∂zF abF = 0, with − − − In this section, we will compute analytically, within a = l +1+ i 2̟, b = l +1 , c =1+ i 2̟ , some approximations, the unstable modes of a scalar field in a black hole mirror system. Due to the presence of a (17) reflecting mirror around the black hole, the scalar wave and its most general solution in the neighborhood of z = is successively impinging back on the black hole, and am- 0 is Az1−cF (a c +1,b c +1, 2 c,z)+ B F (a,b,c,z). plified. Using (15), one− finds that− the most− general solution of We assume that 1/ω M, i.e., that the Compton the near-region equation is wavelength of the scalar≫ particle is much larger than R = Az−i ̟(1 z)l+1F (a c +1,b c +1, 2 c,z) the typical size of the black hole. We will also assume, − − − − for simplicity, that a M. Following [5, 13], we di- +Bzi ̟(1 z)l+1F (a,b,c,z) . (18) vide the space outside≪ the event horizon in two regions, − The first term represents an ingoing wave at the horizon namely, the near-region, r r+ 1/ω, and the far- − ≪ z = 0, while the second term represents an outgoing wave region, r r+ M. We will solve the radial equation (6) in each− one≫ of these two regions. Then, we will match at the horizon. We are working at the classical level, so the near-region and the far-region solutions in the over- there can be no outgoing flux across the horizon, and thus one sets B = 0 in (18). One is now interested in lapping region where M r r+ 1/ω is satisfied. ≪ − ≪ the large r, z 1, behavior of the ingoing near-region Finally, we will insert a mirror around the black hole, → and we will find the properties of the unstable modes. solution. To achieve this aim one uses the z 1 z transformation law for the hypergeometric function→ [14],− F (a c+1,b c+1, 2 c,z)= (1 z)c−a−b 1. Near-region wave equation and solution − − − − Γ(2−c)Γ(a+b−c) F (1 a, 1 b,c a b+1, 1 z) × Γ(a−c+1)Γ(b−c+1) − − − − − Γ(2−c)Γ(c−a−b) In the near-region, r r+ 1/ω, the radial wave + Γ(1−a)Γ(1−b) F (a c+1,b c+1, c+a+b+1, 1 z), equation can be written as− ≪ − − − − (19) 4 2 ∆∂r (∆∂rR)+ r (ω mΩ) R l(l + 1)∆ R =0 . (11) and the property F (a,b,c, 0) = 1. Finally, noting that + − − when r one has 1 z (r+ r−)/r, one obtains To find the analytical solution of this equation, one first the large→r ∞behavior of the− ingoing→ − wave solution in the introduces a new radial coordinate, near-region, −l r r (r r−) Γ(2l + 1) z = − + , 0 z 1 , (12) R A Γ(1 i 2̟) + − rl r r− ≤ ≤ ∼ − Γ(l + 1)Γ(l +1 i 2̟) −  − l+1 (r r−) Γ( 2l 1) with the event horizon being at z = 0. Then, one has + + − − − r−l−1 . (20) Γ( l)Γ( l i 2̟) ∆∂r = (r+ r−)z∂z, and the near-region radial wave − − −  equation can− be written as

1 z l(l + 1) 2. Far-region wave equation and solution z(1 z)∂2R + (1 z)∂ R + ̟2 − R R =0 , − z − z z − 1 z − (13) In the far-region, r r+ M, the effects induced by the black hole can be− neglected≫ (a 0, M 0, ∆ 2 ∼ ∼ ∼ where we have defined the superradiant factor r ) and the radial wave equation reduces to the wave equation of a massless scalar field of frequency ω and 2 angular momentum l in a flat background, r+ ̟ (ω mΩ) . (14) 2 2 2 ≡ − r+ r− ∂ (rR)+ ω l(l + 1)/r (rR)=0 . (21) − r − Through the definition The most general solution of this equation is a linear combination of Bessel functions [14], i ̟ l+1 R = z (1 z) F , (15) −1/2 − R = r αJ l+1/2(ωr)+ βJ−l−1/2(ωr) . (22) the near-region radial wave equation becomes For large r this solution can be written as [14]

2 2 1 z(1 z)∂zF + (1 + i 2̟) [1 + 2(l +1)+ i 2̟] z ∂zF R α sin(ωr lπ/2)+ β cos(ωr + lπ/2) , − − ∼ πω r −   r   (l + 1)2 + i 2̟(l + 1) F =0 . (16) (23) −   4 while for small r it reduces to [14] The solution of (29) can be found in the approximation that applies suitably to this problem, namely, ω 1, (ω/2)l+1/2 (ω/2)−l−1/2 and Re(ω) Im(ω). For very small ω, the r.h.s. of≪ (29) R α rl + β r−l−1. (24) ≫ ∼ Γ(l +3/2) Γ( l +1/2) is very small and can be assumed to be zero in the first − approximation for ω. This yields

3. Matching the near-region and the far-region solutions Jl+1/2(ωr0)=0 , (30) which has well studied (real) solutions [14]. We shall When M r r+ 1/ω, the near-region solution and label the solutions of (30) as j : the far-region≪ solution− ≪ overlap, and thus one can match l+1/2,n the large r near-region solution (20) with the small r far- Jl+1/2(ωr0)=0 ωr0 = jl+1/2,n , (31) region solution (24). This matching yields ⇔ where n is a non-negative integer number. We now as- l l+1/2 (r+ r−) Γ(l + 1) Γ(l +1 i 2̟) ω sume that the complete solution to (29) can be written as A = − − α, ˜ Γ(l +3/2) Γ(2l + 1) Γ(1 i 2̟) 2 ω jl+1/2,n/r0 + iδ/r0, where we have inserted a small −   ∼ (25) imaginary part proportional to δ˜ 1. One then has, from (29) ≪

β Γ( l +1/2) Γ(l + 1) Γ( 2l 1) Γ(l +1 i 2̟) J (j + iδ˜) l! 2 = − − − − l+1/2 l+1/2,n = i( 1)l+1 ̟ α Γ(l +3/2) Γ(2l + 1) Γ( l) Γ( l i 2̟) ˜ − − − J−l−1/2(jl+1/2,n + iδ) − (2l 1)!! 2l+1  −  ω 2l+1 (r+ r−) . (26) 2l+1 l × 2 − 2 (r+ r−) 2 2 2l+1   − (k +4̟ ) ω .(32) × 2l +1 (2l)!(2l + 1)! ! Using the property of the gamma function, Γ(1 + kY=1 Γ(−l+1/2) (−1)l2l x) = xΓ(x), one can show that Γ(1/2) = (2l−1)!! , Now, we can use, for small δ˜ the Taylor expansion of the Γ(l+3/2) (2l+1)!! Γ(−2l−1) (−1)l+1l! Γ(l+1−i 2̟) l.h.s., Γ(1/2) = 2l+1 , Γ(−l) = (2l+1)! and Γ(−l−i 2̟) = l i ( 1)l+12̟ (k2 +4̟2). Then, the matching con- ˜ ′ k=1 Jl+1/2(jl+1/2,n + iδ) Jl+1/2(jl+1/2,n) dition− (26) yields i δ˜ (33) ˜ Q J−l−1/2(jl+1/2,n + iδ) ∼ J−l−1/2(jl+1/2,n) l 2 2l+1 β ( 1) l! (r r−) = i 2̟ − + − The quantities j , J ′ (j ) , J (j ) α 2l +1 (2l 1)!! (2l)!(2l + 1)! l+1/2,n l+1/2 l+1/2,n −l−1/2 l+1/2,n  −  are tabulated in [14], and can also easily be extracted l 2 2 2l+1 using Mathematica. Here it is important to note that (k +4̟ ) ω . (27) ′ l × Jl+1/2(jl+1/2,n) and ( 1) J−l−1/2(jl+1/2,n) always have k=1 ! − Y the same sign. Furthermore, for large overtone n, j (n + l/2)π. The frequencies of the scalar wave l+1/2,n ∼ 4. Mirror condition. Properties of the unstable modes that are allowed by the presence of the mirror located at r = r0 (BQN frequencies) are then

If one puts a mirror near infinity at a radius r = r0, j ω l+1/2,n + iδ , (34) the scalar field must vanish at the mirror surface. Thus, BQN ≃ r setting the radial field (22) to zero yields the extra con- 0 dition between the amplitudes α and β, and the position where n is a non-negative integer number, labelling the of the mirror r0, mode overtone number. For example, the fundamental mode corresponds to n = 0. In (34), δ = Im[ωBQN ] is β J (ωr0) = l+1/2 . (28) obtained by substituting (33) in (32), α −J−l−1/2(ωr0) ( 1)lJ (j ) j /r mΩ − −l−1/2 l+1/2,n l+1/2,n 0 − This mirror condition together with the matching con- δ γ ′ 2(l+1) , (35) ≃− J (jl+1/2,n) r dition (27) yield a condition between the position of the l+1/2 0 mirror and the frequency of the scalar wave, where

2 2 2 2l Jl+1/2(ωr0) 2 l! l! r (r+ r−) = i( 1)l+1 ̟ γ + − J−l−1/2(ωr0) − 2l +1 (2l 1)!! ≡ (2l 1)!! (2l)!(2l + 1)!  −   −  2l+1 l l (r r−) 2 2l+1 + − (k2 +4̟2) ω2l+1 . (29) (k2 +4̟2) j . (36) × (2l)!(2l + 1)! × 2l +1 l+1/2,n k=1 ! k=1 ! Y Y   5

lm Notice that δ is very small for large r0 and thus satis- where 0fi is the expansion coefficient (for the explicit fies the conditions that go with the approximation used, form, see, e.g., [12]). In this study, we keep the terms in Re(ω) Im(ω). Equations (34) and (35) constitute the the expansion up to an order of (aω)2. As mentioned, main results≫ of this section. Two important features of near the horizon, the physically acceptable solution of this system, black hole plus mirror, can already be read equation (40) is the incoming wave solution, given by from the equations above: first, from equations (34) and (35) one has, Y = e−i(ω−mΩ)r∗ [y + y (r r )+ y (r r )2 + ](43), 0 1 − + 2 − + · · · δ (Re[ωBQN ] mΩ) . (37) ∝− − where yi is the expansion coefficient determined by ω and Therefore, δ > 0 for Re[ωBQN ] < mΩ, and δ < 0 for y0. Here, we do not show explicit expression for the yi’s Re[ωBQN ] > mΩ. The scalar field Φ has the time de- because it is straightforward to derive it. pendence e−iωt = e−iRe(ω)teδt, which implies that for In order to obtain the proper solutions numerically, by Re[ω ] < mΩ, the amplitude of the field grows ex- using a Runge-Kutta method, we start integrating the BQN −5 ponentially and the BQNM becomes unstable, with a differential equation (40) outward from r = r+(1+10 ) growth timescale given by τ = 1/δ. In this case, we with the asymptotic solution (43). We then stop the in- see that the system behaves in fact as a bomb, a black tegration at the radius of the mirror, r = r0, and get the hole bomb. Second, value of the wave function at r = r0, which is considered a function of the frequency, Y (r0,ω). If Y (r0,ω) = 0, the jl+1/2,n solution satisfies the boundary condition of perfect re- Re[ωBQN ]= , (38) r0 flection due to the mirror and the frequency ω is a BQN showing that the wave frequency is proportional to the frequency, which we label as ωBQN . In other words, the inverse of the mirror’s radius. As one decreases the dis- dispersion relation of our problem is given by the equa- tance at which the mirror is located, the allowed wave tion Y (r0,ωBQN ) = 0. In order to solve the algebraic frequency increases, and there will thus be a critical ra- equation Y (r0,ωBQN ) = 0 iteratively, we use a secant dius at which the BQN frequency no longer satisfies the method. Here, it is important to note that if the mode is stable or Im(ωBQN ) < 0, the asymptotic solution (43) superradiant condition (10). Notice also that Re[ωBQN ] as given by (38) is equal to the normal mode frequen- diverges exponentially and another independent solution, which is unphysical, damps exponentially as r∗ . cies of a spherical mirror in a flat spacetime [15]. In the → −∞ next subsection, when the numerical results will also be available, we will return to this discussion. 2. Numerical results

C. The black hole bomb. Numerical approach Our numerical results are summarized in Figs. 1-7. As we remarked earlier, we only show the data correspond- 1. Numerical procedure ing to the unstable BQNMs. We have also found the stable modes, but since they lead to no bomb we refrain In the numerical calculation for determining oscillation from presenting them. ¿From the figures we confirm the frequencies of the modes, we use the same function as analytical expectations. In addition we can discuss grow- that defined by Teukolsky [16] given by (see also [17]) ing timescales, oscillation frequencies, energy extracted and efficiencies with great accuracy. Figure 1 plots the Y = (r2 + a2)1/2R . (39) imaginary part of the BQN frequency for the fundamen- tal BQNM as a function of the mirror’s location r0 and Then, Teukolsky equation becomes a canonical equation, the rotation parameter a. In figure 2, we show the real given by part of the BQN frequency for the fundamental BQNM d2 also as a function of r0 and a. Supporting the analytical Y + V Y =0 , (40) dr2 results, figure 1 shows that: (i) The instability is weaker ∗ (the growing timescale τ = 1 is larger) for larger Im[ωBQN ] where mirror radius, meaning that Im[ωBQN ] decreases as r0 in- 2 K λ∆ 2 d creases. This is also expected on physical grounds, as was V = 2 − 2 2 G G , (41) remarked in [7], if one views the process as one of succes- (r + a ) − − dr∗ sive amplifications and reflections on the mirror. (ii) As 2 2 2 2 −2 with K = (r + a )ω am, and G = r∆(r + a ) . one decreases r0 the instability gets stronger, as expected, − 2 2 For the separation constant λ = Alm + a ω 2amω, we but surprisingly, suddenly the BQNM is no longer unsta- − make use of a well known series expansion in aω, given ble. The imaginary component of ωBQN drops from its by maximum value to zero, and the mode becomes stable at crit ∞ a critical radius r0 . Physically, this happens because λ = a2ω2 2amω + f lm(aω)i , (42) superradiance generates wavelengths with λ > 1/Ω. So − 0 i i=0 the mirror at a distance r0 will “see” these wavelengths X 6

FIG. 1: The imaginary part of the fundamental (n = 0) BQN FIG. 2: The real part of the fundamental (n = 0) BQN fre- frequency (ωBQN ) as a function of the mirror’s location r0 is quency (ωBQN ) as a function of the mirror’s location r0 is plotted. The plot refers to a l = m = 1 wave. It is also shown plotted. The plot refers to a l = m = 1 wave. There is the dependence on the rotation parameter a. One sees that no perceptive a-dependence (as matter of fact there is a very crit for r0 greater than a critical value, r0 , depending on a, there small a-dependence but too small to be noticeable). Thus, is the possibility of building a bomb. Moreover, the imaginary the oscillation frequency basically depends only on r0, and component of the BQN frequency decreases abruptly from its for large r0 goes as 1/r0, as predicted by the analytical for- crit crit crit maximum value to zero at r0 . For r0 < r0 the BQNM mula (34). The dots indicate r0 (cf. Fig. 1). is stable. Tracking the mode to yet smaller distances shows that indeed it remains stable (the imaginary part of ωBQN is negative). TABLE I: The fundamental BQN frequencies for a black hole with a = 0.8M and a mirror placed at r0 = 100M. The data corresponds to the l = m modes, and the frequency is measured in units of the mass M of the black hole. We crit N A if r0 > r0 λ 1/Ω. One can improve the esti- present both the numerical (ωBQN ) and analytical (ωBQN ) crit ∼ ∼ results. Notice that the agreement between the two is very mate for r0 using equations (10), (37), and (38), yield- ing rcrit jl+1/2,n . This estimate for the critical radius good, and it gets better as l increases. Also, we have checked 0 mΩ that for very large r the two yield basically the same results. matches∼ very well with our numerical data, even though 0 the analytical calculation is a large wavelength approx- a = 0.8M,r0 = 100M crit imation. In fact, to a great accuracy r0 is given by l ωN : ωA : crit BQN BQN the root of Re[ω(r )] mΩ = 0, as is shown in fig- − − − − 0 − 1 8.75 × 10 2 + 1.19 × 10 7i 8.99 × 10 2 + 1.41 × 10 7i ure 3. Also supporting the analytical results, figure 2 −1 −12 −1 −12 1 2 1.13 × 10 + 6.77 × 10 i 1.15 × 10 + 6.89 × 10 i shows that Re[ωBQN ] behaves as , which is consis- r0 × −1 × −16 × −1 × −16 tent with equation (34). This means that it is indeed 3 1.37 10 + 2.45 10 i 1.39 10 + 2.26 10 i the mirror which selects the allowed vibrating frequen- cies. The results for higher mode number n is shown in figures 4 and 5, and the behaviour agrees with the Let us now take Press and Teukolsky example of a picture provided by the analytical approximation. The black hole with mass M = 1M⊙ [7]. We are now in a agreement between the analytical and numerical results position to make a much improved quantitative analy- is best seen in Table I, where we show the lowest BQN sis. We take a = 0.8 M, a large angular momentum so frequencies obtained using both methods. In figures 6 that we make a full use of our results. In addition, to and 7 we show the numerical results referring to different better take advantage of the whole process, one should values of the angular number l and m. Our numerical place the mirror at a position near the point of maxi- 1 −2(l+1) mum growing rate, but farther. Thus, for the example, results indicate that Im[ω ] r0 , in agreement BQN ∼ r 22M 33Km (see Fig. 1). This gives a grow- with the analytical result, equation (35). The numeri- 0 ∼ ∼ cal results also indicate that the oscillating frequencies ing timescale of about τ 0.6 s (see also Fig. 1), which means that every 0.6 s the∼ amplitude of the field gets ap- (Re[ωBQN ]) scale with l, more precisely, Re[ωBQN] be- proximately doubled. This means that at the end of 13 haves as Re[ωBQN] π/r0(n + l/2). This behavior is also predicted by the∼ analytical study. seconds the initial amplitude of the wave has grown to 107 of its initial value, and that thus the energy content is 1014 times greater than the initial perturbation. We con- 7

FIG. 3: This figure helps in understanding why the instability FIG. 5: Same as Fig. 4, but for the real part of ωBQN . disappears for r0 smaller than a certain critical value. The condition for superradiance is ω − mΩ < 0. Since ω goes as 1/r0 (check Fig. 2), then it is expected that the condition will stop to hold at a critical r0. This is clearly seen here. Note that the critical value of r0 is in excellent agreement with that shown in Fig. 1.

FIG. 6: The imaginary part of the fundamental ωBQN for an a = 0.4 black hole, as a function of the mirror’s location r0 here shown for some values of l,m. Furthermore, as is evident from this figure and also as could be anticipated, the larger m the smaller r0 can be, still displaying instability. Note however that the maximum instability is larger for the m = 1 FIG. 4: The imaginary part of the BQN frequency as a func- mode. This is a general feature. The imaginary part of the ∼ −2(l+1) tion of r0, for a = 0.4 and for the three lowest overtones n, frequency seems to behave as Im[ωBQN ] r0 , which for l = m = 1. As expected from the general arguments pre- agrees with the analytical prediction (35). sented, higher overtones get stable at larger distances, and attain a smaller maximum growing rate. crit we see that r0 increases with decreasing a. At a cer- crit tain stage r0 coincides with the position of the mirror sider there are no losses through the mirror and assume at r0, at which point there is no more possibility of su- the process to be adiabatic. Using the first law of ther- perradiance. The process is finished. Thus from figure 1, modynamics one can then set ∆M Ω∆J, where ∆M or more accurately from our numerics, one can find ∆ a, and ∆J are the changes in mass and∼ angular momentum and thus ∆ J. Then ∆ M follows from ∆ M Ω∆ J. of the black hole in this process, respectively. Now, the In the example this gives a total amount ∆ M ∼ 0.01M black hole is losing angular momentum in each superra- of extracted energy before the bomb stops functioning.∼ diant scattering. Thus a decreases. If we go to figure 1 The process has thus an efficiency of 1%, about the same 8

for the black hole bomb. The imaginary component of ω is δ (Re[ω ] mΩ). The electromagnetic BQN ∝ − BQN − field has the time dependence e−iωt = e−iRe(ω)teδt and thus, for Re[ωBQN ] < mΩ, the amplitude of the field grows exponentially with time and the mode becomes unstable, with a growth timescale given by τ = 1/δ. Second, Re[ωBQN ] 1/r0, i.e., the wave frequency is proportional to the∝ inverse of the mirror’s radius, as it was for the black hole bomb. Therefore, as one decreases the distance at which the mirror is located, the allowed wave frequency increases, and again there will be a crit- ical radius at which the frequency no longer satisfies the superradiant condition (10). If one tries to use the sys- tem as it is, it would be almost impossible to observe superradiance in the laboratory. Take as an example a cylinder with a radius R = 10cm, rotating at a frequency Ω=2π 102 s−1, and a surrounding mirror with radius × r0 = 20cm. For the system to be unstable and experi- FIG. 7: Same as Fig. 6 but for the real part. Note that mentally detectable, the minimum mirror radius is given there is an l-dependence of the real part of the frequency. by r c (where we have reinstated the velocity of crit ∼ mΩ On the other hand, there is no noticeable m-dependence, in light c), which yields rcrit 1000Km, for a m = 1 wave. accordance with the analytical result (34). It seems impossible to use∼ this apparatus to measure su- perradiance experimentally. A way out of this problem may be the one suggested in [19]: to surround the con- order of magnitude as the efficiency of nuclear fusion of ducting mirror by a material with a low velocity of light. hydrogen burning into helium ( 0.7%). If, instead, the In this case the critical radius would certainly decrease, ∼ mirror is placed at r0 200M 300Km one still gets although further investigation is needed in order to as- a good growing timescale∼ of about∼ 15 min. This means certain what kind of material should be used. that at the end of 6 hours the initial amplitude of the wave has grown to 107 of its initial value. In this case ∆ M 0.1 M, with a 10% efficiency, and one can show III. ARE KERR-ADS BLACK HOLES ∼ that the efficiency grows with mirror radius r0. Since the UNSTABLE? 2 cost of mirror construction scales as r0, we see that small mirrors are more effective. One can give other interesting A spacetime with a naturally incorporated mirror in examples. For a black hole at the center of a galaxy, with it is anti-de Sitter (AdS) spacetime, which has attracted 8 mass M 10 M⊙, a =0.8M, and r = 22 M, the max- ∼ 0 a great deal of attention recently due to the AdS/CFT imum growing timescale is of the order of 2 years. An- correspondence and other matters. As is well known, other interesting situation happens when the black hole anti-de Sitter (AdS) space behaves effectively as a box, has a mass of the order of the Earth mass. In this case, in other words, the AdS infinity works as a mirror wall. by placing the mirror at r0 = 1m one gets a growing Thus, one might worry that Kerr-AdS black holes could timescale of about 0.02 s. At the other end of the black behave as the black hole bomb just described, and that hole spectrum one can consider Planck size black holes. they would be unstable. Hawking and Reall [10] have shown that, at least for large Kerr-AdS black holes, this instability is not present. The stability of large Kerr- D. Zel’dovich’s cylinder surrounded by a mirror AdS black holes in four and higher dimensions can be understood in yet another way, using the knowledge one As a corollary, we discuss here electromagnetic super- acquired from the black hole bomb study. The black hole radiance in the presence of a cylindrical rotating body, rotation is constrained to be a<ℓ, where ℓ is the AdS a situation first discussed by Zel’dovich [3]. He noted radius [10]. Large black holes are the ones for which that by surrounding this rotating body with a reflect- r+ >> ℓ. In this case the angular velocity of the horizon a 2 2 Ω= 2 2 (1 a /ℓ ) goes to zero and one expects that ing mirror one could amplify the radiation, much as the r++a − black hole bomb process just described. Bekenstein and the rotation plays a neglecting role in this regime, with Schiffer [19] have recently elaborated on this. An inde- the results found for the non-rotating AdS black hole pendent analytical approximation, similar in all respects [20, 21] still holding approximately when the rotation is to the one we discussed earlier in the black hole bomb non-zero. The characteristic quasinormal frequencies for context, can also be applied here for finding the BQN large, non-rotating AdS black holes were computed in frequencies of this system (conducting cylinder plus re- [20, 21] showing that the real part scales with r+. Now, flecting mirror), and leads to almost the same results as since Ω 0 for large black holes and the QNMs have a → 9 very large real part, one can understand why there is no plays the same instabilities as the black hole bomb. How- instability: superradiant modes are simply not excited, ever, for the instability to be triggered in an Earth based as the condition for superradiance, ωaccretion disk. Once To conclude, we have investigated the black hole bomb there, the inner boundary of the accretion disk might thoroughly, by analytical means in the long wavelength act as a reflecting wall, and waves can then suffer mul- limit, and numerically. We have provided both analytical tiple reflection and superradiant amplification, increas- and numerical accurate estimates for growing timescales ing their energy. This energy increase will continue un- and oscillation frequencies of the corresponding unstable til the magnetosphere that surrounds the system is no Boxed Quasinormal Modes (BQNMs). Both results agree longer capable of supporting the energy pressure in the and yield consistent answers. An important feature born waveguide cavity. The wave energy would then be re- out in this work is that there is a minimum distance at leased in a burst, and collimated into a relativistic jet which the mirror must be located in order for the system by, e.g., the Blandford-Znajek process [25], and finally to become unstable and for the bomb to work. Basi- transferred into the observed γ -ray photons, as described cally this is because the mirror selects the frequencies by the Fireball model [26]. In order to better compare that may be excited. For distances smaller than this, the the energy and timescales derived from the superradiant system is stable and the perturbation dies off exponen- model with the observational data taken from gamma-ray tially. This minimum distance increases as the rotation bursts, it is appropriate to apply the methods of this pa- parameter decreases. We have given an explicit example per to superradiant cavities that have an accreting matter where such a system works yielding a reliable source of configuration of a torus or disk. energy. By using appropriately this extracted energy one could perhaps build a black hole power plant. Although we have worked only with zero spin (scalar) waves, we Acknowledgements expect that the general features for other spins will be the same. Moreover, it is known that a charged black The authors acknowledge stimulating discussions with hole, even in the absence of rotation, provides a back- Ana Sousa and Jorge Dias de Deus, which have inspired ground for superradiance, as long as the impinging wave us to do this work, and thank Emanuele Berti and Jacob is a bosonic charged wave (fermions do not exhibit su- Bekenstein for a critical reading of the manuscript and perradiance). In this case, the critical radius should be for useful suggestions. This work was partially funded 1 of order rcrit , where e is the charge of the scalar by Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT) – Por- ∼ eΦ particle and Φ is the black hole’s electromagnetic poten- tugal through project CERN/FNU/43797/2001. V.C. tial. and O.J.C.D. acknowledge finantial support from FCT We have also shown that a mirror surrounding through grant SFRH/BPD/2003. S.Y. acknowledges fi- Zel’dovich’s rotating cylinder leads to a system that dis- nancial support from FCT through project SAPIENS 10

36280/99.

[1] For a very good review on the subject as well as a careful Teukolsky, Phys. Rev. Lett 29, 1114 (1972). explanation of various misinterpretations that have ap- [12] E. Seidel, Class. Quantum Grav. 6, 1057 (1989). peared over the years concerning the Klein paradox, we [13] W. G. Unruh, Phys. Rev. D 14, 3251 (1976). refer the reader to: C. A. Manogue, Annals of Physics [14] M. Abramowitz and A. Stegun, Handbook of mathemat- 181, 261 (1988). ical functions, (Dover Publications, New York, 1970). [2] W. Greiner, B. M¨uller and J. Rafelski, Quantum electro- [15] L. Landau and E. Lifshitz, Quantum mechanics, non- dynamics of strong fields, (Springer-Verlag, Berlin, 1985). relativistic theory (Mir, Moscow, 1974). [3] Ya. B. Zel’dovich, Pis’ma Zh. Eksp. Teor. Fiz. 14, 270 [16] S.A. Teukolsky, Astrophys. J. 185, 635 (1973). (1971) [JETP Lett. 14, 180 (1971)]; Zh. Eksp. Teor. Fiz [17] S. A. Hughes, Phys. Rev. D 62, 044029 (2000); Erratum- 62, 2076 (1972) [Sov. Phys. JETP 35, 1085 (1972)]. ibid. D 67, 089902 (2003). [4] J. M. Bardeen, W. H. Press and S. A. Teukolsky, Astro- [18] E. W. Leaver, Proc. R. Soc. London A402, 285 (1985). phys. J. 178, 347 (1972). [19] J. D. Bekenstein and M. Schiffer, Phys. Rev. D 58, [5] A. A. Starobinsky, Zh. Eksp. Teor. Fiz 64, 48 (1973) [Sov. 064014 (1998). Phys. JETP 37, 28 (1973)]; A. A. Starobinsky and S. M. [20] G. T. Horowitz, and V. Hubeny, Phys. Rev. D 62, Churilov, Zh. Eksp. Teor. Fiz 65, 3 (1973) [Sov. Phys. 024027(2000); V. Cardoso and J. P. S. Lemos, Phys. Rev. JETP 38, 1 (1973)]. D 64, 084017 (2001); E. Berti, K. D. Kokkotas, Phys. [6] S. A. Teukolsky and W. H. Press, Astrophys. J. 193, 443 Rev. D 67, 064020 (2003). (1974). [21] V. Cardoso, R. Konoplya, J. P. S. Lemos, Phys. Rev. D [7] W. H. Press and S. A. Teukolsky, Nature 238, 211 (1972). 68, 044024 (2003). [8] T. Damour, N. Deruelle and R. Ruffini, Lett. Nuovo Ci- [22] V. Cardoso, unpublished. mento 15, 257 (1976). [23] M. H. P. M. Putten, Science 284, 115 (1999). [9] S. Detweiler, Phys. Rev. D 22, 2323 (1980); T. M. Zouros [24] A. Aguirre, Astrophys. J. 529, L9 (2000). and D. M. Eardley, Annals of Physics 118, 139 (1979); [25] R. D. Blandford and R. L. Znajek, Mon. Not. Roy. As- H. Furuhashi and Y. Nambu, gr-qc/0402037. tron. Soc. 179, 433 (1997). [10] S. W. Hawking and H. S. Reall, Phys. Rev. D 61, 024014 [26] T. Piran, Phys. Rep. 314, 575 (1999); P. M´esz´aros, Ann. (1999). Rev. of Astron. and Astrophys. 40, 137 (2002). [11] D. R. Brill, P. L. Chrzanowski, C. M. Pereira, E. D. Fack- erell and J. R. Ipser, Phys. Rev. D 5, 1913 (1972); S. A.