Gravitational Wave Signatures of the Absence of an Event Horizon: Nonradial Oscillations of a Thin-Shell Gravastar

Home , Gravastar, Schwarzschild radius

PHYSICAL REVIEW D 80, 124047 (2009) Gravitational wave signatures of the absence of an event horizon: Nonradial oscillations of a thin-shell gravastar

Paolo Pani,1,* Emanuele Berti,2,3,† Vitor Cardoso,2,4,‡ Yanbei Chen,3,x and Richard Norte3,k 1Dipartimento di Fisica, Universita` di Cagliari, and INFN sezione di Cagliari, Cittadella Universitaria 09042 Monserrato, Italy 2Department of Physics and Astronomy, The University of Mississippi, University, Mississippi 38677-1848, USA 3Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA 4Centro Multidisciplinar de Astrofı´sica - CENTRA, Departamento de Fı´sica, Instituto Superior Tećnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (Received 1 September 2009; published 30 December 2009) Gravitational waves from compact objects provide information about their structure, probing deep into strong-gravity regions. Here we illustrate how the presence or absence of an event horizon can produce qualitative differences in the gravitational waves emitted by ultracompact objects. In order to set up a straw-man ultracompact object with no event horizon, but which is otherwise almost identical to a black hole, we consider a nonrotating thin-shell model inspired by Mazur and Mottola’s gravastar, which has a Schwarzschild exterior, a de Sitter interior and an infinitely thin shell with finite tension separating the two regions. As viewed from the external space-time, the shell can be located arbitrarily close to the Schwarzschild radius, so a gravastar might seem indistinguishable from a black hole when tests are only performed on its external metric. We study the linearized dynamics of the system, and, in particular, the junction conditions connecting internal and external gravitational perturbations. As a first application of the formalism we compute polar and axial oscillation modes of a thin-shell gravastar. We show that the quasinormal mode spectrum is completely different from that of a black hole, even in the limit when the surface redshift becomes infinite. Polar quasinormal modes depend on the equation of state of matter on the shell and can be used to distinguish between different gravastar models. Our calculations suggest that low-compactness gravastars could be unstable when the sound speed on the shell vs=c * 0:92.

DOI: 10.1103/PhysRevD.80.124047 PACS numbers: 04.30.Db, 04.25.Nx, 04.80.Cc

presence of a ‘‘dark object’’ of mass M ’ð4:1 0:6Þ I. INTRODUCTION 6 10 M [5]. Recent millimeter and infrared observations of Black holes (BHs), once considered an exotic mathe- Sagittarius A, the compact source of radio, infrared, and matical solution to Einstein’s field equations, have now X-ray emission at the center of the Milky Way, infer an þ16 been widely accepted as astronomical objects [1–4]. intrinsic diameter of 3710 microarcseconds, even smaller Stellar-mass BHs are believed to be the final stage of the than the expected apparent size of the event horizon of the evolution of sufficiently massive stars. Massive BHs seem presumed BH [6]. Some of the exotic alternatives to a BH to populate the center of many galaxies at low redshift, and (such as ‘‘fermion balls’’) are incompatible with the ob- must have played an important role in the formation of servations [7] and any distribution of individual objects structure in the Universe. Most evidence supporting the within such a small region (with the possible exception of astrophysical reality of BHs comes from the weak-gravity dark matter particles or asteroids, which however should be region, i.e. from observations probing the space-time sev- ejected by three-body interactions with stars) would be eral Schwarzschild radii away from the event horizon. gravitationally unstable [8,9]. In a recent attempt to probe Attempts to rule out possible alternatives to BHs usually the event horizon, Broderick and Narayan have analyzed rely on general relativity being the correct theory of grav- the observations of Ref. [6]. If the object at the center of ity, and/or on constraints on the equation of state of matter our Galaxy had a surface it would be hot enough to glow at high densities. For massive BHs the most precise mea- with a steady emission of infrared light, but no such glow surements so far come from observations of stellar proper has been detected [10]. This and similar arguments are motion at the center of our own Galaxy, indicating the inevitably dependent on the gas distribution and on details of the accretion process, and they really set lower limits on *[email protected] the gravitational redshift corresponding to the hypothetical Currently at Centro Multidisciplinar de Astrofı´sica - CENTRA, surface replacing the event horizon (see e.g. [11] for a Departamento de Fı´sica, Instituto Superior Tećnico, Av. Rovisco review). Indeed, some hold the view that an observational Pais 1, 1049-001 Lisboa, Portugal proof of the existence of event horizons based on electro- †[email protected] ‡[email protected] magnetic observations is fundamentally impossible [12]. [email protected] For these reasons, model-independent tests of the strong- [email protected] field dynamics of BHs and studies of the gravitational

1550-7998=2009=80(12)=124047(20) 124047-1 Ó 2009 The American Physical Society PANI et al. PHYSICAL REVIEW D 80, 124047 (2009) radiation signatures of event horizons are necessary to QNM spectrum of boson stars, showing that it is remark- confirm or disprove the BH paradigm [13,14]. ably different from the QNM spectrum of BHs and lending Gravitational wave detectors offer a new way of observ- support to the feasibility of no-hair tests using QNM ing BHs, complementing the wealth of information from measurements [29–31]. present electromagnetic observations [4]. As first proposed Another proposed alternative to massive BHs, which we by Ryan [15,16], an exquisite map of the external space- shall focus on in this paper, are the so-called gravastars time of BHs (outside the innermost stable orbit, if there is [32]. The gravastar model assumes that the space-time any) can be constructed by observing the gravitational undergoes a quantum vacuum phase transition in the vi- waveform emitted when a small compact object spirals cinity of the BH horizon. The model effectively replaces into the putative supermassive BH at the center of a galaxy the BH event horizon by a transition layer (or shell) and the with the Laser Interferometer Space Antenna (LISA). As BH interior by a segment of de Sitter space [33,34]. Mazur an extension of Ryan’s work, Li and Lovelace considered and Mottola argued for the thermodynamic stability of the the small object’s tidal coupling with the central object and model. A dynamical stability analysis by Visser and showed that by studying details of radiation reaction, in- Wiltshire [35] confirmed that a simplified version of the formation about the space-time region within the orbit can gravastar model by Mazur and Mottola is also stable under also be obtained [17]. Ryan’s proposal to map space-times radial perturbations for some physically reasonable equa- using inspiral waveforms is promising, but the data analy- tions of state for the transition layer. Chirenti and Rezzolla sis task is affected by a ‘‘confusion problem’’: the possi- [36] first considered nonradial perturbations of gravastars, bility of misinterpreting truly non-Kerr waveforms by Kerr restricting attention to the relatively simple case of oscil- waveforms with different orbital parameters [18,19]. This lations with axial parity. They computed the dominant ambiguity was shown to be resolvable if the orbit is known axial oscillation modes and found no instabilities. In anal- to be circular [15] or if one only probes the mass, spin, and ogy with previous studies of the oscillation modes of boson quadrupole moment of the object using waveforms gener- stars [29–31], they confirmed that the axial QNM spectrum ated in the weak-gravity region [20]. A different approach of gravastars can be used to discern a gravastar from a BH. to test the BH nature of ultracompact objects is based on In the thin-shell limit, the axial QNM frequencies of measuring several of their free oscillation frequencies Ref. [36] and our own calculations recover Fiziev’s calcu- (‘‘ringdown waves’’) and comparing them with the quasi- lation of the axial QNMs of ultracompact objects with a normal mode (QNM) spectrum of BHs [21]. These tests of totally reflecting surface [37]. the ‘‘no-hair theorem’’ require a signal-to-noise ratio which In this paper we study the stability with respect to non- could be achieved by advanced Earth-based gravitational radial oscillations of a simplified ‘‘thin-shell’’ gravastar, wave interferometers and they are one of the most prom- retaining most of the essential features of the original ising science goals of LISA [22–24]. model. We consider both axial perturbations (reproducing In this paper we explore how to test possible alternatives and extending the results of Ref. [36]) and polar perturba- to the BH paradigm. We retain the ‘‘conservative’’ assump- tions. In the polar case, the matching of interior and tion that general relativity is the correct theory of gravity exterior perturbations at the gravastar shell requires a and we focus on possible tests of the existence (or absence) more careful analysis because (unlike the axial case) polar of an event horizon. perturbations of spherical objects actually induce motions Theorists conceived several families of compact objects of matter, which in turn couple back to gravitational per- with no event horizons. For example, boson stars are turbations. For this reason, nonspherical polar perturba- horizonless compact objects based on plausible models tions provide a more stringent test on the gravastar’s of particle physics at high densities, and they are (still) overall stability. Polar perturbations are also crucial in compatible with astrophysical observations [25,26]. Being studying the dynamics of objects orbiting the hypothetical indistinguishable from BHs in the Newtonian regime, bo- gravastar. In order to treat polar perturbations of a non- son stars are good ‘‘straw men’’ for supermassive BHs. The rotating thin-shell gravastar we set up a rather generic space-time of nonrotating spherical boson stars can ap- formalism combining standard perturbation theory (in the proximate arbitrarily well a Schwarzschild geometry Regge-Wheeler gauge) with Israel’s junction conditions even close to the event horizon, and being very compact [38]. The formalism can be applied to gravitational pertur- it is not easily distinguishable from a BH by electromag- bations of any spherically symmetric space-time charac- netic observations [26,27]. Building on Ryan’s proposal, terized by regions with different cosmological constants Kesden et al. showed that the inspiral of a small compact separated by infinitely thin shells with finite surface energy object into a nonrotating boson star will emit a rather and tension. Because matter is concentrated on these different gravitational waveform at the end of the evolu- shells, the junction conditions deduced here will be suffi- tion, when the small object falls into the central potential cient in describing the linear dynamics of matter. Quite well of the boson star instead of disappearing into the event predictably, these conditions depend on the equation of horizon of a BH [28]. Several authors have computed the state of the shell. As an application of the formalism we

124047-2 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) study the QNM spectrum of polar and axial perturbations 2M 8a2 1 ¼ 1 : (2.3) of gravastars, exploring the nonradial stability of these a 3 objects. Polar QNMs (unlike axial QNMs) depend on the equation of state of matter on the shell: they can be used In this paper we shall usually drop the dependence of fðrÞ not only to distinguish a gravastar from a BH, but also to and hðrÞ on r. From the jump in the radial derivatives of f distinguish between different gravastar models. We also we could easily obtain the two defining properties of this find that the imaginary part of some QNMs seems to have a gravastar model: the surface energy density and surface zero crossing when the gravastar is not very compact and tension . The junction conditions read [38] the speed of sound on the shell is close to the speed of light, S ½½K ¼ 8 S ; (2.4) suggesting that some gravastar models may be unstable ij ij ij 2 under nonradial perturbations. In a companion paper we will apply the formalism developed in this paper to study where the symbol ½½... gives the ‘‘jump’’ in a given gravitational radiation from compact objects inspiraling quantity across the spherical shell (or r ¼ a), i.e. into nonrotating gravastars. ½½A AðaþÞAðaÞ: (2.5) The plan of the paper is as follows. In Sec. II we review our static thin-shell gravastar model. Section III sketches The indices i and j correspond to coordinates t, , and the calculation of axial and polar gravitational perturba- ’ which parametrize curves tangential to the spherical ¼r ¼ tions and of the matching conditions at the gravastar shell. shell, Kij pffiffiffiffiffiffiffi inj is the extrinsic curvature, n rr Details of the matching procedure are provided in ð0; 1; 0; 0Þ= g is the unit normal vector, and Sij is the Appendix A, and details of the QNM calculation are given surface stress-energy tensor in Appendix B. Our numerical results for the polar and ¼ð Þ axial QNM spectra are presented in Sec. IV and supported Sij uiuj ij; (2.6) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by analytical calculations in the high-compactness limit in where u ¼ 1=gttð1; 0; 0; 0Þ (or u~ ¼ 1=gtt@~t) is the Appendix C. four-velocity of mass elements on the shell and ¼ We use geometrical units (G ¼ c ¼ 1). The Fourier g n n is the induced three-metric on the shell. We transform of the perturbation variables is performed by assuming a time dependence of the form ei!t. Greek in- then have dices ð; ; ...Þ refer to the four-dimensional space-time S ð Þ S ¼ð Þu u þ : (2.7) metric. Latin indices i; j; ... refer to the three- ij ij 2 i j 2 ij dimensional space-time metric on the shell. Latin indices at the beginning of the alphabet ða; b; ...Þ refer to the In the static, spherically symmetric case, pffiffiffiffiffiffiffi spatial metric on a two-sphere. grr ½½K ¼ g : (2.8) ij 2 ij;r II. EQUILIBRIUM MODEL Discontinuities in the metric coefficients are then related to The metric for a static thin-shell gravastar has the form the surface energy and surface tension as [35] pffiffiffi [35,36] pffiffiffi 0 ½½ ¼ f h ¼ ð Þ 1 h 4a; 8 2 : 2 ¼ ð Þ 2 þ 2 þ 2ð 2 þ 2 2Þ f ds0 f r dt dr r d sin d’ : hðrÞ (2.9) (2.1) In order to summarize the above relations and reveal the Here independent parameter space of a thin-shell gravastar we define hðrÞ¼1 2M ;r>a; ð Þ¼ r f r 8 (2.2) 3 ð Þ¼ ð 2Þ 4a 2 h r 1 3 r ;ra. The M ¼ Mv þ Ms 1 þ ; (2.11) junction conditions on the r ¼ a surface have already a 2a induced metric partially been chosen by requiring the to be continuous across the shell, which also dictates that fðrÞ ¼ 1 2M=a ; (2.12) be continuous at r ¼ a,or 1 2Mv=a

124047-3 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009) 1 1 4M =a 1 M=a enough, because nonradial oscillations of a gravastar will ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (2.13) in general produce nonzero variations of the surface energy 8a 1 2M =a 1 2M=a v (i.e., 0). As a consequence, gravastar types can be specified by the dimensionless parameters M =a and M =a. In this paper v s III. GRAVITATIONAL PERTURBATIONS we only consider a simplified version of the original Mazur-Mottola gravastar, which has vanishing surface en- In both the interior (de Sitter) and exterior ergy ( ¼ 0) and (Schwarzschild) background space-times we consider perturbations in the Regge-Wheeler (RW) gauge [39], writing 3M=a M ¼ M ; a ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : v 8 (2.14) 2 ¼ 2 þð Þ 1 2M=a ds ds0 RWg dx dx (3.1) However it is convenient to keep our notation general with

2 3 ð Þ ð Þ ð Þ ð Þ 1 @Ylm ð Þ @Ylm f r H0 t; r Ylm H1 t; r Ylm h0 t; r sin @’ h0 t; r sin @ 6 ð Þ 7 6 H2 t;r Ylm ð Þ 1 @Ylm ð Þ @Ylm 7 k k¼6 ð Þ h1 t; r h1 t; r sin 7 RWg 6 h r sin @’ @ 7; (3.2) 4 2 5 r Kðt; rÞYlm 0 2 2 r sin Kðt; rÞYlm

3 where Ylmð; Þ denotes the ordinary spherical harmonics rameter C ð2M=aÞ , related to the parameter ¼ M=a and stands for terms obtainable by symmetry. In this of Ref. [36]byC ¼ 83. Then we have (assuming ¼ 0) gauge the perturbations split into two independent sets: the 8 2M metric functions h and h are axial or odd-parity pertur- fðrÞ¼1 r2 ¼ 1 r2 1 Cðr=2MÞ2: 0 1 3 a3 bations, while H0, H1, H2, K are polar or even-parity perturbations. The linearized Einstein equations automati- (3.3) ¼ cally require H0 H2 H. In the de Sitter interior both axial and polar perturbations In the rest of this section we work out perturbations of can be reduced to the study of the (frequency-domain) the gravastar space-time, including the dynamics of the master equation shell itself, in three steps. In Sec. III A, we present the well- d2in known solution of the perturbation equations in de Sitter þ½!2 V ðrÞ in ¼ ;r

2pM!ffiffiffi 2pM!ffiffiffi pffiffiffi l þ 2 i 1 þ l i 3 Cr2 in ¼ rlþ1ð1 Cðr=2MÞ2ÞiðM!= CÞF C ; C ;lþ ; ; (3.8) 2 2 2 4M2

124047-4 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) where Fða; b; c; zÞ is the hypergeometric function [44]. lðl þ 1Þ rf0 rf0 !2r2 in 1 þ 1 K From and its derivative we can get the axial perturba- 2 2 2f f tion variables in the frequency domain [45]: ð þ Þ ð þ Þ 0 þ þ l l 1 0 l l 1 þ rf r i d i!r i f H1 f H ¼ in ¼ ð inÞ 4! 2 2 h1 ;h0 r : (3.9) f ! dr ¼ 0: (3.18) The polar metric functions K and H can be obtained 1 fðrÞ immediately from Note that if we make the appropriate choice for , Eqs. (3.15), (3.16), and (3.17) and the algebraic relation lðl þ 1Þ din also apply to the interior de Sitter spacetime [cf. Equa- ¼ in þ K ; (3.10) tion (3.12)]. 2r dr The Zerilli function ZoutðrÞ [45], which satisﬁes a wave equation, and its spatial derivative are also constructed in in ¼ i!r þ d from H and K as H1 : (3.11) 1 f r dr Hout A Kout ð¼ ¼ Þ Zout ¼ 1 3 ; (3.19) The quantity H H0 H2 and its derivatives can then A A A be found from the algebraic relation 2 1 3 lðl þ 1Þ 1 !2r2 lðl þ 1ÞCr out out out dZ A2K A1H K þ i!r i H1 ¼ 1 2 f f 8M2! ; (3.20) dr A2 A1A3 ðl 1Þðl þ 2Þ H ¼ 0: (3.12) 2 where This procedure ﬁxes all metric quantities and their deriva- 6M2 þ =2ð1 þ =2Þr2 þ 3=2Mr A ¼ ; (3.21) tives in the interior. Here and henceforth in the paper we 1 r2ð3M þ r=2Þ drop the dependence of h0, h1, H, H1, and K on ! and r. i!ð3M2 3=2Mr þ r2=2Þ B. The Schwarzschild exterior A ¼ ; (3.22) 2 rð3M þ r=2Þf In the Schwarzschild exterior, axial and polar perturbations obey different master equations [46]. The determination of the axial perturbation variables can still be ¼ r A3 i! ; (3.23) reduced to the solution of the RW equation [39], a f Schro¨dinger-like ODE identical to Eq. (3.7): and ¼ðl 1Þðl þ 2Þ. The metric perturbations can then 2 out 0 out 2 ð Þ @ f @ ! Vout r be obtained by integrating the Zerilli equation outward, þ þ out ¼ 0; (3.13) ¼ @r2 f @r f2 starting from r aþ. It is also useful to recall that (in the exterior) we can switch from the Zerilli function ZoutðrÞ where (3.20) to the RW function outðrÞ by using a differential relation between polar and axial variables discussed in lðl þ 1Þ 6M V ¼ f : (3.14) Chandrasekhar’s book [46]: out r2 r3 out out out out dZ d out out The metric can then be obtained from Eqs. (3.9), with fðrÞ ¼ Z ; ¼ þZ ; given by Eq. (2.2). dr dr The perturbed Einstein equations relate the polar varia- where bles (K, H, H1) via three differential equations: ð þ 2Þ ð þ 2Þ 6MfðrÞ d ð Þ ð þ Þ¼ ¼ i!; ¼ þ : fH1 i! H K 0; (3.15) 12M 12M rðr þ 6MÞ dr out out out From Z and dZ =dr we can easily compute and 0 0 0 þ ð Þþ ¼ out i!H1 f H K f H 0; (3.16) d =dr outside the shell and use them as initial conditions to integrate the RW equation outward. Leins et al. H 1 f0 lðl þ 1Þ [47] showed that this procedure is convenient to compute K0 þ K þ i H ¼ 0; (3.17) r r 2f 2!r2 1 polar oscillation modes by the continued fraction method; more details on the QNM calculation are given in and an algebraic relation: Appendix B.

124047-5 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009) C. The matching conditions u ¼½fðaÞ1=2ð1 þ t;_ r;_ ;_ ’_ Þ: (3.25) In this section we discuss the most delicate part of the perturbation problem, namely, the junction conditions for We now carry out a gauge transformation which maps the the RW perturbation variables across the shell. Here we shell to a fixed location (note that two different gauge only outline our strategy and present the results; more transformations are required for the exterior and for the details are given in Appendix A. Technically, the applica- interior). For any general gauge transformation x ¼ x tion of Israel’s junction conditions is easier if the shell’s ðxÞ we have, to first order in , world tube happens to coincide with a fixed coordinate ¼ ð Þ ð Þ¼ ð Þþ ð Þ sphere at constant radius. However this is incompatible g g x g x ; x ; x ; with choosing the RW gauge in both the interior and the (3.26) exterior, which is convenient to cast the perturbation equations into simple forms. In fact, such a choice of gauge where the semicolon represents a covariant derivative with does not leave any freedom. We must explicitly parame- respect to the four-metric and gðx Þ is the metric tensor trize the three-dimensional motion of each mass element in the new coordinate system. We impose that, when on the shell and then perform the matching on a moving evaluated at ðt; r; ; ’Þ¼ðt;^ a; ;’Þ, the vector co- shell. In order to take advantage of both the simplicity of incides with ðt; r; ; ’Þ, so that in the new coordinate the field equations and the convenience of matching on a system we will have fixed coordinate sphere, we carry out the matching in the qffiffiffiffiffiffiffiffiffi following way. We first construct a particular coordinate t ¼ = fðaÞ; r ¼ a; ¼ ; ’ ¼ ’; transformation (for both the exterior and interior space- times) such that in the new coordinate system, any mass on (3.27) the shell remains static on the coordinate sphere with r ¼ a. In this new coordinate system the metric perturbations where we are ignoring second-order corrections. The full will no longer be RW, but will be augmented by quantities metric in the new coordinate system is that carry information about how masses on the shell move ¼ ð0Þ þ þ in the RW gauge. The stress-energy tensor of masses on the g g RWg g; (3.28) shell will correspondingly be modified. We then carry out ¼ ð0Þ the matching at r a and obtain junction conditions relat- where g is the static gravastar background metric, given ing the interior and exterior metric perturbations, plus by Eq. (2.1). The explicit form of and the corresponding equations of motion for matter on the shell. As could be changes in the metric components, Eq. (3.26), are given in anticipated from the general features of oscillations of Appendix A, where the equations of motion, as well as the nonrotating stars, axial perturbations do not couple to gauge transformation, are presented systematically in a matter motion and the axial junction conditions are very multipole expansion. We then match components of g simple, basically imposing continuity of the master vari- along the shell, which now simply sits at ðt;^ a; ;’Þ, and able and of its first derivative. Polar perturbations, on the apply Israel’s junction conditions to the extrinsic curvature other hand, do couple to matter motion, so polar junction given by g. For axial perturbations these matching con- conditions do involve the shell dynamics, i.e. its equation ditions read of state. pffiffiffi We parametrize the world line of matter elements on the ½½h ¼ 0; ½½ hh ¼ 0: (3.29) shell in terms of the coordinates ðt; r; ; ’Þ as follows: 0 1 qffiffiffiffiffiffiffiffiffi For thin shell gravastars hðrÞ is continuous across the shell, t ¼ = fðaÞ þ tð; ;’Þ;r¼ a þ rð; ;’Þ; implying continuity of the RW function and its derivative 0 across the shell [cf. Equation (3.9)]. In more general ¼ þ ð; ;’Þ;’¼ ’ þ ’ð; ;’Þ; cases where hðrÞ may have a discontinuity across the shell (3.24) the axial junction conditions (3.29) show that must also be discontinuous. where and ’ identify physical mass elements on the The treatment of polar perturbations is more involved sphere, while parametrizes their proper time. Note that and it yields the following relations, determining the jump the Lagrangian equations of motion will not be the same of the polar metric functions at the shell: for the interior and exterior space-times. Therefore, points ½½ ¼ with the same t, , and ’ coordinates are not in general the K 0; (3.30) same when viewed from the interior and from the exterior. As shown in Appendix A, the four-velocity of thepffiffiffiffiffiffiffiffiffi mass 0 element ð;’Þ at the scaled proper time t^ = fðaÞ is, ½½K ¼ 8 pffiffiffiffiffiffiffiffiffi ; (3.31) to leading order in the perturbation variables, fðaÞ

124047-6 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) 2M (after eliminating ). Alternatively, the number of junc- ½½H ½½Hf0 2fðaÞ½½H0 þ 4i!½½H a2 1 tion conditions can be obtained considering that all metric qffiffiffiffiffiffiffiffiffi perturbations can be expressed in terms of in and out, ¼ ð Þð þ 2Þ 16 f a 1 2vs : (3.32) and that each of these master variables satisfies a second- order ODE. More specifically, we use Eqs. (3.30), (3.31), Note that fðrÞ is continuous across the shell [cf. Equa- in in and (3.32) to determine two relations among ð ;@r Þ tion (2.3)]. The parameter vs in the equations above de- out out and ð ;@r Þ, plus the corresponding . pends on the equation of state on the thin shell, ¼ ðÞ: 2 @ IV. NUMERICAL RESULTS vs ; (3.33) @ ¼ 0 Some axial and polar QNM frequencies for a static thin- and it has the dimensions of a velocity. One might naively shell gravastar, as computed by the continued fraction interpret vs as the speed of sound on the thin shell and method, are plotted in Figs. 1 and 2. The C++ code used require vs 1, i.e. that the speed of sound cannot exceed for the calculations is an update of the FORTRAN code used the speed of light. Furthermore, for a shell of ordinary, in Ref. [50] to verify and extend results on stellar oscil- 2 stable matter we would have vs > 0. The standard argu- lations by Kokkotas [51] and Leins, Nollert, and Soffel 2 ment used to deduce that vs > 0 does not necessarily hold [47]. For axial modes, our numerical results are in excel- when one deals with exotic matter (as in the case of lent agreement with the thin-shell limit of the QNM fre- gravastars and wormholes), so the specification of upper quencies computed by Chirenti and Rezzolla [36] and with and lower bounds on vs may require a detailed micro- Fiziev’s calculation of the axial QNMs of ultracompact physical model of the exotic matter itself [48,49]. In our objects with a totally reflecting surface (compare Figs. 3 discussion of gravastar stability we will consider the whole and 4 of Ref. [37]). range of vs, but we will primarily focus on the range 0 To find the QNM frequencies we adopt the following 2 ¼ vs 1. The application of the polar junction conditions is numerical procedure. We usually fix 0:4 and (for more involved than the axial case due to their complexity, polar perturbations) we choose a constant value of vs.In which arises from the fact that polar perturbations couple the calculations leading to Figs. 1 and 2 we chose, some- 2 to oscillations of the shell. what arbitrarily, vs ¼ 0:1. Later in this section we will Here we note that even though we have three quantities discuss the dependence of polar modes on vs. ð Þ K; H; H1 that satisfy a coupled system of first-order As explained in Appendix B, within the continued frac- ODEs in both the interior and exterior, in each region there tion method the complex QNM frequencies can be deter- is an algebraic relation relating the three quantities. For this mined as the roots of any of the n equations fnð!Þ¼0 reason we only need to impose two independent junction [cf. Equation (B19)], where n is the ‘‘inversion index’’ of conditions, and Eqs. (3.30), (3.31), and (3.32) provide the continued fraction. To locate QNMs we first fix a value ¼ exactly two independent relations among K, H, and H1 of (usually 0:4). We compute the real and imagi-

2.0 2.0 2 2 v =0.1 v =0.1 l=2 l=3 1.5 3 1.5 3 4

I 4 I ω 2 ω 1.0 1.0 2

2M µ=0.10 2M µ=0.20 µ 1 0.5 1 =0.30 0.5 µ=0.40 µ=0.49 0.0 0.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 4.0 ω ω 2M R 2M R

2 FIG. 1 (color online). First few axial (continuous lines) and polar (dashed lines) QNMs of a thin-shell gravastar with vs ¼ 0:1. In the left panel we follow modes with l ¼ 2 as the compactness varies. In the right panel we do the same for modes with l ¼ 3. Along each track we mark by different symbols (as indicated in the legend) the points where ¼ 0:1, 0.2, 0.3, 0.4, and 0.49. Our numerical method becomes less reliable when 2M!I is large and when the modes approach the pure-imaginary axis. Numbers next to the polar and axial modes refer to the overtone index N (N ¼ 1 being the fundamental mode).

124047-7 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009)

2 2.0 2 v =0.1 v =0.1 3 l=2 l=2 1.5 3 4

I R 4 ω ω 2 3 1.0 2 2M 2M 1 0.5 1 2 1 0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 µ µ

2 FIG. 2 (color online). Real (left) and imaginary (right) part of polar and axial QNMs with l ¼ 2 as functions of for vs ¼ 0:1. Line styles are the same as in Fig. 1. Numbers refer to the overtone index.

nary parts of fnð!Þ for a given inversion index n on a Modes emerging from the imaginary axis at some gener- ð Þ suitably chosen grid of !R;!I values, and make contour ally nonvanishing compactness are wII-modes [47]orin- plots of the curves along which the two functions are zero. terface modes, and they are similar in nature to acoustic The intersections of these curves are used as initial guesses waves scattered off a hard sphere [55,56]. The only quali- for the quasinormal frequencies; more precise values are tative difference with Fig. 3 of Ref. [53] are the loops then obtained using Mu¨ller’s method [52]. For fixed n (say, appearing for higher-order w-modes, for which we have n ¼ 0) this method singles out some spurious roots besides no analytical understanding. the physical QNM frequencies. The spurious roots can The fact that w-modes are effectively waves ‘‘trapped easily be ruled out, since they are not present for different inside the star’’, while wII-modes are interface modes, values of n. Looking for roots with n ¼ 0 is usually similar to acoustic waves scattered off a hard sphere, is sufficient, but sometimes we get more stable numerical also clear from the behavior of their wave functions in the solutions for n ¼ 1 and n ¼ 2 when the QNMs have a stellar interior. The real and imaginary parts of the wave large imaginary part (!I * 1:5 or so). functions are shown in Fig. 3 (see Ref. [47] for comparison The QNM spectrum with ¼ 0:4 corresponds to the with the wave functions of ordinary stars). This figure empty (polar) and filled (axial) squares in the left panel of shows eigenfunctions computed at the polar QNM frequen- ¼ ¼ Fig. 1, respectively. Starting from 0:4, we follow each cies for the first four w-modes and wII-modes with l 2 ! ! 2 QNM as 0 and as 1=2 to produce the tracks and vs ¼ 0:1. The plot shows that w-modes can be thought displayed in the figure. For this value of vs some of these of as standing waves inside the gravastar, and that the tracks end at the origin in the limit ! 0, while others hit overtone number corresponds to the number of nodes in the pure-imaginary axis at some finite limiting compact- the real and imaginary parts of the eigenfunction. The ’ ness imag (e.g. imag 0:24 for the first polar mode). The situation is different for wII-modes, where the wave func- mode frequencies usually move clockwise in the complex tion has a maximum close to the shell, as expected for plane (with the exception of QNMs displaying ‘‘loops’’) as interface modes. is increased. The imaginary part of both axial modes One of our most important conclusions is that neither (continuous lines) and polar modes (dashed lines) becomes axial nor polar modes of a gravastar reduce to the QNMs very small as ! 1=2, i.e. when the gravastar most of a Schwarzschild BH when ! 1=2. In this limit, the closely approximates a BH. The behavior is perhaps real part of most modes is extremely small (much smaller clearer from Fig. 2, where we separately show the real than the Schwarzschild result, 2M!R ’ 0:74734 for the and imaginary parts as functions of . fundamental mode with l ¼ 2 [4]). Indeed, the QNM spec- For both axial and polar spectra the dependence of the trum is drastically different from the QNM spectrum of a mode frequencies on the gravastar compactness resembles Schwarzschild BH: when ! 1=2 the entire spectrum that of ‘‘ordinary’’ ultracompact stars: see e.g. Fig. 3 of seems to collapse toward the origin. This is in sharp con- Ref. [53]. Intuitive models that capture most of the physics trast with the Schwarzschild BH case and, as first noted in of this problem have been presented in Refs. [54,55]. In Ref. [36], it can be used to discern a very compact gravastar their terminology, modes that emerge from the origin in from a BH. In Appendix C we prove analytically that the Fig. 1 when 0 are w-modes or curvature modes, QNM frequencies of a gravastar do not reduce to those of a roughly corresponding to waves trapped inside the star. Schwarzschild BH as ! 1=2. The proof is based on the

124047-8 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) 1.0 0.8 ω=1.44+1.37i ω=1.44+1.37i 0.8 ω=3.42+1.38i 0.6 ω=3.42+1.38i ω=5.11+1.52i ω=5.11+1.52i 0.6 ω=6.73+1.63i ω=6.73+1.63i 0.4 (a)) 0.4 (a)) Ψ Ψ 0.2 (r)/ 0.2 (r)/ Ψ Ψ 0.0 0.0 Im( Re( -0.2 -0.2

-0.4 l=2, µ=0.4 -0.4 l=2, µ=0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 r r

ω=0.90+0.32i l=2, µ=0.4 ω=0.90+0.32i l=2, µ=0.4 ω=1.16+1.20i 0.15 ω=1.16+1.20i 0.8 ω=1.08+2.48i ω=1.08+2.48i ω=1.04+3.63i ω=1.04+3.63i (a)) 0.6 (a)) Ψ

Ψ 0.10 (r)/ 0.4 (r)/ Ψ Ψ 0.05 Im( Re( 0.2

0.0 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.5 1.0 r r

FIG. 3 (color online). Top row: real and imaginary part of the wave function in the interior for the first four w-modes. Bottom row: real and imaginary part of the wave function in the interior for the first four wII-modes. In both cases we consider polar QNMs with 2 l ¼ 2 and vs ¼ 0:1. observation that the Zerilli wave function for polar modes However, Figs. 1, 2, and 4 provide evidence that axial is continuous in this limit. and polar QNMs do become isospectral when the gravastar It is clear from the figures that axial and polar modes do compactness approaches that of a Schwarzschild BH ( ! not have the same spectra for general values of . 0:5). An analytic proof of isospectrality in the high-

0.8 0.6 -0.10 l=2 0.10 -0.99 l=2 0.50 -2.00 0.7 0.70 -4.00 0.80 0.5 Axial 0.90 0.6 0.99 1.00 0.5 1.013 I 0.4 I 1.014 ω ω 1.02 0.4 0.3 2M 2M 0.3 0.2 Axial 0.2 0.1 0.1 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ω ω 2M R 2M R

FIG. 4 (color online). Tracks of the fundamental polar and axial w-modes for different values of the ‘‘sound speed’’ parameter vs 2 2 2 when vs < 0 (left) and when vs > 0 (right). Different line styles correspond to different values of vs , as indicated in the legend.

124047-9 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009) 0 0 10 10 l=2 l=2 -1 -1 10 ISCO 10

-2 -2 10 10

I I ω ω 0.10 -3 -3 0.50 10 10 0.70 2M 2M 0.80 0.10 0.90 0.50 0.99 -4 0.70 -4 1.00 10 0.80 10 1.013 0.90 1.014 1.00 1.02 -5 -5 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ω ω 2M R 2M R

FIG. 5 (color online). Left: spectrum of the weakly damped family of QNMs. The vertical line corresponds to twice the orbital frequency of a particle in circular orbit at the innermost stable circular orbit (ISCO), as we will discuss in a follow-up paper, only QNMs to the left of the line can be excited by a compact object inspiraling into the gravastar along quasicircular orbits. In the case vs ¼ 0:8 the mode ‘‘turns around’’ describing a loop in the complex plane. For vs & 0:8 the modes move clockwise in the complex plane as increases. For vs * 0:8 they move counterclockwise and they cross the real axis at ﬁnite compactness. To facilitate comparison, in the right panel we show again the right panel of Fig. 4 using a logarithmic scale for the imaginary part. compactness limit is given in Appendix C, where we show calculations extend the considerations of that paper to that in this limit both the Zerilli and RW functions are nonradial oscillations. continuous at the shell. The vs-dependence of the modes is studied in Figs. 4–6. From the matching conditions (3.29), (3.30), (3.31), and In Fig. 4 we show the tracks described in the complex plane (3.32) it is quite clear that polar QNMs (unlike axial by the fundamental polar and axial w-mode as we vary the QNMs) should depend on vs, i.e. [by Eq. (3.33)] on the compactness parameter . The fundamental axial mode equation of state on the shell. This is a new feature that does not depend on the equation of state parameter, as does not arise in the case of axial perturbations [36]. The expected, but the polar modes do change as a function of situation closely parallels the ordinary stellar perturbation vs. The standard argument used to deduce that the speed of 2 problem [42,57]. The role played by the equation of state in sound vs > 0 does not necessarily hold when one deals the dynamical stability of gravastars against spherically with exotic matter (as in the case of gravastars and worm- symmetric perturbations was discussed in Ref. [35]. Our holes). Therefore, for completeness, in the left panel of

0 1.4 10 0.10 l=2 0.50 l=2 1.2 0.90 1.00 -1 1.013 10 1.0 1.014 1.10 I R ω ω 0.8 -2 10

2M 2M 0.6 0.10 -3 0.50 0.4 10 0.90 1.00 1.013 1.014 0.2 -4 10 1.10 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 µ µ

FIG. 6 (color online). Real (left) and imaginary parts (right) of the fundamental polar w-mode for different values of the equation of 2 state parameter vs. Different line styles correspond to different values of vs , as indicated in the legend. The horizontal line in the left panel corresponds to twice the orbital frequency of a particle in circular orbit at the ISCO: only modes below the line can be excited during a quasicircular inspiral.

124047-10 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) 2 Fig. 4 we compute polar QNMs when vs < 0. Different pact, the weakly damped modes of a thin-shell gravastar 2 line styles correspond to different values of vs, as indicated only exist for small compactness <crit. in the legend. The solid black line reproduces the funda- In the left (right) panel of Fig. 6 we show the real mental axial w-mode of Figs. 1 and 2. The dashed black (imaginary) parts of both families of QNMs as functions line corresponds to the fundamental polar w-mode for a of for selected values of vs. Both the real and imaginary 2 shell with low sound speed (vs ¼ 0:1), corresponding to part of the weakly damped modes tend to zero at some the fundamental polar w-mode of Figs. 1 and 2. The dash- finite, vs-dependent compactness . The range where 2 dash-dotted (red), dash-dotted (blue), and dotted (green) weakly damped modes exist increases with vs. The fact lines represent a marginally subluminal, imaginary sound that both the mode frequency and its damping tend to zero 2 2 speed (vs ¼0:99) and superluminal sound speeds (vs ¼ at the critical compactness crit suggests that the mode 2 2 and vs ¼4, respectively). Nothing particularly strik- somehow ‘‘disappears’’ there, rather than undergoing a ing happens in this regime: QNM frequencies for polar and nonradial instability, but this conjecture deserves a more axial perturbations are different in all cases, but for large careful analytical study. Plots similar to those shown in compactness the results become vs-independent and Fig. 6 show that the imaginary part of ordinary modes with 2 modes of different parity become approximately isospec- vs * 0:84 (vs * 0:92) rapidly approaches zero at some tral, in agreement with the analytical results of finite compactness while the real part of the modes stays Appendix C. Furthermore, as jvsj!1the polar modes finite. We are unable to track QNMs numerically when 5 seem to approach the axial mode. We can perhaps under- 2M!I & 10 using the continued fraction method, and in stand this behavior if we think that the shell is effectively any case we cannot really prove by numerical methods that 2 becoming so stiff that matter decouples from the space- !I ! 0 at some finite compactness >0. When vs ¼ time dynamics, and only the ‘‘space-time’’ character of the 1:0134 the two families of modes approach each other oscillations survives. along a cusp, but their frequencies and damping times 2 The situation is more interesting in the case vs > 0, never cross. The ordinary family of modes exists all the displayed in the right panel of Fig. 4. At first (when way up to ¼ 0:5 but it becomes unstable (in the sense 2 vs 0:5 or so) the modes show a behavior similar to that the imaginary part of the mode crosses the real axis 2 that seen for vs < 0, albeit in the opposite direction (i.e. with the real part remaining finite) at some finite, the real and imaginary parts of the QNM frequencies vs-dependent value of the compactness . increase rather than decreasing when jvsj increases). Summarizing, our numerical results suggest that 2 When vs ¼ 0:7 a cusp develops, and as the speed of sound (1) weakly damped QNMs only exist when their compact- 2 * approaches the speed of light (for vs 0:9 in the figure) ness is smaller than some vs-dependent critical threshold 2 the modes ‘‘turn around’’ describing a loop in the complex and (2) when vs > 0 and vs * 0:84, the imaginary part of a plane. The area of this loop in the complex plane increases QNM crosses the real axis at another critical threshold until the sound speed reaches a critical value 1:013 whose value can be estimated by extrapolation. 2 ’ vcrit 1:014 (corresponding to vcrit 1:007). For vs > Nonrotating thin-shell gravastars should be unstable vcrit the QNM behavior changes quite drastically. The against nonradial perturbations when their compactness complex mode frequencies still approach the !I ¼ 0 axis is smaller than this critical value. clockwise as ! 0:5.However,as decreases the modes The two thresholds are plotted in Fig. 7. Weakly damped ¼ approach the axis !I ¼ 0 very rapidly along tracks which QNMs with l 2 exist only below the dashed line and, are now tangent to the lower branch of the fundamental according to our extrapolation of numerical results, thin- axial mode. shell gravastars should be unstable to nonradial perturba- 2 ¼ Even more interestingly, when vs > 0 there is also a tions with l 2 below the solid line (we verified that the second family of QNMs with a very small imaginary part. instability condition for l ¼ 3 is less stringent than for l ¼ 2 A plot of the eigenfunctions shows that these modes are 2). The dashed line extends up to vs < 1:0134, where our similar in nature to the wII-modes. The trajectories de- numerical search for weakly damped QNMs becomes im- scribed in the complex plane by some of these weakly practical (the modes trace smaller and smaller loops in the damped QNMs are shown in the left panel of Fig. 5.For complex plane and they seem to disappear when the com- comparison, the right panel of Fig. 5 shows some of the pactness is still smaller than ¼ 0:5). ordinary modes. These ordinary modes are the same as in To validate results from the continued fraction method the right panel of Fig. 4, except that this time we use a we used an independent numerical approach: the resonance logarithmic scale for the imaginary part. The second fam- method [57–59], which is applicable to QNMs with ily of QNMs plotted in the left panel has very long damp- !I !R. The resonance method was first used by ing, and in this sense it is similar to the s-modes of ordinary Chandrasekhar and Ferrari in their analysis of gravitational ultracompact stars discussed by Chandrasekhar and Ferrari wave scattering by constant-density, ultracompact stars [57]. There is, however, an important difference: unlike the [57,58]. Chandrasekhar and Ferrari showed that the radial s-modes, which exist only when a star is extremely com- potential describing odd-parity perturbations of these stars

124047-11 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009) 0.5 A straightforward analysis of Eq. (3.5) and an inspection of Fig. 8 show that the axial potential for a gravastar 0.4 develops a minimum when * 0:43. The compactness of ordinary stars is limited by the Buchdal limit (< 4=9 ’ 0:4444), but since gravastars can be considerably 0.3 more compact than this limit, s-modes can exist all the way µ down to the ‘‘Schwarzschild limit’’ ( ! 0:5). These 0.2 modes can be computed via the continued fraction method, but since they are long-lived the resonance method, which 0.1 is computationally very simple, provides very accurate estimates of their frequencies and damping times. We 0.0 find that the resonance method and the continued fraction 0.0 0.5 1.0 1.5 2.0 2.5 3.0 method are in very good agreement whenever the reso- 2 nance method is applicable. In the limit ! 1=2 we have vs Imð!ÞReð!Þ, and all QNM frequencies can be interpreted as trapped states. ð 2Þ FIG. 7 (color online). Significant thresholds in the ; vs The resonance method essentially confirms our contin- plane. Thin-shell gravastars should be unstable with respect to ued fraction results for modes with !I !R. In particu- nonradial perturbations below the solid line, corresponding to lar, it provides additional numerical evidence for the ordinary QNM frequencies whose imaginary part crosses the conjectured nonradial instability of thin-shell gravastars zero axis while their real part stays finite. The dashed line corresponds to the ‘‘vanishing point’’ of weakly damped QNM with low compactness. Despite the numerical evidence, frequencies, i.e. to the point where both their real and imaginary an analytic confirmation of our estimates of the instability parts have a zero crossing. We could not find weakly damped threshold would be highly desirable. modes in the region above the dashed line. V. CONCLUSIONS AND OUTLOOK displays a local minimum as well as a maximum when the * In this work we have studied the nonradial perturbations stellar compactness 0:39. If this minimum is suffi- on nonrotating, thin-shell gravastars. It should not be too ciently deep, quasistationary, ‘‘trapped’’ states can exist: hard to extend our formalism to the more complex case of gravitational waves can only leak out to infinity by ‘‘tun- five-layer gravastars of the type originally proposed by neling’’ through the potential barrier. The damping time of Mazur and Mottola (see [36] for a treatment of the axial these modes is very long, so they were dubbed ‘‘slowly case). damped’’ modes (or s-modes) [57]. A presumably less trivial extension concerns rotating gravastars. Slowly rotating gravastars may be unstable against scalar perturbations because of an exponential µ growth of the perturbations due to superradiance, the so- 2 =0.40 10 µ=0.45 called ‘‘ergoregion instability’’ [60,61]. An extension of µ=0.49 the present formalism can be used to study nonradial 1 10 µ=0.4999 gravitational perturbations of slowly rotating gravastars and to discuss their ergoregion instability, which is be- 0 10 lieved to be stronger for gravitational perturbations [60]. V(r) For gravastars this instability is due to superradiant gravi- -1 10 tational wave scattering in the ergoregion, so it is essentially the same as the ‘‘w-mode instability’’ discussed by -2 10 Kokkotas et al. for ultracompact stars [62]. The main l=2 difference is that gravastars can be more compact than -3 10 constant-density stars, so we may expect the instability to 01234be stronger. r Finally, our formalism can be applied to explore nonradial gravitational perturbations of nonrotating worm- FIG. 8 (color online). The potential governing axial and polar holes, where the position of the throat plays a role perturbations for different values of the gravastar compactness M=a, where a is the location of the shell (see [57], showing similar to the thin shell of a gravastar. Such an analysis a similar plot for axial perturbations of constant-density stars). could confirm or disprove some conjectures on the simi- The potential develops a minimum when * 0:43. Note that larity of the QNM spectra of wormholes and BHs [63]. the polar and axial perturbations in the interior are both governed In follow-up work we will extend our study to explore by the same potential, given in Eq. (3.5) below. QNM excitation by compact objects in closed orbits

124047-12 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) around thin-shell gravastars (see [64,65] for similar studies reduce the problem to a wave equation. This means that considering the excitation of axial modes by scattering we only need two conditions relating these quantities from orbits). According to preliminary estimates by Norte, fol- the two sides. Similarly, for axial perturbations we have h0 lowing the weak-gravity expansion of Poisson and Sasaki and h1, which satisfy a coupled wave equation, and again [66] (extended by Li and Lovelace [17] to general bound- we need only two conditions connecting the interior and ary conditions), the gravitational wave luminosity can exterior. change dramatically at the resonances, while being very Now we go directly to step (ii), and parametrize the shell close to the BH value in nonresonant conditions. A weak- position as in Eqs. (3.24). Let us define field study should provide a reasonably accurate estimate qffiffiffiffiffiffiffiffiffi of the orbital frequency at which the resonance occurs, but t^ = fðaÞ; (A1) it can only provide a leading-order estimate of the radiated energy. A more precise characterization requires the nu- qffiffiffiffiffiffiffiffiffi merical integration of the perturbation equations [67–70]. F_ ð@F=@t^Þ¼ fðaÞð@F=@Þ: (A2) In a follow-up paper we will compare gravitational radiation from extreme mass-ratio inspirals in thin-shell grav- Note that t^ is simply a rescaling of proper time of the mass astar space-times to the BH results of Refs. [71–74]. element, and that fðaÞ is common to the interior and exterior, due to the requirement that the induced metric is ^ ACKNOWLEDGMENTS continuous. In the absence of perturbations, t coincides with t. Henceforth, we will use ðt;^ ;’Þ, to parametrize This work was partially supported by FCT-Portugal the mass element. As a consequence, the four-velocity u through projects PTDC/FIS/64175/2006, PTDC/FIS/ of the mass element ð;’Þ would be as in Eq. (3.25). 098025/2008, and PTDC/FIS/098032/2008. Y.C. was sup- Imposing that gu u ¼1, we actually have to require ported by NSF Grant Nos. PHY-0653653 and PHY- that 0601459, and the David and Barbara Groce Start-up ð1 þ 2t_Þg ðt^ þ t; a þ r; þ ; ’ þ ’Þ Fund at Caltech. E. B.’s research was supported by NSF tt ¼1; Grant No. PHY-0900735. fðaÞ (A3) APPENDIX A: PERTURBATION EQUATIONS AND which, to leading order, is MATCHING CONDITIONS 0 f ðaÞrðt;^ ;’Þþ2fðaÞt_ðt;^ ;’Þ In this appendix we develop the formalism to study polar ð Þ¼ and axial nonradial (linear) oscillations of an object con- RWgtt t;^ ;’ 0: (A4) sisting of a thin spherical shell separating two spherically symmetric regions. Though we are mainly interested in Here RWgtt is the tt-component of the metric perturbation thin-shell gravastars, we shall keep the discussion as gen- in the RW gauge. eral as possible. We focus on a background metric of the We will now carry out a gauge transformation, both in form (2.1), keeping fðrÞ and hðrÞ generic. the exterior and in the interior, which maps the shell to a The RW gauge is incompatible with the requirement that fixed sphere. As explained in the main text, for any general ¼ ð Þ the shell’s world tube sits at a fixed location. We deal with gauge transformation x x x we have this issue in two steps: (i) we choose a coordinate system @x @x ð Þ¼ ð Þ (system A) such that perturbations in both the interior and g x g x @x @x exterior are in the RW gauge; (ii) we write down the ¼½ equations of motion of mass elements on the shell in this g g; g; x ; (A5) coordinate system (separately for the interior and the exterior), and try to apply matching conditions on the bound- where we have ignored terms of second order in . Noting ary of this moving shell. We then transform to a new that coordinate system (system B) in which the shell is fixed, gðx Þ¼gðx þ Þ; (A6) simplifying the matching procedure. System B is an aux- iliary tool for matching: when we consider perturbations of we obtain Eq. (3.26). This is the desired form, because we the gravastar induced (say) by orbiting particles we will want to use to transform to a coordinate system x mostly use system A, which is the usual RW gauge. where mass elements on the shell are fixed in spatial Step (i) is a straightforward adaptation of formulas in location and move uniformly in the time direction: i.e., Sec. III. In the RW gauge, polar perturbations are defined for mass elements on the shell, x ¼ðt;^ a; ;’Þ. Then by three functions K, H, and H1 for the interior and Israel’s junction conditions will be applied to the RW exterior, respectively. These functions satisfy an algebraic metric evaluated at a fixed coordinate location relation and two coupled ODEs, which can be used to ðt;^ a; ;’Þ, and in terms of the transformation generators

124047-13 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009)

(which are related to the motion of the shell in the RW where m, n, and a run through and ’. Here ab is again gauge). deﬁned with respect to the metric (A7), so that Let us make four consecutive transformations ’ ¼’ ¼ sin: (A13) ¼½f1ðrÞyðtÞY ; 0; 0; 0; ð0Þ lm In terms of y and z, the normalization of the four- ¼½ ð Þ ð Þ velocity would be written as ð1Þ 0;h r z t Ylm; 0; 0 ; ð Þ ð Þ ¼f0ðaÞhðaÞzðt^ÞþfðaÞHðt;^ aÞ fðaÞ½f1ðaÞyðt^Þ w t w t 0 2 _ ð Þ ¼ 0; 0; Ylm;; Ylm;’ ; 2 r2 r2sin2 ¼ 0ð Þ ð Þ ð Þþ ð Þ ð Þþ ð Þ f a h a z t^ f a H t;^ a 2y_ t^ : ð Þ ð Þ x t x t ¼ 0; 0; Y ; Y ; Here we note again that fþ ¼ f. In this new coordinate ð3Þ r2 lm;’ r2 lm; sin sin system the metric is given by Eq. (3.28), where or, by lowering indices, ¼ þ þ g ð1Þg ð2Þg ð3Þg: ð Þ ¼½yðtÞY ; 0; 0; 0;ð Þ ¼½0;zðtÞY ; 0; 0; 0 lm 1 lm We are now in a position to match metric components ¼½ ð Þ ð Þ ð Þ ð2Þ 0; 0;w t Ylmj;w t Ylmj’ ; along the shell, which simply sits at t;^ a; ;’ .Of course, all of the matching conditions will be expressed ¼½ ð Þ j’ ð Þ j ð3Þ 0; 0;x t ’Ylm ;x t ’Ylm ; in terms of the RW metric perturbations, and the motion of where the covariant derivative j is deﬁned with respect to the shell in the (internal and external) RW gauges. In the the two-dimensional metric and ’ directions, we have

2 2 2 xþðt^Þ¼xðt^Þ; (A14) Gab ¼ d þ sin d’ : (A7) ð Þ¼ð We impose that, when evaluated at t; r; ; ’ t;^ a; wþðt^Þ¼wðt^Þ; (A15) ;’Þ, the vector coincides with ðt; r; ; ’Þ,so that in the new coordinate system Eqs. (3.27) will be valid. 2hþðaÞzþðt^ÞþaKþðt;^ aÞ¼2hðt^Þzðt^ÞþaKðt;^ aÞ: Such a transformation leads to the following changes in the (A16) metric components: 2 0 3 In the t and t’ directions we have, in addition, 2y_ f yy@ y@ 6 f ’ 7 ð Þ¼ ð Þ ð Þ¼ ð Þ 6 7 yþ t^ y t^ ;h0þ t;^ a h0 t;^ a ; (A17) ð Þg ¼ 6 7Y ; 0 4 5 lm (A8) while in the tt direction the matching condition is auto- matically satisfied, with 2 3 0 ð Þ¼ ð Þ f hz z_ g tt t;^ a; ;’ f a ; (A18) 6 0 7 6 h zz@ z@ 7 ¼ 6 h ’ 7 accurate up to first order in the perturbations. This is a ð1Þg 4 5Ylm; 2rhz consistency check, since we have imposed that the four- 2 2rhzsin velocity of mass elements on the shell is (A9) qffiffiffiffiffiffiffiffiffi u ¼ð1= fðaÞ; 0; 0; 0Þ; (A19) which are purely polar perturbations, 2 3 wY_ wY_ which should have a norm of 1. In simplified form, we 6 lm; lm;’ 7 6 2w Y 2w Y 7 have 6 r lm; r lm;’ 7 ð2Þg ¼ ; (A10) 4 2wY j 2wY j 5 ½½ ¼ ½½ ¼ ½½ ¼ ½½ ¼ ½½ þ lm lm ’ x w y h0 2hz aK 2wYlmj’’ ¼½½2y_ þ fH f0hz ¼ 0: (A20) which are also polar perturbations, and finally The symbol ½½..., as defined by Eq. (2.5), gives the jump 2 j j 3 x_ Y ’ x_ Y of any given quantity across the shell. 6 ’ lm ’ lm 7 6 2x j’ 2x j 7 ¼ 6 r ’Ylm r ’Ylm 7 The four-velocity of mass elements on the shell is given ð3Þg 4 5; x x’ by Eq. (A19). The surface stress-energy tensor of the shell x’ x’’ is

(A11) Sjk ¼½ þð ÞYlmujuk which are axial perturbations. Here we have deﬁned ½ þ Ylmjk; (A21) ¼ a þ a mn m Ylmjna n Ylmjma; (A12) where j and k go through t, and ’. Now let us try to

124047-14 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) 2 3 evaluate the extrinsic curvature of the shell at the location fðaÞ ð Þ 6 7 t;^ a; ;’ . First of all, we note that k k¼6 2 7 pffiffiffiffiffiffiffi Sjk 4 a 5; n ¼ð0; 1; 0; 0Þ= grr; a2sin2 2 3 2f and the extrinsic curvature is 2 6 a 7 pffiffiffiffiffiffiffiffiffi6 0 7 ¼r ¼ þ 6 af 7 Kij inj nj;i n; kK k¼a hðaÞ ½1 þ ; ji jk 46 2f 57 0 where i, j run through t, , ’. Since n only has a nonzero ½ þ af 2 1 2f sin r-component the first term vanishes, and we obtain ¼ 1 where j, k t, , ’. From this we have relations (2.9) for K ¼ pffiffiffiffiffiffiffi r : static gravastars. ij grr ij Now let us focus on first-order quantities. The tensors From the static configuration of the star, it is easy to Sjk and Kjk can each be decomposed into six terms, for obtain that example,

2 3 2 3 2 3 ð1Þ @ @ @ @ 6 S 7 6 ’ 7 6 csc ’ sin 7 4 ð2Þ 5 ð3Þ4 5 ð4Þ4 5 Sjk ¼ S Ylm þ S @ Ylm þ S csc@’ Ylm Sð2Þsin2 @ sin@ 2 3 2 ’ 3 0 0 ð5Þ6 7 ð6Þ6 7 þ S 4 Yj Yj’ 5 þ S 4 ’ 5: (A22) Yj’ Yj’’ ’ ’’

We have pffiffiffi x_ 2h af0 h0 h_ K ð4Þ ¼ h 0 1 þ þ 0 1 ; (A32) Sð1Þ ¼ f þ½ð 2ÞðfH hzf0 þ 2y_ Þ; (A23) a 2f 2

ð2Þ pffiffiffi S ¼aðaK þ 2hz þ aÞ; (A24) 0 2 h az af K ð5Þ ¼ þ w 1 þ ; (A33) ð Þ a 2 2f S 3 ¼ðy þ w_ Þ; (A25) Sð4Þ ¼ ðh x_ Þ; (A26) pffiffiffi h x af0 0 K ð6Þ ¼ h 1 þ 1 þ ; (A34) 2 a 2f Sð5Þ ¼2w; (A27) ð4Þ ð6Þ where L ¼lðl þ 1Þ. Here K and K are axial quan- ð Þ Sð6Þ ¼x; (A28) tities. The junction condition on K 6 yields pffiffiffi and ½½ ¼ pffiffiffi hh1 0; (A35) f h pffiffiffi þ 2h ah0 ð1Þ ¼ ð 0Þþ L ½½ ¼ K H aK f hz 2 ; (A29) which together with h0 0 completes the junction a a conditions for axial perturbations (the junction condition pffiffiffi ð4Þ 0 on K yields an equation of motion for the variable x). For a2 h 2ðH_ þ z€Þ af 2K H ð Þ ð Þ K ð2Þ ¼ K0 H0 þ 1 þ 1 þ polar quantities, from ½½K 5 ¼ 8S 5 we have 2 f 2f a pffiffiffi ð Þ 0 þ 0 ½½ ¼ þ 2 h L þ 2f h fh hz 0: (A36) a2 af ½½ ð3Þ ¼ ð3Þ 0 0 0 2 00 Then matching K 8S gives us an equation of ff h 2ðf Þ h þ 2hff ð1;2Þ ð1;2Þ þ z ; (A30) motion for w, while matching ½½K ¼ 8S yields 2f2 pffiffiffi 0pffiffiffi pffiffiffi H 0 2h h 0 h K þ hz ¼ 8; h aH af a a2 a K ð3Þ ¼ 1 þ 2y þ az_ þ 1 þ w_ ; (A31) a 2 2f (A37)

124047-15 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009) pffiffiffi 2H_ pffiffiffi af0 H same problem occurs also in the case of QNMs of BHs, and h K0 H0 þ 1 h 1 þ f 2f a was solved by Leaver [75]. Leaver found a continued fraction relation that can be regarded as an implicit equa- h0 2h f00h f0h0 pffiffiffi þ þ hz ¼16: (A38) tion which identifies the quasinormal frequencies, thus a a2 f 2f circumventing the need to perform an integration out to The remaining equations are large values of r. This method was subsequently adapted pffiffiffi pffiffiffi pffiffiffi to the polar and axial oscillations of a star [47,50]. The RW ½½K ¼ 2½½ h hz=a ¼ 8 hz; (A39) equation, which describes the perturbed space-time outside pffiffiffi the gravastar, becomes 0 pffiffiffi pffiffiffi f h 2 ½½H ¼ hz ¼ 8ð 2Þ hz; (A40) d þ½ 2 ¼ f 2 ! Vout 0; (B1) dr ¼v2; (A41) with where v is defined as in Eq. (3.33). 2M lðl þ 1Þ 6M V ¼ 1 (B2) The formalism described above is more general than we out r r2 r3 need for a static thin-shell gravastar. It can easily be ¼ þ ð Þ adapted to more general horizonless space-times and to and the tortoise coordinate r r 2M ln r=2M 1 . We shall now write the solution of the RW equation in a static wormholes. For the Mazur-Mottolla gravastar, we have power-series form as follows. Defining z 1 R2=r, where r ¼ R is some point outside the shell of the grav- 0 2 ½½f astar, and introducing a function ðzÞ, related to ðrÞ by ¼ 0; ¼ pffiffiffiffiffiffiffiffiffi ; ½½f ¼ 0 ¼½½f00; ð Þ 16 f a ðrÞ¼ðr 2MÞi2M!ei!rðzÞðrÞðzÞ; (B3) 6M 12M2 ½½f0 ¼ ; ½½f02 ¼ : one finds that satisfies the differential equation a2 a4 d2 d ð þ þ 2 þ 3Þ þð þ þ 2Þ Using the equations of Sec. III A above together with these c0 c1z c2z c3z d0 d1z d2z junction conditions we obtain continuity conditions for the dz2 dz ½½ ¼ ½½ ¼ ½½ ¼ ½½ ¼ þð þ Þ ¼ shell position, x y w z 0, and the e0 e1z 0: (B4) matching conditions for the axial and polar perturbations functions presented in the main text [Eqs. (3.29), (3.30), The constants depend only on !, l, and R2 through the (3.31), and (3.32)]. relations ¼ 2M ¼ 6M ¼ 6M c0 1 ;c1 2;c2 1 ; APPENDIX B: THE CONTINUED FRACTION R2 R2 R2 METHOD ¼ 2M ¼ þ 6M c3 ;d0 2i!R2 2; Our numerical search for the QNMs of gravastars is R R 2 2 based on the continued fraction method, as modified in 6M 6M d ¼ 2 1 ;d¼ ; [47,50]. The QNMs of an oscillating gravastar are solu- 1 R 2 R tions of Eqs. (3.7) and (3.13) satisfying the boundary con- 2 2 ditions imposed by physical requirements: should be ¼ 6M ð þ Þ ¼6M e0 l l 1 ;e1 : regular at the origin, have the behavior of a purely outgoing R2 R2 wave at infinity, and satisfy the junction conditions dis- Let us now perform a power-series expansion of ðzÞ: cussed in Sec. III C. The QNM frequencies are the (complex) frequencies ! ¼ ! þ i! for which these X1 R R ð Þ¼ n requirements are satisfied. z anz : (B5) ¼ The numerical determination of the QNM frequencies is n 0 nontrivial, especially for modes with large imaginary parts By substituting this expression in Eq. (B4), the expansion (strongly damped modes). The reason is simple to under- coefficients an are found to satisfy a four-term recurrence stand. Solutions of Eq. (3.13) representing outgoing and relation of the form ingoing waves at infinity have the asymptotic behavior þ þ ¼ ¼ out r= in r= a a a 0;n1; e and e as r !1, where ¼ 1 2 1 1 1 0 þ þ þ ¼ 1=!I is the damping time. Therefore, identifying by nu- nanþ1 nan nan1 nan2 0;n2; merical integration the purely outgoing solutions (that is, (B6) those solutions for which in is zero) becomes increas- ingly difficult as the damping of the mode increases. The where

124047-16 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) ¼ ð þ Þ n n n 1 c0;n1; The elimination step is not as trivial as it may seem, because in the process one of the three independent solu- ¼ð Þ þ n n 1 nc1 nd0;n1; tions to Eq. (B6) is lost. It can be shown that this solution is ¼ð Þð Þ þð Þ þ n n 2 n 1 c2 n 1 d1 e0;n1; not relevant for our purposes [47]. ¼ð Þð Þ þð Þ þ We now turn to investigating the asymptotic behavior of n n 3 n 2 c3 n 2 d2 e1;n2: the coefﬁcients an in the expansion (B5). Let us make the (B7) ansatz

The coefficient a is a normalization constant, and it is a þ h k 0 lim n 1 ¼ 1 þ þ þ (B14) irrelevant from the point of view of imposing outgoing- !1 1=2 n an n n wave boundary conditions. The ratio a1=a0 can simply be 0 determined by imposing the continuity of and at r ¼ 2 3=2 Dividing Eq. (B6)byn an, keeping terms up to n , R2, since from Eq. (B3) it follows that and equating to zero the various terms in the expansion in ð Þ powers of n1=2 we find the relations ¼ j ¼ R2 a0 z¼0 ð Þ ; (B8) R2 þ þ þ ¼ þ ¼ c0 c1 c2 c3 0; 2c0 c1 c3 0; a R i!R 3 (B15) 1 ¼ 2 ½ 0ð Þþ 2 ð Þ h2 ¼ 2i!R ;k¼ þ i!ðR þ 2MÞ: ð Þ R2 R2 : (B9) 2 2 a0 R2 R2 2M 4 ð Þ 0ð Þ In the axial case, the values of R2 and R2 can be The first two of these equations are identities. Substituting obtained by the taking the interior solution (3.8)atr ¼ a the second pair of equations in Eq. (B14) we get and applying the junction conditions (3.29) to determine pffiffiffiffiffiffiffiffiffiffiffiffi the wave function in the exterior, i.e. at r ¼ aþ. From then 3=4þi!ðR þ2MÞ 2 2i!R2n lim an ¼ n 2 e : (B16) onward, we can numerically integrate the RW equat- n!1 ¼ ion (3.13)uptor R2. The remaining coefficients can then be determined by recursion from Eq. (B6). In the polar According to a definition given by Gautschi [77], the case we proceed in a similar way: we obtain the Zerilli solution of Eq. (B14) corresponding to the plus sign in out Eq. (B16) is said to be dominant, whereas that correspond- function Z and its derivative at r ¼ aþ by imposing the matching conditions (3.30), (3.31), and (3.32). Then we use ing to the minus sign is said to be minimal [77]. If we select Eq. (3.24) to obtain the corresponding RW function at r ¼ the minimal solution the expansion (B5) is absolutely and uniformly convergent outside the star, provided that we aþ, integrate forward to find a0 and a1, and finally obtain the remaining coefficients by recursion. choose R2 such that R2=2 2. Fur- To apply the continued fraction technique, it is easier to thermore, according to Eq. (B3), the solution to Eq. (B1) consider three-term recurrence relations. Leaver has shown behaves as a pure outgoing wave at infinity, i.e. it is the that the four-term recurrence relation (B6) can be reduced QNM wave function. Thus, the key point is to identify the to a three-term recurrence relation by a Gaussian elimina- minimal solutions of Eq. (B14). According to a theorem tion step [76]. Define due to Pincherle [77], if Eq. (B14) has a minimal solution then the following continued fraction relation holds: ¼ a1 ¼ ^ 0 ; ^ 0 1; (B10) a a1 ^ ^ ^ ^ ^ 0 ¼ 1 1 2 2 3 ...; (B17) a0 ^ ^ ^ where a1=a0 is obtained numerically from Eq. (B9). Now 1 2 3 set where the continued fraction on the right-hand side is ^ ^ n ¼ n; n ¼ n; ðn ¼ 0; 1Þ; convergent and completely determined since the coefficients ^ , ^ , and ^ ; defined in Eqs. (B11) and (B12) ^ ¼ ; ðn ¼ 1Þ; (B11) n n n n n are known functions of !. Moreover, from Eqs. (B9) and and for n 2 (B10) it is apparent that the dependence on the stellar model is all contained in the ratio a1=a0. Keeping in ^ ^ n1n mind the definitions (B10), Eq. (B17) can be recast in the ^ n ¼ n; n ¼ n ; ^ n1 form (B12) ^ n1n ^ ^ n ¼ n ; n ¼ 0: ^ 0^ 1 ^ 1^ 2 ^ 2^ 3 ^ 0 ¼ f ð!Þ¼^ ...: (B18) n 1 0 0 ^ ^ ^ 1 2 3 By this Gaussian elimination, Eq. (B6) reduces to Using the inversion properties of continued fractions [78], þ ^ þ ¼ ^ nanþ1 nan ^ nan1 0: (B13) the latter equation can be inverted n times to yield:

124047-17 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009)

0 ¼ fnð!Þ Solving for the metric quantities we find, up to dominant terms in a 2M, ^ ^ ^ ^ ^ ^ ^ ^ þ ¼ ^ n 1 n n 2 n 1 ... 0 1 n n 1 n ^ ^ ^ ^ ð þ Þþ n1 n2 0 nþ1 l l 1 2ia! KðaÞ¼ ðaÞ; (C4) ^ þ ^ þ ^ þ ^ þ 2a n 1 n 2 n 2 n 3 ... (B19) ^ ^ nþ2 nþ3 0 ilðl þ 1Þþ2!a ¼ K ðaÞ¼! ðaÞ; (C5) for n 1; 2; :::. These n conditions are analytically equiva- 2ða 2MÞ lent to Eq. (B18). However, since the functions fnð!Þ have different convergence properties, each of them is best suited to find the quasinormal frequencies in a given region !að!a iÞ H ðaÞ¼ ðaÞ; (C6) of the complex ! plane. This is the main reason for the 1 a 2M accuracy and flexibility of the continued fraction technique. 2 0 !ð!a iÞð4M þ i!a Þ H ðaÞ¼ ðaÞ; (C7) 1 ða 2MÞ2 APPENDIX C: HIGH-COMPACTNESS LIMIT To investigate the behavior at the surface of the gravastar ð Þ ð Þ¼!a !a i ð Þ ! H0 a a ; (C8) in the high-compactness limit, we use the z 1 z trans- a 2M formation law for the hypergeometric function [44], ð Þð þ 2Þ ðcÞða þ b cÞ 0 ! !a 2i 2M i!a Fða; b; c; zÞ¼ð1 zÞc a b H ðaÞ¼ ðaÞ: (C9) ðaÞðbÞ 0 ða 2MÞ2 ð þ Þ F c a; c b; c a b 1; 1 z In the exterior we get ðcÞðc a bÞ þ lðl þ 1Þþ2ia! ðc aÞðc bÞ KðaþÞ¼ ðaÞ; (C10) 2a Fða; b; c þ a þ b þ 1; 1 zÞ: (C1) ð þ Þþ Using Eq. (3.8) in the limit when C ! 1 and r ! a we get 0 il l 1 4!a K ðaþÞ¼! ðaÞ; (C11) 2ða 2MÞ 2ða rÞ iM! ðl þ 3Þði2M!Þ 2 a ð2þli2M!Þð1þli2M!Þ 2 2 !Mði þ 4M!Þ H ðaþÞ¼ ðaÞ; (C12) 2ða rÞ iM! ðl þ 3Þði2M!Þ 1 a 2M þ 2 : (C2) ð1þlþi2M!Þ ð2þlþi2M!Þ a 2 2 0 M!ð2M! þ iÞð1 4iM!Þ H ðaþÞ¼ ðaÞ; (C13) Within our conventions the first term is ingoing, while the 1 ða 2MÞ2 second term is outgoing near the surface. So it is clear that in this regime both in- and outgoing modes are present, and QNMs do not reduce to the Schwarzschild QNMs (which M!ði þ 4M!Þ H ðaþÞ¼ ðaÞ; (C14) require only ingoing waves). Furthermore, we can clearly 0 a 2M see that ingoing and outgoing waves always have the same magnitude: the gravastar appears like a reflecting object, as 0 0 H ðaþÞ¼H ðaþÞ: (C15) suggested by [79]. Because this reflection happens in a 0 1 polar coordinate system, it can simply be interpreted as due ¼ Notice that, even though H1 is not continuous at r a, the to the fact that waves going into a lossless gravastar will Zerilli function is. Indeed, we get reemerge without loss. Nevertheless, such a behavior al- out ready supports the conclusions of Ref. [60,80], which Z ðaþÞ¼ðaÞ; ðaÞ¼ðaþÞ: (C16) showed that (for scalar fields) the ergoregion instability is more effective when the surfaces of the compact objects Thus, we conclude that in the high-compactness limit, the behave like a ‘‘perfect mirror’’ in this sense. master wave function for polar perturbations is continuous It is easy to show that in the high-compactness limit across the shell. A trivial extension of the known Schwarzschild results then shows that polar and axial 0 i!a perturbations are isospectral for large compactness, i.e. ðaÞ¼ ðaÞ: (C3) a 2M when a ! 2M and ! 1=2.

124047-18 GRAVITATIONAL WAVE SIGNATURES OF THE ABSENCE ... PHYSICAL REVIEW D 80, 124047 (2009) [1] F. Melia, arXiv:0705.1537. [39] T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 [2] R. Narayan, New J. Phys. 7, 199 (2005). (1957). [3] A. Mueller, Proc. Sci. P2GC (2006) 017 [arXiv:astro-ph/ [40] L. Lindblom and S. L. Detweiler, Astrophys. J. Suppl. Ser.. 0701228]. 53, 73 (1983). [4] E. Berti, V. Cardoso, and A. O. Starinets, Classical [41] S. Detweiler and L. Lindblom, Astrophys. J. 292,12 Quantum Gravity 26, 163001 (2009). (1985). [5] A. M. Ghez et al., Astrophys. J. 689, 1044 (2008). [42] S. Chandrasekhar and V. Ferrari, Proc. R. Soc. A 432, 247 [6] S. Doeleman et al., Nature (London) 455, 78 (2008). (1991). [7] R. Schodel et al., Nature (London) 419, 694 (2002). [43] V. Cardoso and M. Cavaglia, Phys. Rev. D 74, 024027 [8] E. Maoz, Astrophys. J. 494, L181 (1998). (2006). [9] M. Coleman Miller, Mon. Not. R. Astron. Soc. Lett. 367, [44] M. Abramowitz and I. A. Stegun, Handbook of L32 (2006). Mathematical Functions with Formulas, Graphs, and [10] A. E. Broderick, A. Loeb, and R. Narayan, Astrophys. J. Mathematical Tables (Dover, New York, 1972). 701, 1357 (2009). [45] F. J. Zerilli, Phys. Rev. D 2, 2141 (1970). [11] D. Psaltis, arXiv:0806.1531. [46] S. Chandrasekhar, The Mathematical Theory of Black [12] M. A. Abramowicz, W. Kluzniak, and J.-P. Lasota, Astron. Holes (Clarendon, Oxford, 1992). Astrophys. 396, L31 (2002). [47] M. Leins, H. P. Nollert, and M. H. Soffel, Phys. Rev. D 48, [13] W. Kundt, Gen. Relativ. Gravit. 41, 1967 (2009). 3467 (1993). [14] M. Visser, arXiv:0901.4365. [48] M. Visser, Lorentzian Wormholes: From Einstein to [15] F. D. Ryan, Phys. Rev. D 52, 5707 (1995). Hawking (AIP Publishing, Melville, NY, 1995). [16] F. D. Ryan, Phys. Rev. D 56, 1845 (1997). [49] E. Poisson and M. Visser, Phys. Rev. D 52, 7318 [17] C. Li and G. Lovelace, Phys. Rev. D 77, 064022 (2008). (1995). [18] N. A. Collins and S. A. Hughes, Phys. Rev. D 69, 124022 [50] O. Benhar, E. Berti, and V. Ferrari, Mon. Not. R. Astron. (2004). Soc. 310, 797 (1999). [19] K. Glampedakis and S. Babak, Classical Quantum Gravity [51] K. D. Kokkotas, Mon. Not. R. Astron. Soc. 268, 1015 23, 4167 (2006). (1994). [20] L. Barack and C. Cutler, Phys. Rev. D 75, 042003 [52] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, (2007). Numerical Recipes in C++ (Cambridge University Press, [21] O. Dreyer et al., Classical Quantum Gravity 21, 787 Cambridge, England, 2007), 3rd ed.. (2004). [53] N. Andersson, Y. Kojima, and K. D. Kokkotas, Astrophys. [22] E. Berti, V. Cardoso, and C. M. Will, Phys. Rev. D 73, J. 462, 855 (1996). 064030 (2006). [54] K. D. Kokkotas and B. F. Schutz, Gen. Relativ. Gravit. 18, [23] E. Berti, J. Cardoso, V. Cardoso, and M. Cavaglia, Phys. 913 (1986). Rev. D 76, 104044 (2007). [55] N. Andersson, Gen. Relativ. Gravit. 28, 1433 (1996). [24] B. S. Sathyaprakash and B. F. Schutz, Living Rev. [56] E. Berti, Laurea thesis, 1998 (unpublished). Relativity 12, 2 (2009). [57] S. Chandrasekhar and V. Ferrari, Proc. R. Soc. A 434, 449 [25] F. E. Schunck and E. W. Mielke, Classical Quantum (1991). Gravity 20, R301 (2003). [58] S. Chandrasekhar and V. Ferrari, Proc. R. Soc. A 437, 133 [26] D. F. Torres, S. Capozziello, and G. Lambiase, Phys. Rev. (1992). D 62, 104012 (2000). [59] E. Berti, V. Cardoso, and P. Pani, Phys. Rev. D 79, 101501 [27] F. S. Guzman, Phys. Rev. D 73, 021501 (2006). (2009). [28] M. Kesden, J. Gair, and M. Kamionkowski, Phys. Rev. D [60] V. Cardoso, P. Pani, M. Cadoni, and M. Cavaglia, Phys. 71, 044015 (2005). Rev. D 77, 124044 (2008). [29] S. Yoshida, Y. Eriguchi, and T. Futamase, Phys. Rev. D 50, [61] C. B. M. H. Chirenti and L. Rezzolla, Phys. Rev. D 78, 6235 (1994). 084011 (2008). [30] J. Balakrishna, R. Bondarescu, G. Daues, F. Siddhartha [62] K. D. Kokkotas, J. Ruoff, and N. Andersson, Phys. Rev. D Guzman, and E. Seidel, Classical Quantum Gravity 23, 70, 043003 (2004). 2631 (2006). [63] T. Damour and S. N. Solodukhin, Phys. Rev. D 76, 024016 [31] E. Berti and V. Cardoso, Int. J. Mod. Phys. D 15, 2209 (2007). (2006). [64] Z. Andrade and R. H. Price, Phys. Rev. D 60, 104037 [32] P.O. Mazur and E. Mottola, arXiv:gr-qc/0109035. (1999). [33] G. Chapline, E. Hohlfeld, R. B. Laughlin, and D. I. [65] V. Ferrari and K. D. Kokkotas, Phys. Rev. D 62, 107504 Santiago, Int. J. Mod. Phys. A 18, 3587 (2003). (2000). [34] P.O. Mazur and E. Mottola, Proc. Natl. Acad. Sci. U.S.A. [66] E. Poisson and M. Sasaki, Phys. Rev. D 51, 5753 (1995). 101, 9545 (2004). [67] Y. Kojima, Prog. Theor. Phys. 77, 297 (1987). [35] M. Visser and D. L. Wiltshire, Classical Quantum Gravity [68] L. Gualtieri, E. Berti, J. A. Pons, G. Miniutti, and V. 21, 1135 (2004). Ferrari, Phys. Rev. D 64, 104007 (2001). [36] C. B. M. H. Chirenti and L. Rezzolla, Classical Quantum [69] J. A. Pons, E. Berti, L. Gualtieri, G. Miniutti, and V. Gravity 24, 4191 (2007). Ferrari, Phys. Rev. D 65, 104021 (2002). [37] P.P. Fiziev, Classical Quantum Gravity 23, 2447 (2006). [70] E. Berti, J. A. Pons, G. Miniutti, L. Gualtieri, and V. [38] W. Israel, Nuovo Cimento B 44, 1 (1966). Ferrari, Phys. Rev. D 66, 064013 (2002).

124047-19 PANI et al. PHYSICAL REVIEW D 80, 124047 (2009) [71] T. Tanaka, M. Shibata, M. Sasaki, H. Tagoshi, and T. [76] E. W. Leaver, Phys. Rev. D 41, 2986 (1990). Nakamura, Prog. Theor. Phys. 90, 65 (1993). [77] W. Gautschi, SIAM Rev. 9, 24 (1967). [72] E. Poisson, Phys. Rev. D 47, 1497 (1993). [78] H. S. Wall, Analytic Theory of Continued Fractions (Van [73] C. Cutler, E. Poisson, G. J. Sussman, and L. S. Finn, Phys. Nostrand, Princeton, 1948). Rev. D 47, 1511 (1993). [79] H. Pﬁster, Classical Quantum Gravity 13, 2267 (1996). [74] C. Cutler, D. Kenneﬁck, and E. Poisson, Phys. Rev. D 50, [80] V. Cardoso, P. Pani, M. Cadoni, and M. Cavaglia, Classical 3816 (1994). Quantum Gravity 25, 195010 (2008). [75] E. W. Leaver, Proc. R. Soc. A 402, 285 (1985).

124047-20