Ergoregion instability of black hole mimickers
aa r X i v : . [ g r- q c ] J a n Ergoregion instability of black hole mimickers ∗ Paolo Pani † Dipartimento di Fisica, Universit`a di Cagliari, and INFN sezione di Cagliari, Cittadella Universitaria 09042 Monserrato, ItalyCurrently at Centro Multidisciplinar de Astrof´ısica - CENTRA, Dept. de F´ısica,Instituto Superior T´ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Vitor Cardoso ‡ Centro Multidisciplinar de Astrof´ısica - CENTRA, Dept. de F´ısica,Instituto Superior T´ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal &Department of Physics and Astronomy, The University of Mississippi, University, MS 38677-1848, USA
Mariano Cadoni § Dipartimento di Fisica, Universit`a di Cagliari, and INFN sezione di Cagliari, Cittadella Universitaria 09042 Monserrato, Italy
Marco Cavagli`a ¶ Department of Physics and Astronomy, The University of Mississippi, University, MS 38677-1848, USA
Ultra-compact, horizonless objects such as gravastars, boson stars, wormholes and superspinarscan mimick most of the properties of black holes. Here we show that these “black hole mimickers”will most likely develop a strong ergoregion instability when rapidly spinning. Instability timescalesrange between ∼ − s and ∼ weeks depending on the object, its mass and its angular momentum.For a wide range of parameters the instability is truly effective. This provides a strong indicationthat astrophysical ultra-compact objects with large rotation are black holes. I. INTRODUCTION
Black holes (BHs) in Einstein-Maxwell theory are char-acterized by three parameters [1]: mass M , electriccharge Q and angular momentum J ≡ aM M .BHs are thought to be abundant objects in the Uni-verse. Their mass is estimated to vary between 3 M ⊙ and 10 . M ⊙ or higher [2], their electrical charge is neg-ligible because of the effect of surrounding plasma [3] andtheir angular momentum is expected to be close to theextremal limit because of accretion and merger events[4]. A non-comprehensive list of some astrophysical BHcandidates [2, 5, 6, 7] is shown in Table I.Despite the wealth of circumstantial evidence, thereis no definite observational proof of the existence of as-trophysical BHs due to the difficulty to detect an eventhorizon in astrophysical BH candidates [2, 8]. Thus as-trophysical objects without event horizon, yet observa-tionally indistinguishable from BHs, cannot be excludeda priori. Some of the most viable alternative modelsdescribing an ultra-compact astrophysical object includegravastars, boson stars, wormholes and superspinars.Dark energy stars or gravastars are compact objects withde Sitter interior and Schwarzschild exterior [9]. These ∗ Black Holes in General Relativity and String TheoryAugust 24-30 2008, Veli Loˇsinj, Croatia † Electronic address: [email protected] ‡ Electronic address: vcardoso@fisica.ist.utl.pt § Electronic address: [email protected] ¶ Electronic address: [email protected] two regions are glued together by a model-dependent in-termediate region. In the original model [9] the inter-mediate region is an ultra-stiff thin shell. Models with-out shells or discontinuities have also been investigated[10, 11].
Boson stars are macroscopic quantum states which areprevented from undergoing complete gravitational col-lapse by Heisenberg uncertainty principle [12]. Theirmodels differ in the scalar self-interaction potential whichalso set the allowed maximum compactness for a bosonstar.An exhaustive description of wormholes can be found inthe monograph [13] (see also Ref. [14]). In this work weshall consider particular wormholes which are infinitesi-mal variations of BH spacetimes. These wormholes maybe indistinguishable from ordinary BHs [15].
Superspinars are solutions of the gravitational field equa-tions that violate the Kerr bound. These geometriescould be created by high energy corrections to Einsteingravity such as those present in string-inspired models[16].
TABLE I: Mass, M , radius, R , angular momentum, J , andcompactness, µ = M/R , for some BH candidates (from [2, 5,6, 7]). Mass and radius are in solar units.
Candidate
M R × − J/M µ = M/R
GRO J1655-40 6 . . − . . − .
80 0 . − . . − . . − .
00 0 . − . . − . . − .
00 0 . − . × .
27 0 . − . & . The objects described above can be almost as com-pact as a BH and thus they are virtually indistinguish-able from BHs in the Newtonian regime, hence the name“BH mimickers”. Although exotic these objects provideviable alternatives to astrophysical BHs. BH mimick-ers being horizonless, no information loss paradox [17]arises in these spacetimes. Moreover they can be regu-lar at the origin, avoiding the problem of singularities.By Birkhoff’s theorem, the vacuum exterior of a spheri-cally symmetric object is described by the Schwarzschildspacetime. Thus the motion of orbiting objects botharound a static BH and around a static ultra-compactobject is the same and it makes virtually impossible todiscern between a Schwarzschild BH and a static neutralBH mimicker. Instead for rotating objects deviations inthe properties of orbiting objects occur. Since BH mim-ickers are very compact these deviations occur close tothe horizon and are not easily detectable electromagnet-ically. To ascertain the true nature of ultra-compact ob-jects it is thus important to devise observational tests todistinguish rotating BH mimickers from ordinary KerrBHs. The traditional way to distinguish a BH from aneutron star is to measure its mass. If the latter is largerthan the Chandrasekhar limit, the object is believed tobe a BH. However, this method cannot be used for theBH mimickers discussed above, because of their broadmass spectrum. The main difference between a BH anda BH mimicker is the presence of an event horizon in theformer. Some indirect experimental methods to detectthe event horizon has been proposed [18, 19]. Anothervery promising observational method to probe the struc-ture of ultra-compact objects is gravitational wave as-tronomy. From the gravitational waveform it is expectedto detect the presence of an event horizon in the source[20]. Some other BH mimickers (for example electricallycharged quasi-BHs [21]) are already ruled out by exper-iments. Moreover there are evidences that some modelfor BH mimickers is plagued by a singular behavior inthe near-horizon limit [22].Here, we describe a method originally proposed in[23, 24] for discriminating rotating BH mimickers fromordinary BHs. This method uses the fact that compactrotating objects without event horizon are unstable whenan ergoregion is present. This ergoregion instability ap-pears in any system with ergoregions and no horizons[25]. The origin of this instability can be traced back tosuperradiant scattering. In a scattering process, super-radiance occurs when scattered waves have amplitudeslarger than incident waves. This leads to extraction ofenergy from the scattering body [26, 27, 28]. Instabilitymay arise whenever this process is allowed to repeat itselfad infinitum. This happens, for example, when a BH issurrounded by a “mirror” that scatters the superradiantwave back to the horizon, amplifying it at each scatter-ing, as in the
BH bomb process [29, 30]. If the mirroris inside the ergoregion, superradiance may lead to aninverted BH bomb. Some superradiant waves escape toinfinity carrying positive energy, causing the energy in-side the ergoregion to decrease and eventually generating an instability. This may occur for any rotating star withan ergoregion: the mirror can be either its surface or,for a star made of matter non-interacting with the wave,its center. On the other hand BHs could be stable dueto the absorption by the event horizon being larger thansuperradiant amplification. Indeed Kerr BHs are stableaganist small scalar, electromagnetic and gravitationalperturbations [31].Rapidly rotating stars do possess an ergoregion andthus they are unstable. However typical instabilitytimescales are shown to be larger than the Hubble time[32]. Thus the ergoregion instability is too weak to pro-duce any effect on the evolution of stars. This conclusionchanges drastically for BH mimickers due to their com-pactness [23, 24]. For some of the rotating BH mimick-ers described above, instability timescales range between ∼ − s and ∼ weeks depending on the object, its massand its angular momentum.This paper is organized as follows. In Section II wedeal with gravastars and boson stars. We describe rotat-ing models for these objects and discuss their instabilitytimescale. In Section III a toy model for both rotat-ing wormholes and superspinars is presented. SectionIV contains a brief discussion of the results and con-cludes the paper. Throughout the paper geometrizedunits ( G = c = 1) are used, except during the discus-sion of results for rotating boson stars when we set theNewton constant to be G = 0 . / (4 π ) as in Ref. [33]. II. GRAVASTARS AND BOSON STARS
This section discusses the main properties of gravas-tars and boson stars as well as the method to computethe ergoregion instability for these objects. For a moredetailed discussion see [23].
A. Nonrotating Gravastars
Although exact solutions for spinning gravastars arenot known, they can be studied in the limit of slow rota-tion by perturbing the nonrotating solutions [34]. Thisprocedure was used in Ref. [35] to study the existenceof ergoregions for ordinary rotating stars with uniformdensity. In the following, we omit the discussion for theoriginal thin-shell model by Mazur and Mottola [9] andwe focus on the anisotropic fluid model by Chirenti andRezzolla [10, 11].The model assumes a thick shell with continuous pro-file of anisotropic pressure to avoid the introduction ofan infinitesimally thin shell. The stress-energy tensor is T µν = diag[ − ρ, p r , p t , p t ], where p r and p t are the ra-dial and tangential pressures, respectively. The sphericalsymmetric metric is ds = − f ( r ) dt + B ( r ) dr + r d Ω (2.1)and it consists of three regions: an interior ( r < r )described by a de Sitter metric, an exterior ( r > r )described by the Schwarzschild metric and a model-dependent intermediate ( r < r < r ) region. In thefollowing we shall indicate with δ = r − r the thicknessof the intermediate region and with µ = M/r the com-pactness of the gravastar. In the model by Chirenti andRezzolla the density function is ρ ( r ) = ρ , ≤ r ≤ r interior ar + br + cr + d , r < r < r intermediate0 , r ≤ r exteriorwhere a , b , c and d are found imposing continuity condi-tions ρ (0) = ρ ( r ) = ρ , ρ ( r ) = ρ ′ ( r ) = ρ ′ ( r ) = 0 and ρ is found fixing the total mass, M. The metric coeffi-cients are f = (cid:18) − Mr (cid:19) e Γ( r ) − Γ( r ) , B = 1 − m ( r ) r , (2.2)where m ( r ) = Z r πr ρdr , Γ( r ) = Z r m ( r ) + 8 πr p r r ( r − m ( r )) dr . (2.3)The above equations and some closure relation, p r = p r ( ρ ), completely determine the structure of the gravas-tar [10]. The behaviors of the metric coefficients for atypical gravastar are shown in Fig. 1. M e t r i c r/r B(r)
FIG. 1: Metric coefficients for the anisotropic pressure model( r = 2 . r = 1 . M = 1).
1. Slowly rotating gravastars and ergoregions
Slowly rotating solutions can be obtained using themethod developed in Ref. [34]. A rotation of order Ωgives corrections of order Ω in the diagonal coefficientsof the metric (2.1) and introduces a non-diagonal term of order Ω, g tφ ≡ − ωg φφ , where φ is the azimuthal co-ordinate and ω = ω ( r ) is the angular velocity of framedragging. The full metric is ds = − f dt + Bdr + r dθ + r sin θ ( dφ − ωdt ) , (2.4)where f , B and ω are radial functions. If the gravastarrotates rigidly, i.e. Ω = constant, from the ( t, φ ) compo-nent of Einstein equations we find a differential equationfor ω ( r ) [23] ω ′′ + ω ′ (cid:18) r + j ′ j (cid:19) = 16 πB ( r )( ω − Ω) ( ρ + p t ) , (2.5)where j ≡ ( f B ) − / is evaluated at zeroth order and ρ , p t are given in terms of the nonrotating geometry. Theabove equation reduces to the corresponding equationfor isotropic fluids [34]. Solutions of Eq. (2.5) describerotating gravastars to first order in Ω.The ergoregion can be found by computing the surfaceon which g tt vanishes [35]. An approximated relation forthe location of the ergoregion in very compact gravastarsis 0 = − f ( r ) + ω r sin θ . (2.6)The existence and the boundaries of the ergoregions canbe computed from the above equations. We integrateequation (2.5) from the origin with initial conditions(Ω − ω ) ′ = 0 and (Ω − ω ) finite. The exterior solutionsatisfies ω = 2 J/r , where J is the angular momentum ofthe gravastar. Demanding the continuity of both (Ω − ω ) ′ and (Ω − ω ), Ω and J are uniquely determined. The rota-tion parameter Ω depends on the initial condition at theorigin. Figure 2 shows the results the gravastar model r/r J/M FIG. 2:
J/M and angular frequency Ω for the anisotropicpressure model with r = 2 . r = 1 . M = 1. described in the previous sections. The ergoregion canbe located by drawing an horizontal line at the desiredvalue of J/M . The minimum of the curve is the mini-mum values of J/M which are required for the existenceof the ergoregion. Comparison with the results for starsof uniform density [35], shows that ergoregions form moreeasily around gravastars due to their higher compactness.The slow-rotation approximation is considered valid forΩ / Ω K < M Ω K = µ / is the Keplerian fre-quency.Depending on the compactness, µ , the angular momen-tum, J , and the thickness, δ , a spinning gravastar does ordoes not develop an ergoregion. The formation of an er-goregion for rotating gravastar is exhaustively discussedin the whole parameters space in Ref. [36]. A delicateissue is the strong dependence on the thickness, δ , whichcannot be directly measured by experiments. Figure 3shows how the ergoregion width is sensitive to δ . ∆ (cid:144) M FIG. 3: Ergoregion width (in units of M) as function of thethickness, δ = r − r , for r = 2 . M = 1 and for different J values. From top to bottom: J/M = 0 .
95, 0 .
90, 0 .
85, 0 . .
75, 0 .
70, 0 .
65 and 0 .
60. The ergoregion width decreases as δ → B. Rotating boson stars
A example of rotating boson star is the model by Klei-haus, Kunz, List and Schaffer (KKLS) [33]. The KKLSsolution is based on the Lagrangian for a self-interactingcomplex scalar field L KKLS = − g µν (cid:0) Φ ∗ , µ Φ , ν + Φ ∗ , ν Φ , µ (cid:1) − U ( | Φ | ) , (2.7)where U ( | Φ | ) = λ | Φ | ( | Φ | − a | Φ | + b ). The mass ofthe boson is given by m B = √ λb . The ansatz for theaxisymmetric spacetime is ds = − f dt + kgf (cid:20) dr + r dθ + r sin θg ( dϕ − ζ ( r ) dt ) (cid:21) (2.8)and Φ = φ e iω s t + inϕ , where the metric components andthe real function φ depend only on r and θ . The require-ment that Φ is single-valued implies n = 0 , ± , ± , . . . .The solution has spherical symmetry for n = 0 and axial symmetry otherwise. Since the Lagrangian density is in-variant under a global U (1) transformation, the current, j µ = − i Φ ∗ ∂ µ Φ+c . c . , is conserved and it is associated to acharge Q , satisfying the quantization condition with theangular momentum J = nQ [37]. The numerical proce-dure to extract the metric and the scalar field is describedin Ref. [33]. Throughout the paper we will consider solu-tions with n = 2, b = 1 . λ = 1 . a = 2 . J , M ) corresponding to J/ ( GM ) ∼ . .
731 and 0 . g tt one can prove that boson starsdevelop ergoregions deeply inside the star. For this par-ticular choice of parameters, the ergoregion extends from r/ ( GM ) ∼ . . r/(GM) g(r)l(r)f(r) (r) r/(GM) f/f g/gl/l/ FIG. 4: Left panel: Metric coefficients for a rotating bo-son star along the equatorial plane, with parameters n = 2, b = 1 . λ = 1 . a = 2 . J/ ( GM ) ∼ . θ = π/ θ = π/ C. Ergoregion instability for rotating gravastarsand boson stars
The stability of gravastars and boson stars can be stud-ied perturbatively by considering small deviations aroundequilibrium. Due to the difficulty of handling gravita-tional perturbations for rotating objects, the calculationsbelow are mostly restricted to scalar perturbations. How- ever the equation for axial gravitational perturbations ofgravastars is identical to the equation for scalar pertur-bations in the large l = m limit [23]. There are alsogeneric arguments suggesting that the timescale of grav-itational perturbations is smaller than the timescale ofscalar perturbations for low m [38]. Thus, scalar pertur-bations should provide a lower bound on the strength ofthe instability. TABLE II: WKB results for the instability of rotating gravastars with r = 2 . r = 1 . M = 1. τ /MJ/M = 0 . J/M = 0 . J/M = 0 . J/M = 0 . J/M = 1 . m Ω / Ω K = 0 .
33 Ω / Ω K = 0 .
49 Ω / Ω K = 0 .
65 Ω / Ω K = 0 .
74 Ω / Ω K = 0 .
821 1 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . ×
1. Scalar field instability for slowly rotating gravastars:WKB approach
Consider now a minimally coupled scalar field in thebackground of a gravastar. The metric of gravastars isgiven by Eq. (2.4). In the large l = m limit, which isappropriate for a WKB analysis [32, 39], the scalar fieldcan be expanded in spherical armonics, Y lm = Y lm ( θ , φ )asΦ = X lm ¯ χ lm exp (cid:20) − Z (cid:18) r + f ′ f + B ′ B (cid:19) dr (cid:21) e − iωt Y lm . (2.9)The functions ¯ χ lm = ¯ χ lm ( r ) are determined by theKlein-Gordon equation which, dropping terms of order O (cid:0) /m (cid:1) , yields¯ χ ′′ lm + m T ( r , Σ) ¯ χ lm = 0 , (2.10)where Σ ≡ − ω/m and T = B ( r ) f ( r ) (Σ − V + ) (Σ − V − ) , V ± = − ω ± p f ( r ) r . Equation (2.10) can be shown to be identical for the axialgravitational perturbations of perfect fluid stars [24].The WKB method [32] for computing the eigenfre-quencies of Eq. (2.10) is in excellent agreement with fullnumerical results [39]. The quasi-bound unstable modesare determined by m Z r b r a p T ( r ) dr = π nπ , n = 0 , , , . . . (2.11) and have an instability timescale τ = 4 exp (cid:20) m Z r c r b p | T | dr (cid:21) Z r b r a dd Σ √ T dr , (2.12)where r a , r b are solutions of V + = Σ and r c is determinedby the condition V − = Σ.Table II shows the WKB results for the anisotropicpressure model for different values of J/M . Althoughthe WKB approximation breaks down at low m values,these results still provide reliable estimates [32]. Thisclaim has be verified with a full numerical integration ofthe Klein-Gordon equation. The results show that theinstability timescale decreases as the star becomes morecompact. Larger values of J/M make the star more un-stable. The maximum growth time of the instability canbe of the order of a few thousand M , but it crucially de-pends on J , µ and δ [36]. For a large range of parametersthis instability is crucial for the star evolution. Gravita-tional perturbations are expected to be more unstable.Moreover it is worth to notice that the slowly rotatingapproximation allows only for µ < .
5, while for rotatingBHs 0 . < µ < µ , we expectthat instability timescales for realistic gravastars shouldbe much shorter than the ones computed. For most of theBH mimickers models to be viable we require J/M ∼ µ ∼
1. It would be interesting to study whetherthe ergoregion instability is or is not always effective inthis case. Possible future developments include: (i) a fullrotating gravastar model, which allows for µ > .
5; (ii)the stability analysis against gravitational perturbationsfor rotating gravastars; (iii) a gravavastar model which isnot strongly dependent on the thickness, δ .The ergoregion instability of a rotating boson staris straightforwardly computed following the method de-scribed above for spinning gravastars. We refer the readerto [23] and we only summarize the results in Table III.The maximum growth time for this boson star model isof the order of 10 M for J/GM = 0 . TABLE III: Instability for rotating boson stars with parame-ters n = 2, b = 1 . λ = 1 . a = 2 . J (from [23]). The Newton constant is defined as 4 πG = 0 . τ / ( GM ) m J/GM = 0 . J/GM = 0 . J/GM = 0 . . × . × − . × . × . × . × . . × . × . × . × . × . × . × . × III. A TOY MODEL FOR KERR-LIKE OBJECTS
This section discusses Kerr-like objects such as partic-ular solutions of rotating wormholes and superspinars. A rigorous analysis of the ergoregion instability for thesemodels is a non-trivial task. Indeed known wormholesolutions are special non-vacuum solutions of the gravi-tational field equations, thus their investigation requiresa case-by-case analysis of the stress-energy tensor. More-over exact solutions of four-dimensional superspinars arenot known. To overcome these difficulties, the follow-ing analysis will focus on a simple model which capturesthe essential features of most Kerr-like horizonless ultra-compact objects. Superspinars and rotating wormholeswill be modeled by the exterior Kerr metric down to theirsurface, where mirror-like boundary conditions are im-posed. This problem is very similar to Press and Teukol-sky’s “BH bomb” [29, 30], i.e. a rotating BH surroundedby a perfectly reflecting mirror with its horizon replacedby a reflecting surface. For a more detailed discussionsee [24].
2. Superspinars and Kerr-like wormholes
A superspinar of mass M and angular momentum J = aM can be modeled by the Kerr geometry [16] ds = − (cid:18) − M r Σ (cid:19) dt + Σ∆ dr + (cid:20) ( r + a )sin θ + 2 M r Σ a (cid:21) sin θdφ − M r Σ a sin θdφdt + Σ dθ , (3.1)where Σ = r + a cos θ and ∆ = r + a − M r . Un-like Kerr BHs, superspinars have a > M and no horizon.Since the domain of interest is −∞ < r < + ∞ , thespace-time possesses naked singularities and closed time-like curves in regions where g φφ < ds = ds + δg ab dx a dx b , (3.2)where δg ab is infinitesimal. In general, Eq. (3.2) de-scribes an horizonless object with a excision at somesmall distance of order ǫ from the would-be horizon[15]. Wormholes require exotic matter and/or divergentstress tensors, thus some ultra-stiff matter is assumedclose to the would-be horizon. In the following, bothsuperspinars and wormholes will be modeled by the Kerr metric with a rigid “wall” at finite Boyer-Lindquistradius r , which excludes the pathological region. A. Instability analysis
If the background geometry of superspinars and worm-holes is sufficiently close to the Kerr geometry, its per-turbations is determined by the equations of perturbedKerr BHs [24]. Thus the instability of superspinars andwormholes is studied by considering Kerr geometries witharbitrary rotation parameter a and a “mirror” at someBoyer-Lindquist radius r . Using the Kinnersley tetradand Boyer-Lindquist coordinates, it is possible to sepa-rate the angular variables from the radial ones, decou-pling all quantities. Small perturbations of a spin- s fieldare reduced to the radial and angular master equations[41]∆ − s ddr (cid:18) ∆ s +1 dR lm dr (cid:19) + (cid:20) K − is ( r − M ) K ∆ + 4 isωr − λ (cid:21) R lm = 0 , (3.3) (cid:2) (1 − x ) s S lm,x (cid:3) ,x + (cid:20) ( aωx ) − aωsx + s + s A lm − ( m + sx ) − x (cid:21) s S lm = 0 , (3.4)where x ≡ cos θ , ∆ = r − M r + a and K =( r + a ) ω − am . Scalar, electromagnetic and gravita-tional perturbations correspond to s = 0, ± ± λ and s A lm are relatedby λ ≡ s A lm + a ω − amω .
1. Analytic results I m () a=0.998M, s=2l=m=2 l=m=4 R e ()- m / m l=m=2l=m=3l=m=4a=0.998M, s=2 FIG. 5: Imaginary and real parts of the characteristic gravi-tational frequencies for an object with a = 0 . M , accordingto the analytic calculation for rapidly-spinning objects. Themirror location is at r = (1 + ǫ ) r + . The real part is ap-proximately constant and close to m Ω, in agreement with theassumptions used in the analytic approach.
Following Starobinsky [27], equations (3.3)-(3.4) canbe analytically solved in the slowly-rotating and low-frequency regime, ωM ≪
1, and in the rapidly-spinningregime, where r + ∼ r − and ω ∼ m Ω h , where Ω h ≡ a/ (2 M r + ) is the angular velocity at the horizon. Thedetails of the analytic approximation are described inRef. [24]. Analytic solutions for a star with a = 0 . M are shown in Fig. 5 where gravitational perturbations areconsidered. The instability timescale for gravitationalperturbations is about five orders of magnitude smallerthan the instability timescale for scalar perturbations. B. Instability analysis: numerical results
The oscillation frequencies of the modes can be foundfrom the canonical form of Eq. (3.3) d Ydr ∗ + V Y = 0 , (3.5)where Y = ∆ s/ ( r + a ) / R ,V = K − is ( r − M ) K + ∆(4 irωs − λ )( r + a ) − G − dGdr ∗ , and K = ( r + a ) ω − am , G = s ( r − M ) / ( r + a ) + r ∆( r + a ) − . The separation constant λ is related tothe eigenvalues of the angular equation by λ ≡ s A lm + a ω − amω . The eigenvalues s A lm are expanded inpower series of aω as [42] s A lm = X k =0 f ( k ) slm ( aω ) k . (3.6)Terms up to order ( aω ) are included in the calculation.Absence of ingoing waves at infinity implies Y ∼ r − s e iωr ∗ . (3.7)Numerical results are obtained by integrating Eq. (3.5)inward from a large distance r ∞ . The integration isperformed with the Runge-Kutta method with fixed ω starting at M r ∞ = 400, where the asymptotic behav-ior (3.7) is imposed. (Choosing a different initial pointdoes not affect the final results.) The numerical integra-tion is stopped at the radius of the mirror r , where thevalue of the field Y ( ω, r ) is extracted. The integrationis repeated for different values of ω until Y ( ω, r ) = 0 isobtained with the desired precision. If Y ( ω, r ) vanishes,the field satisfies the boundary condition for perfect re-flection and ω = ω is the oscillation frequency of themode.
1. Objects with a < M
The regime a < M requires a surface or mirror at r = r + (1 + ǫ ) > r + . Thus the compactness is M/r ∼ (1 − ǫ ) M/r + and, in the limit ǫ →
0, it is infinitesi-mally close to the compactness of a Kerr BH. Numericalresults for scalar and gravitational perturbations of ob-jects with a < M are summarized in Table IV and arein agreement with the analytic results [24]. The insta-bility is weaker for larger m . This result holds also for TABLE IV: Characteristic frequencies and instabilitytimescales for a Kerr-like object with a = 0 . M . The mir-ror is located at ǫ = 0 .
1, corresponding to the compactness µ ∼ . µ Kerr . (Re( ω ) M ,
Im( ω ) M ) l = m s = 0 s = 21 (0 . , . × − ) − . , . × − ) (0 . , . . , . × − ) (0 . , . . , . × − ) (1 . , . l = m and s = 0, ± ±
2. The minimum instabilitytimescale is of order τ ∼ M for a wide range of mir-ror locations. Figure 6 shows the results for gravitationalperturbations. Instability timescales are of the order of τ ∼ ÷ M . Thus gravitational perturbations lead to aninstability about five orders of magnitude stronger thanthe instability due to scalar perturbations (see Table IV).Figure 6 shows that the ergoregion instability remainsrelevant even for values of the angular momentum as lowas a = 0 . M . I m ( M ) l=m=2l=m=4l=m=3a=0.998M, s=2 R e ( M ) l=m=2l=m=3 l=m=4 a=0.998M, s=2 I m ( M ) a=0.6M a=0.8M a=0.9Ml=m=2, s=2 R e ( M ) a=0.6M a=0.8Ma=0.9Ml=m=2, s=2 FIG. 6: Details of the instability for gravitational perturbations, for different l = m modes and a/M = 0 .
998 (top panels) andfor l = m = 2 and different a/M <
2. Objects with a > M
Objects with a > M could potentially describe su-perspinars. Several arguments suggest that objects ro-tating above the Kerr bound are unstable. Firstly, ex-tremal Kerr BHs are marginally stable. Thus the ad-dition of extra rotation should lead to instability. Sec-ondly, fast-spinning objects usually take a pancake-likeform [43] and are subject to the Gregory-Laflamme in-stability [44, 45]. Finally, Kerr-like geometries, like nakedsingularities, seem to be unstable against a certain classof gravitational perturbations [46, 47] called algebraically special perturbations [40]. For objects with a > M thesurface or mirror can be placed anywhere outside r = 0.In general the instability is as strong as in the a < M regime. An example in shown in Fig. 7 for the surface at r /M = 0 . s=2l=m=4l=m=3 I m ( M ) a/Ml=m=2 l=m=3l=m=4 s=2l=m=2 R e ( M ) a/M FIG. 7: The fundamental l = m = 2 , , r /M = 0 . IV. CONCLUSION
We investigated the ergoregion instability of someultra-compact, horizonless objects which can mimick thespacetime of a rotating black hole. We studied some ofthe most viable BH mimickers: gravastars, boson stars,wormhole and superspinars.If rotating, boson stars and gravastars may develop er-goregion instabilities. Analytical and numerical resultsindicate that these objects are unstable against scalarfield perturbations for a large range of the parameters. Slowly rotating gravastars can develop an ergoregion de-pending on their angular momentum, their compactnessand the thickness of their intermediate region. In a recentwork [36] it has pointed out that slowly rotating gravas-tars may not develop an ergoregion. In the formation ofthe ergoregion for rotating gravastars an important roleis played by the thickness (see Figure 3) which is not eas-ily detectable. Thus further investigations are needed tobetter understand the ergoregion formation in physicalresonable gravastar models.The instability timescale for both boson stars andgravastars can be many orders of magnitude strongerthan the instability timescale for ordinary stars with uni-form density. In the large l = m approximation, suitablefor a WKB treatment, gravitational and scalar pertur-bations have similar instability timescales. In the low- m regime gravitational perturbations are expected to haveeven shorter instability timescales than scalar perturba-tions. Instability timescales can be as low as ∼ . M = 1 M ⊙ objects and about a week for su-permassive BHs, M = 10 M ⊙ , monotonically decreasingfor larger rotations and a larger compactness.The essential features of wormholes and superspinarshave been captured by a simple model whose physicalproperties are largely independent from the dynamicaldetails of the gravitational system. Numerical and ana-lytic results show that the ergoregion instability of theseobjects is extremely strong for any value of their angularmomentum, with timescales of order 10 − seconds for a1 M ⊙ object and 10 seconds for a M = 10 M ⊙ object.Therefore, high rotation is an indirect evidence for hori-zons.Although further studies are needed, the above investi-gation suggests that exotic objects without event horizonare likely to be ruled out as viable candidates for as-trophysical ultra-compact objects. This strengthens therole of BHs as candidates for astrophysical observationsof rapidly spinning compact objects. Acknowledgements
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