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Ergoregion instability of black hole mimickers∗ Paolo Pani† Dipartimento di Fisica, Università di Cagliari, andINFNsezione di Cagliari, CittadellaUniversitaria 09042 Monserrato, Italy Currently at Centro Multidisciplinar de Astrof´ısica - CENTRA, Dept. de F´ısica, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Vitor Cardoso‡ Centro Multidisciplinar de Astrof´ısica - CENTRA, Dept. de F´ısica, Instituto Superior Técnico, 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, Università di Cagliari, andINFNsezione di Cagliari, CittadellaUniversitaria 09042 Monserrato, Italy 9 0 Marco Cavaglià¶ 0 Department of Physics and Astronomy, The University of Mississippi, University, MS 38677-1848, USA 2 n Ultra-compact, horizonless objects such as gravastars, boson stars, wormholes and superspinars a can mimick most of the properties of black holes. Here we show that these “black hole mimickers” J willmostlikelydevelopastrongergoregion instabilitywhenrapidlyspinning. Instabilitytimescales 3 rangebetween∼10−5sand∼weeksdependingontheobject,itsmassanditsangularmomentum. 1 For a wide range of parameters the instability is truly effective. This provides a strong indication that astrophysical ultra-compact objects with large rotation are black holes. ] c q - I. INTRODUCTION two regionsare glued together by a model-dependent in- r termediate region. In the original model [9] the inter- g mediate region is an ultra-stiff thin shell. Models with- [ Blackholes(BHs)inEinstein-Maxwelltheoryarechar- out shells or discontinuities have also been investigated 2 acterized by three parameters [1]: mass M, electric [10, 11]. v charge Q and angular momentum J aM 6 M2. Boson stars are macroscopic quantum states which are 0 BHs are thought to be abundant obje≡cts in the Uni- prevented from undergoing complete gravitational col- 5 verse. Their mass is estimated to vary between 3M 8 ⊙ lapse by Heisenberg uncertainty principle [12]. Their and 109.5M or higher [2], their electrical charge is neg- 0 ⊙ modelsdifferinthescalarself-interactionpotentialwhich ligiblebecauseoftheeffectofsurroundingplasma[3]and . also set the allowed maximum compactness for a boson 1 their angular momentum is expected to be close to the star. 0 extremal limit because of accretion and merger events 9 An exhaustive description of wormholes can be found in [4]. A non-comprehensive list of some astrophysical BH 0 the monograph [13] (see also Ref. [14]). In this work we : candidates [2, 5, 6, 7] is shown in Table I. shall consider particular wormholes which are infinitesi- v i Despite the wealth of circumstantial evidence, there mal variations of BH spacetimes. These wormholes may X is no definite observational proof of the existence of as- be indistinguishable from ordinary BHs [15]. r trophysical BHs due to the difficulty to detect an event Superspinars aresolutionsofthegravitationalfieldequa- a horizon in astrophysical BH candidates [2, 8]. Thus as- tions that violate the Kerr bound. These geometries trophysical objects without event horizon, yet observa- could be created by high energy corrections to Einstein tionally indistinguishable from BHs, cannot be excluded gravity such as those present in string-inspired models a priori. Some of the most viable alternative models [16]. describing anultra-compactastrophysicalobjectinclude gravastars,boson stars, wormholes and superspinars. Darkenergystarsorgravastars arecompactobjectswith TABLE I: Mass, M, radius, R, angular momentum, J, and de Sitter interior and Schwarzschild exterior [9]. These compactness, µ=M/R,for some BH candidates (from [2, 5, 6, 7]). Mass and radius are in solar units. Candidate M R×10−5 J/M2 µ=M/R GRO J1655-40 6.3 1.6−2.6 0.65−0.80 0.47−0.83 ∗BlackHolesinGeneralRelativityandStringTheory XTEJ1550-564 10 2.1−8.4 0.90−1.00 0.25−0.99 August24-302008,VeliLoˇsinj,Croatia GRS 1915+105 14 2.9−9.7 0.98−1.00 0.30−0.99 †Electronicaddress: [email protected] SGR A* 4×106 .27 0.50−1.00 &0.31 ‡Electronicaddress: vcardoso@fisica.ist.utl.pt §Electronicaddress: [email protected] ¶Electronicaddress: [email protected] The objects described above can be almost as com- 2 pact as a BH and thus they are virtually indistinguish- aninstability. This may occur for any rotating star with ablefromBHsinthe Newtonianregime,hence the name an ergoregion: the mirror can be either its surface or, “BH mimickers”. Although exotic these objects provide for a star made ofmatter non-interactingwith the wave, viable alternatives to astrophysical BHs. BH mimick- its center. On the other hand BHs could be stable due ers being horizonless, no information loss paradox [17] to the absorptionby the event horizonbeing largerthan arises in these spacetimes. Moreover they can be regu- superradiant amplification. Indeed Kerr BHs are stable lar at the origin, avoiding the problem of singularities. aganist small scalar, electromagnetic and gravitational By Birkhoff’s theorem, the vacuum exterior of a spheri- perturbations [31]. cally symmetricobjectis describedby the Schwarzschild Rapidly rotating stars do possess an ergoregion and spacetime. Thus the motion of orbiting objects both thus they are unstable. However typical instability around a static BH and around a static ultra-compact timescales are shown to be larger than the Hubble time object is the same and it makes virtually impossible to [32]. Thus the ergoregion instability is too weak to pro- discernbetweenaSchwarzschildBHanda static neutral duceanyeffectontheevolutionofstars. Thisconclusion BH mimicker. Instead for rotating objects deviations in changes drastically for BH mimickers due to their com- the properties of orbiting objects occur. Since BH mim- pactness [23, 24]. For some of the rotating BH mimick- ickers are very compact these deviations occur close to ersdescribedabove,instabilitytimescalesrangebetween the horizon and are not easily detectable electromagnet- 10−5s and weeks depending on the object, its mass ∼ ∼ ically. To ascertain the true nature of ultra-compact ob- and its angular momentum. jects it is thus important to devise observationaltests to This paper is organized as follows. In Section II we distinguish rotating BH mimickers from ordinary Kerr dealwithgravastarsandbosonstars. We describerotat- BHs. The traditional way to distinguish a BH from a ing models for these objects and discuss their instability neutronstaristomeasureitsmass. Ifthelatterislarger timescale. In Section III a toy model for both rotat- than the Chandrasekhar limit, the object is believed to ing wormholes and superspinars is presented. Section be a BH. However, this method cannot be used for the IV contains a brief discussion of the results and con- BH mimickers discussed above, because of their broad cludes the paper. Throughout the paper geometrized mass spectrum. The main difference between a BH and units (G = c = 1) are used, except during the discus- a BHmimickeris the presenceofaneventhorizoninthe sion of results for rotating boson stars when we set the former. Some indirect experimental methods to detect Newton constant to be G=0.05/(4π) as in Ref. [33]. the event horizon has been proposed [18, 19]. Another verypromisingobservationalmethod to probe the structure of ultra-compact objects is gravitational wave as- II. GRAVASTARS AND BOSON STARS tronomy. Fromthe gravitationalwaveformitisexpected to detect the presence of an event horizon in the source [20]. Some other BH mimickers (for example electrically This section discusses the main properties of gravas- charged quasi-BHs [21]) are already ruled out by exper- tars and boson stars as well as the method to compute iments. Moreover there are evidences that some model the ergoregion instability for these objects. For a more for BH mimickers is plagued by a singular behavior in detailed discussion see [23]. the near-horizonlimit [22]. Here, we describe a method originally proposed in A. Nonrotating Gravastars [23, 24] for discriminating rotating BH mimickers from ordinary BHs. This method uses the fact that compact rotatingobjectswithouteventhorizonareunstablewhen Although exact solutions for spinning gravastars are an ergoregion is present. This ergoregion instability ap- notknown,they canbe studied inthe limit ofslow rota- pears in any system with ergoregions and no horizons tion by perturbing the nonrotating solutions [34]. This [25]. The origin of this instability can be traced back to procedure was used in Ref. [35] to study the existence superradiant scattering. In a scattering process, super- of ergoregions for ordinary rotating stars with uniform radiance occurs when scattered waves have amplitudes density. In the following, we omit the discussion for the larger than incident waves. This leads to extraction of original thin-shell model by Mazur and Mottola [9] and energy from the scattering body [26, 27, 28]. Instability we focus on the anisotropic fluid model by Chirenti and mayarisewheneverthisprocessisallowedtorepeatitself Rezzolla [10, 11]. ad infinitum. This happens, for example, when a BH is The model assumes a thick shell with continuous pro- surrounded by a “mirror”that scatters the superradiant file of anisotropic pressure to avoid the introduction of wave back to the horizon, amplifying it at each scatter- an infinitesimally thin shell. The stress-energy tensor is ing, as in the BH bomb process [29, 30]. If the mirror Tµ = diag[ ρ,p ,p ,p ], where p and p are the ra- ν r t t r t − is inside the ergoregion, superradiance may lead to an dialandtangentialpressures,respectively. Thespherical inverted BH bomb. Some superradiant waves escape to symmetric metric is infinity carrying positive energy, causing the energy in- sidetheergoregiontodecreaseandeventuallygenerating ds2 = f(r)dt2+B(r)dr2+r2dΩ2 (2.1) − 2 3 and it consists of three regions: an interior (r < r ) of order Ω, g ωg , where φ is the azimuthal co- 1 tφ φφ ≡ − described by a de Sitter metric, an exterior (r > r ) ordinate and ω = ω(r) is the angular velocity of frame 2 described by the Schwarzschild metric and a model- dragging. The full metric is dependent intermediate (r < r < r ) region. In the 1 2 followingwe shallindicate with δ =r r the thickness ds2 = fdt2+Bdr2+r2dθ2+r2sin2θ(dφ ωdt)2 , of the intermediate region and with µ2−= M1 /r the com- − − (2.4) 2 pactness of the gravastar. In the model by Chirenti and where f, B and ω are radial functions. If the gravastar Rezzolla the density function is rotates rigidly, i.e. Ω =constant, from the (t,φ) compo- nentofEinsteinequations wefind a differentialequation ρ , 0 r r interior for ω(r) [23] 0 1 ≤ ≤ ρ(r)= ar3+br2+cr+d, r1 <r<r2 intermediate  0, r2 ≤r exterior ω′′+ω′ 4 + j′ =16πB(r)(ω Ω)(ρ+pt) , (2.5) (cid:18)r j (cid:19) −  where a, b, c and d are found imposing continuity conditions ρ(0)= ρ(r ) =ρ , ρ(r ) = ρ′(r )= ρ′(r ) = 0 and where j (fB)−1/2 is evaluated at zeroth order and ρ, 1 0 2 1 2 ≡ ρ is found fixing the total mass, M. The metric coeffi- p are given in terms of the nonrotating geometry. The 0 t cients are above equation reduces to the corresponding equation for isotropic fluids [34]. Solutions of Eq. (2.5) describe 2M 1 2m(r) f = 1 eΓ(r)−Γ(r2), =1 , (2.2) rotating gravastars to first order in Ω. (cid:18) − r2 (cid:19) B − r Theergoregioncanbefoundbycomputingthesurface onwhich g vanishes [35]. An approximatedrelationfor tt where thelocationoftheergoregioninverycompactgravastars r r 2m(r)+8πr3p is m(r)= 4πr2ρdr, Γ(r)= rdr. Z0 Z0 r(r−2m(r)) 0= f(r)+ω2r2sin2θ. (2.6) (2.3) − The above equations and some closure relation, pr = The existence and the boundaries of the ergoregionscan pr(ρ), completely determine the structure of the gravas- be computed from the above equations. We integrate tar [10]. The behaviors of the metric coefficients for a equation (2.5) from the origin with initial conditions typical gravastar are shown in Fig. 1. (Ω ω)′ = 0 and (Ω ω) finite. The exterior solution sati−sfiesω =2J/r3,wh−ereJ istheangularmomentumof thegravastar. Demandingthecontinuityofboth(Ω ω)′ − and(Ω ω),ΩandJ areuniquelydetermined. Therota- − tion parameter Ω depends on the initial condition at the 10f(r) B(r) origin. Figure 2 shows the results the gravastar model 10 c etri M 1 1.4 1.2 1.0 0.1 0.8 5M 1 2 3 4 2 r/r2 0.6 J/M 0.4 0.2 FIG.1: Metriccoefficientsfortheanisotropicpressuremodel (r2=2.2, r1=1.8 and M =1). 0.2 0.4 0.6 0.8 1.0 r/r2 FIG. 2: J/M2 and angular frequency Ω for the anisotropic 1. Slowly rotating gravastars and ergoregions pressure model with r2=2.2, r1=1.8 and M =1. Slowly rotating solutions can be obtained using the described in the previous sections. The ergoregion can method developed in Ref. [34]. A rotation of order Ω be located by drawing an horizontal line at the desired gives corrections of order Ω2 in the diagonal coefficients value of J/M2. The minimum of the curve is the mini- of the metric (2.1) and introduces a non-diagonal term mumvaluesofJ/M2 whicharerequiredfortheexistence 4 of the ergoregion. Comparison with the results for stars symmetry otherwise. Since the Lagrangiandensity is in- ofuniformdensity[35],showsthatergoregionsformmore variant under a global U(1) transformation, the current, easilyaroundgravastarsduetotheirhighercompactness. jµ = iΦ∗∂µΦ+c.c.,isconservedanditisassociatedtoa − The slow-rotation approximation is considered valid for charge Q, satisfying the quantization condition with the Ω/Ω < 1 where MΩ = µ3/2 is the Keplerian fre- angular momentum J = nQ [37]. The numerical proce- K K quency. duretoextractthemetricandthescalarfieldisdescribed Depending on the compactness, µ, the angular momen- inRef.[33]. Throughoutthepaperwewillconsidersolu- tum,J,andthethickness,δ,aspinninggravastardoesor tions with n = 2, b = 1.1, λ = 1.0, a = 2.0 and different does not develop an ergoregion. The formation of an er- values of (J,M) corresponding to J/(GM2) 0.566, ∼ goregion for rotating gravastar is exhaustively discussed 0.731 and 0.858. In Fig. 4 the metric functions for a bo- in the whole parameters space in Ref. [36]. A delicate son star along the equatorial plane are shown. By com- issue is the strongdependence onthe thickness, δ,which puting the coefficient g one can prove that boson stars tt cannot be directly measured by experiments. Figure 3 develop ergoregions deeply inside the star. For this par- shows how the ergoregionwidth is sensitive to δ. ticularchoiceofparameters,theergoregionextendsfrom r/(GM) 0.0471 to 0.770. A more complete discussion ∼ on the ergoregions of rotating boson stars can be found in Ref. [33]. 2.0 1.5 1.0 1.0 g(r) 0.5 0.8 0.6 l(r) 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ∆(cid:144)M 0.4 0.2 FIG. 3: Ergoregion width (in units of M) as function of the thickness,δ=r2−r1,forr2=2.3,M =1andfordifferentJ 0.0 f(r) (r) values. From top to bottom: J/M2 = 0.95, 0.90, 0.85, 0.80, 0.75, 0.70, 0.65 and 0.60. The ergoregion width decreases as 0.1 1 10 δ→0. r/(GM) B. Rotating boson stars 1.2 f/f 1.0 AexampleofrotatingbosonstaristhemodelbyKlei- haus, Kunz, List and Schaffer (KKLS) [33]. The KKLS 0.8 solution is basedon the Lagrangianfor a self-interacting 0.6 l/l complex scalar field 0.4 1 LKKLS =−2gµν Φ∗,µΦ,ν +Φ∗,νΦ,µ −U(|Φ|), (2.7) 0.2 g/g (cid:0) (cid:1) 0.0 / where U(Φ) = λΦ2(Φ4 aΦ2 + b). The mass of | | | | | | − | | 0.01 0.1 1 10 the boson is given by m = √λb. The ansatz for the B r/(GM) axisymmetric spacetime is kg r2 sin2θ FIG. 4: Left panel: Metric coefficients for a rotating bo- ds2 = fdt2+ dr2+r2dθ2+ (dϕ ζ(r)dt)2 son star along the equatorial plane, with parameters n = 2, − f (cid:20) g − (cid:21) b = 1.1, λ = 1.0, a = 2.0, J/(GM2) ∼ 0.566. Right panel: (2.8) Fractionaldifferenceofthemetricpotentialsbetweenθ=π/2 and Φ = φ eiωst+inϕ, where the metric components and and θ=π/4 for thesame star. the realfunctionφdependonly onr andθ. The require- ment that Φ is single-valued implies n = 0, 1, 2,.... ± ± The solutionhas sphericalsymmetry for n=0 andaxial 5 C. Ergoregion instability for rotating gravastars everthe equationfor axialgravitationalperturbationsof and boson stars gravastars is identical to the equation for scalar perturbations in the large l = m limit [23]. There are also Thestabilityofgravastarsandbosonstarscanbestud- generic arguments suggesting that the timescale of grav- iedperturbativelybyconsideringsmalldeviationsaround itational perturbations is smaller than the timescale of equilibrium. Due to the difficulty of handling gravita- scalar perturbations for low m [38]. Thus, scalar pertur- tionalperturbationsforrotatingobjects,thecalculations bations should provide a lower bound on the strength of belowaremostlyrestrictedtoscalarperturbations. How- the instability. TABLE II: WKBresults for theinstability of rotating gravastars with r2 =2.2, r1=1.8 and M =1. τ/M J/M2 =0.40 J/M2 =0.60 J/M2 =0.80 J/M2 =0.90 J/M2 =1.0 m Ω/ΩK =0.33 Ω/ΩK =0.49 Ω/ΩK =0.65 Ω/ΩK =0.74 Ω/ΩK =0.82 1 1.33×107 2.78×104 5.99×103 3.58×103 2.34×103 2 8.25×107 1.14×106 1.11×105 4.81×104 2.33×104 3 1.31×1010 5.65×107 2.25×106 6.82×105 2.45×105 4 2.50×1012 2.95×109 4.81×107 1.02×107 2.73×106 5 5.06×1014 1.59×1011 1.02×109 1.52×108 3.07×107 1. Scalar field instability for slowly rotating gravastars: and have an instability timescale WKB approach rc rb d τ =4exp 2m T dr √Tdr, (2.12) (cid:20) Z | | (cid:21)Z dΣ Consider now a minimally coupled scalar field in the rb p ra background of a gravastar. The metric of gravastars is wherera,rb aresolutionsofV+ =Σandrc isdetermined given by Eq. (2.4). In the large l = m limit, which is by the condition V− =Σ. appropriate for a WKB analysis [32, 39], the scalar field Table II shows the WKB results for the anisotropic can be expanded in spherical armonics, Y =Y (θ,φ) pressure model for different values of J/M2. Although lm lm as the WKB approximation breaks down at low m values, these results still provide reliable estimates [32]. This Φ= χ¯ exp 1 2 + f′ + B′ dr e−iωtY . claim has be verified with a full numerical integration of lm lm (cid:20)−2Z (cid:18)r 2f 2B(cid:19) (cid:21) the Klein-Gordon equation. The results show that the Xlm (2.9) instability timescale decreases as the star becomes more The functions χ¯ = χ¯ (r) are determined by the compact. LargervaluesofJ/M2 makethe starmoreun- lm lm Klein-Gordon equation which, dropping terms of order stable. The maximum growthtime of the instability can 1/m2 , yields be of the order of a few thousand M, but it cruciallyde- O pendsonJ,µandδ [36]. Foralargerangeofparameters (cid:0) (cid:1) χ¯′′ +m2T(r,Σ)χ¯ =0, (2.10) this instability is crucial for the star evolution. Gravita- lm lm tional perturbations are expected to be more unstable. where Σ ω/m and Moreover it is worth to notice that the slowly rotating ≡− approximationallowsonlyforµ<0.5,while forrotating B(r) f(r) BHs 0.5 < µ < 1 (see Table I). The ergoregion insta- T = (Σ V )(Σ V ) , V = ω . f(r) − + − − ± − ± pr bility being monotonically increasing with µ, we expect that instability timescales for realistic gravastars should Equation(2.10)canbeshowntobeidenticalfortheaxial bemuchshorterthantheonescomputed. Formostofthe gravitationalperturbations of perfect fluid stars [24]. BH mimickers models to be viable we require J/M2 1 ∼ The WKB method [32] for computing the eigenfre- and µ 1. It would be interesting to study whether ∼ quencies of Eq. (2.10) is in excellent agreement with full the ergoregion instability is or is not always effective in numerical results [39]. The quasi-bound unstable modes thiscase. Possiblefuturedevelopmentsinclude: (i)afull are determined by rotating gravastar model, which allows for µ > 0.5; (ii) the stability analysis against gravitationalperturbations rb π forrotatinggravastars;(iii)agravavastarmodelwhichis m T(r)dr = +nπ, n=0,1,2,... (2.11) Z 2 not strongly dependent on the thickness, δ. ra p 6 The ergoregion instability of a rotating boson star Arigorousanalysisofthe ergoregioninstability forthese is straightforwardly computed following the method de- models is a non-trivial task. Indeed known wormhole scribedaboveforspinninggravastars. Wereferthereader solutions are special non-vacuum solutions of the gravi- to [23] and we only summarize the results in Table III. tational field equations, thus their investigation requires The maximum growth time for this boson star model is acase-by-caseanalysisofthestress-energytensor. More- of the order of 106M for J/GM2 = 0.857658. Thus the overexactsolutionsoffour-dimensionalsuperspinarsare instability seems to be truly effective for rotating boson not known. To overcome these difficulties, the follow- stars. ing analysis will focus on a simple model which captures the essentialfeatures of most Kerr-likehorizonlessultra- compact objects. Superspinars and rotating wormholes TABLEIII:Instabilityforrotatingboson starswithparame- willbemodeledbytheexteriorKerrmetricdowntotheir ters n=2, b=1.1, λ=1.0, a=2.0 and different valuesof J surface, where mirror-like boundary conditions are im- (from [23]). The Newton constant is definedas 4πG=0.05. posed. This problemisverysimilartoPressandTeukol- τ/(GM) sky’s “BH bomb” [29, 30], i.e. a rotating BH surrounded m J/GM2 =0.5661 J/GM2 =0.7307 J/GM2 =0.8577 1 8.847×102 6.303×103 − by a perfectly reflecting mirror with its horizon replaced 2 7.057×103 5.839×104 1.478×106 by a reflecting surface. For a more detailed discussion 3 6.274×104 9.274.×105 2.815×108 see [24]. 4 5.824×105 1.603×107 2.815×1010 5 5.554×106 2.915×108 1.717×1012 2. Superspinars and Kerr-like wormholes III. A TOY MODEL FOR KERR-LIKE OBJECTS This section discussesKerr-likeobjects suchas partic- AsuperspinarofmassM andangularmomentumJ = ular solutions of rotating wormholes and superspinars. aM can be modeled by the Kerr geometry [16] 2Mr Σ (r2+a2) 2Mr 4Mr ds2 = 1 dt2+ dr2+ + a2 sin4θdφ2 asin2θdφdt+Σdθ2, (3.1) Kerr −(cid:18) − Σ (cid:19) ∆ (cid:20) sin2θ Σ (cid:21) − Σ where Σ = r2+a2cos2θ and ∆ = r2 +a2 2Mr. Un- Kerrmetric with a rigid“wall”at finite Boyer-Lindquist − like KerrBHs,superspinarshavea>M andno horizon. radius r , which excludes the pathological region. 0 Since the domain of interest is < r < + , the −∞ ∞ space-timepossessesnakedsingularitiesandclosedtime- A. Instability analysis like curves in regions where g < 0 [40]. High energy φφ modifications (i.e. stringy corrections) in the vicinity of the singularity are also expected. Ifthebackgroundgeometryofsuperspinarsandworm- Kerr-like wormholes are described by the metric holes is sufficiently close to the Kerr geometry, its perturbations is determined by the equations of perturbed ds2 =ds2 +δg dxadxb, (3.2) Kerr BHs [24]. Thus the instability of superspinars and wormhole Kerr ab wormholesisstudiedbyconsideringKerrgeometrieswith where δg is infinitesimal. In general, Eq. (3.2) de- arbitrary rotation parameter a and a “mirror” at some ab scribes an horizonless object with a excision at some Boyer-Lindquist radius r . Using the Kinnersley tetrad 0 small distance of order ǫ from the would-be horizon and Boyer-Lindquist coordinates, it is possible to sepa- [15]. Wormholes require exotic matter and/or divergent rate the angular variables from the radial ones, decou- stress tensors, thus some ultra-stiff matter is assumed pling all quantities. Small perturbations of a spin-s field close to the would-be horizon. In the following, both are reduced to the radial and angular master equations superspinars and wormholes will be modeled by the [41] d dR K2 2is(r M)K ∆−s ∆s+1 lm + − − +4isωr λ R =0, (3.3) lm dr (cid:18) dr (cid:19) (cid:20) ∆ − (cid:21) 7 (m+sx)2 (1 x2) S + (aωx)2 2aωsx+s+ A S =0, (3.4) − s lm,x ,x (cid:20) − s lm− 1 x2 (cid:21)s lm (cid:2) (cid:3) − where x cosθ, ∆ = r2 2Mr + a2 and K = B. Instability analysis: numerical results (r2 +a2)ω≡ am. Scalar, elec−tromagnetic and gravita- − tionalperturbationscorrespondtos=0, 1, 2respec- The oscillation frequencies of the modes can be found ± ± tively. The separation constants λ and sAlm are related from the canonical form of Eq. (3.3) by λ A +a2ω2 2amω. s lm ≡ − d2Y +VY =0, (3.5) dr2 ∗ 1. Analytic results where Y = ∆s/2(r2+a2)1/2R, K2 2is(r M)K+∆(4irωs λ) dG V = − − − G2 , (r2+a2)2 − − dr ∗ 0.085 and K = (r2 +a2)ω am, G = s(r M)/(r2 +a2)+ r∆(r2 +a2)−2. The s−eparation consta−nt λ is related to ) l=m=2 Im( 0.080 l=m=4 tah2eω2eigen2vaamluωe.s oTfhteheeiagnenguvalalureesquAationabreyeλxp≡ansdAeldmi+n s lm − power series of aω as [42] 0.075 a=0.998M, s=2 A = f(k)(aω)k. (3.6) s lm slm 0.0002 0.0004 0.0006 0.0008 0.0010 kX=0 Terms up to order (aω)2 are included in the calculation. Absence of ingoing waves at infinity implies 0.07 Y r−seiωr∗. (3.7) ∼ 0.06 l=m=2 m Numerical results are obtained by integrating Eq. (3.5) / m 0.05 inward from a large distance r∞. The integration is )- l=m=3 performed with the Runge-Kutta method with fixed ω e( R 0.04 l=m=4 starting at Mr∞ = 400, where the asymptotic behavior (3.7) is imposed. (Choosing a different initial point 0.03 a=0.998M, s=2 does not affect the final results.) The numerical integration is stopped at the radius of the mirror r , where the 0.0002 0.0004 0.0006 0.0008 0.0010 0 value of the field Y(ω,r ) is extracted. The integration 0 is repeated for different values of ω until Y(ω,r )=0 is 0 FIG. 5: Imaginary and real parts of the characteristic gravi- obtainedwiththe desiredprecision. IfY(ω,r0)vanishes, tationalfrequenciesforanobjectwitha=0.998M,according the field satisfies the boundary condition for perfect re- to the analytic calculation for rapidly-spinning objects. The flection and ω = ω is the oscillation frequency of the 0 mirror location is at r0 = (1+ǫ)r+. The real part is ap- mode. proximatelyconstantandclosetomΩ,inagreementwiththe assumptions used in the analytic approach. 1. Objects with a<M Following Starobinsky [27], equations (3.3)-(3.4) can be analytically solved in the slowly-rotating and low- frequency regime, ωM 1, and in the rapidly-spinning ≪ regime, where r r and ω mΩ , where Ω The regime a < M requires a surface or mirror at + − h h ∼ ∼ ≡ a/(2Mr ) is the angular velocity at the horizon. The r = r (1+ǫ) > r . Thus the compactness is M/r + 0 + + 0 ∼ details of the analytic approximation are described in (1 ǫ)M/r and, in the limit ǫ 0, it is infinitesi- + − → Ref. [24]. Analytic solutions for a star with a=0.998M mally close to the compactness of a Kerr BH. Numerical areshowninFig.5wheregravitationalperturbationsare results for scalar and gravitational perturbations of ob- considered. The instability timescale for gravitational jects with a < M are summarized in Table IV and are perturbations is about five orders of magnitude smaller in agreement with the analytic results [24]. The insta- than the instability timescale for scalar perturbations. bility is weaker for larger m. This result holds also for 8 l = m and s = 0, 1 and 2. The minimum instability TABLE IV: Characteristic frequencies and instability ti6mescale is of ord±er τ 1±05M for a wide range of mir- timescales for a Kerr-like object with a=0.998M. The mir- ∼ rorlocations. Figure6showstheresultsforgravitational ror is located at ǫ = 0.1, corresponding to the compactness µ∼0.9µKerr. perturbations. Instability timescales are of the order of τ 2 6M. Thusgravitationalperturbationsleadtoan (Re(ω)M,Im(ω)M) ∼ ÷ instability about five orders of magnitude stronger than l=m s=0 s=2 1 (0.1120,0.6244×10−5) − theinstabilityduetoscalarperturbations(seeTableIV). 2 (0.4440,0.5373×10−5) (0.4342,0.2900) Figure 6 shows that the ergoregion instability remains 3 (0.7902,0.1928×10−5) (0.7803,0.2977) relevantevenforvaluesoftheangularmomentumaslow 4 (1.1436,0.5927×10−6) (1.1336,0.3035) as a=0.6M. 0.45 1.8 a=0.998M, s=2 0.40 1.6 0.35 1.4 ) 0.30 l=m=4 ) 1.2 M M Im( 0.25 l=lm=m=2=3 Re( 01..80 l=m=3l=m=4 0.20 0.6 l=m=2 0.15 0.4 0.10 a=0.998M, s=2 0.2 0.05 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 l=m=2, s=2 l=m=2, s=2 a=0.9M ) ) M M 0.1 a=0.8M Im( Re( 0.1 a=0.6M a=0.6M a=0.8M a=0.9M 0.01 0.0 0.2 0.4 1E-4 1E-3 0.01 FIG. 6: Details of theinstability for gravitational perturbations, for different l=m modes and a/M =0.998 (top panels) and for l=m=2 and different a/M <1. 2. Objects with a>M special perturbations [40]. For objects with a > M the surface or mirror can be placed anywhere outside r =0. In general the instability is as strong as in the a < M Objects with a > M could potentially describe su- regime. Anexample inshowninFig.7for the surfaceat perspinars. Several arguments suggest that objects ro- r /M = 0.001. This result confirms other investigations tating above the Kerr bound are unstable. Firstly, ex- 0 suggestingthatultra-compactobjectsrotatingabovethe tremal Kerr BHs are marginally stable. Thus the ad- Kerr bound are unstable [48]. dition of extra rotation should lead to instability. Sec- ondly, fast-spinning objects usually take a pancake-like form [43] and are subject to the Gregory-Laflamme instability[44,45]. Finally,Kerr-likegeometries,likenaked singularities, seem to be unstable against a certain class ofgravitationalperturbations[46,47]calledalgebraically 9 Slowly rotating gravastarscan developanergoregionde- pending on their angular momentum, their compactness andthethicknessoftheirintermediateregion. Inarecent 0.100 work [36] it has pointed out that slowly rotating gravas- 0.096 l=m=2 tars may not develop an ergoregion. In the formation of 0.092 the ergoregion for rotating gravastars an important role 0.088 isplayedbythethickness(seeFigure3)whichisnoteas- ) ily detectable. Thus further investigations are needed to M 0.084 Im( 0.080 l=m=3 better understand the ergoregion formation in physical resonable gravastar models. 0.076 The instability timescale for both boson stars and 0.072 l=m=4 gravastars can be many orders of magnitude stronger 0.068 thantheinstabilitytimescaleforordinarystarswithuni- s=2 formdensity. In the largel =m approximation,suitable 0.064 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 for a WKB treatment, gravitational and scalar pertur- a/M bations have similar instability timescales. In the low-m regime gravitational perturbations are expected to have even shorter instability timescales than scalar perturba- 1.4 tions. Instability timescales can be as low as 0.1 sec- ∼ 1.3 l=m=4 s=2 onds for a M = 1M⊙ objects and about a week for su- 1.2 permassive BHs, M =106M , monotonically decreasing 1.1 ⊙ for larger rotations and a larger compactness. 1.0 ) 0.9 l=m=3 The essential features of wormholes and superspinars M 0.8 have been captured by a simple model whose physical Re( 0.7 properties are largely independent from the dynamical 0.6 details of the gravitational system. Numerical and ana- 0.5 l=m=2 lytic results show that the ergoregioninstability of these 0.4 0.3 objectsisextremely strongforanyvalue oftheir angular 0.2 momentum, with timescales of order 10−5 seconds for a 0.1 1M object and 10 seconds for a M = 106M object. ⊙ ⊙ 0.0 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 Therefore, high rotation is an indirect evidence for hori- a/M zons. Althoughfurtherstudiesareneeded,theaboveinvesti- FIG. 7: The fundamental l =m=2,3,4 modes of an object gationsuggeststhatexoticobjectswithouteventhorizon spinning above the Kerr bound as function of rotation. The are likely to be ruled out as viable candidates for as- surface is located at r0/M =0.001. trophysical ultra-compact objects. This strengthens the role of BHs as candidates for astrophysical observations of rapidly spinning compact objects. IV. 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