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PoS(BHs, GR and Strings)027 http://pos.sissa.it/ ce. that these “black hole mimick- ruly effective. This provides a ty when rapidly spinning. Instability son stars, wormholes and superspinars issippi, University, MS issippi, University, MS ísica, Instituto Superior Técnico, ts with large rotation are black holes. zione di Cagliari, Cittadella zione di Cagliari, Cittadella RA, Dept. de Física, Instituto Superior Técnico, Av. weeks depending on the object, its mass and its angular ∼ ive Commons Attribution-NonCommercial-ShareAlike Licen † s and 5 − 10 ∼ ∗ [email protected] [email protected] [email protected] [email protected] can mimick most of the propertiesers” of will black most holes. likely Here develop wetimescales show a range strong between ergoregion instabili Ultra-compact, horizonless objects such as gravastars, bo Speaker. Presently at Centro Multidisciplinar de Astrofísica - CENT momentum. For a widestrong range indication that of astrophysical parameters ultra-compact objec the instability is t ∗ † Copyright owned by the author(s) under the terms of the Creat c Rovisco Pais 1, 1049-001 Lisboa, Portugal Universitaria 09042 Monserrato, Italy E-mail: Marco Cavaglià Department of Physics and Astronomy, The University of Miss Av. Rovisco Pais 1, 1049-001 Lisboa,Department Portugal of Physics and Astronomy, The38677-1848, University USA of Miss E-mail: Mariano Cadoni Dipartimento di Fisica, Università di Cagliari, and INFN se 38677-1848, USA E-mail: Paolo Pani Dipartimento di Fisica, Università di Cagliari,Universitaria and 09042 INFN Monserrato, se Italy E-mail: Vitor Cardoso Centro Multidisciplinar de Astrofísica - CENTRA, Dept. de F Ergoregion instability of black hole mimickers Black Holes in General Relativity andAugust String 24-30 Theory 2008 Veli Lošinj,Croatia PoS(BHs, GR and Strings)027 R / ld ex- M Paolo Pani = 83 99 99 µ . . . 0 0 0 31 . 0 − − − & 47 25 30 or higher [2], their . . . ⊙ ergoing complete grav- , for some BH candidates M 5 d thus they are virtually R . 9 / ate the Kerr bound. These Compactness M ost viable alternative models = µ 80 0 00 0 00 0 00 and 10 . . . . efinite observational proof of the 2 ependent intermediate region. In ized by three parameters [1]: mass 0 1 1 1 ⊙ M e of accretion and merger events [4]. zschild spacetime. Thus the motion − − − − ary BHs [15]. he name “BH mimickers”. Although es [2, 5, 6, 7] is shown in Table 1. / M mpactness for a boson star. nstein gravity such as those present in khoff’s theorem, the vacuum exterior e infinitesimal variations of BH space- J t an event horizon in astrophysical BH t horizon, yet observationally indistin- rounding plasma [3] and their angular acetimes. Moreover they can be regular 65 90 98 50 tiff thin shell. Models without shells or gravastars, boson stars, wormholes and . BHs are thought to be abundant objects . . . . 2 hysical BHs. BH mimickers being hori- 0 0 0 2]. Their models differ in the scalar self- M 5 5 5 6 − − − ) 10 10 10 ⊙ aM 2 , and compactness, R × × × J ≡ ) ) ) 27 0 J 6 4 7 . . . 2 8 9 . can be found in the monograph [13] (see also Ref. [14]). − − − 6 1 9 Radius ( . . . 1 2 2 ( ( ( are compact objects with de Sitter interior and Schwarzschi ) ⊙ 6 wormholes M ( 3 10 . , angular momentum, × R gravastars and angular momentum Q , radius, M are solutions of the gravitational field equations that viol are macroscopic quantum states which are prevented from und Mass, SGRA* 4 Candidate Mass Black holes (BHs) in Einstein-Maxwell theory are character Despite the wealth of circumstantial evidence, there is no d The objects described above can be almost as compact as a BH an GRO J1655-40 6 GRS 1915+105 14 XTE J1550-564 10 , electric charge (from [2, 5, 6, 7]) in the Universe. Their mass is estimated to vary between 3 M Ergoregion instability of black hole mimickers 1. Introduction superspinars. Dark energy stars or existence of astrophysical BHs due tocandidates the [2, difficulty 8]. to detec Thusguishable astrophysical objects from without BHs, even cannotdescribing be an excluded ultra-compact a astrophysical priori. object include Some of the m electrical charge is negligiblemomentum because is of expected to the be effect close toA of the non-comprehensive sur extremal list limit of becaus some astrophysical BH candidat Table 1: indistinguishable from BHs in the Newtonianexotic regime, these hence t objects provide viablezonless, alternatives no information to loss astrop paradox [17]at arises in the these origin, sp avoiding theof problem a of spherically singularities. symmetric object By is Bir described by the Schwar geometries could be created by highstring-inspired energy models corrections [16]. to Ei In this work we shall consider particulartimes. wormholes which These ar wormholes may be indistinguishableSuperspinars from ordin discontinuities have also been investigated [10,Boson 11]. stars terior [9]. These twothe original regions model are [9] glued the together intermediate region by is a an model-d ultra-s itational collapse by Heisenberg uncertaintyinteraction principle potential [1 which also set theAn allowed exhaustive maximum description co of PoS(BHs, GR and Strings)027 Paolo Pani appears in process [29, 30]. If weeks depending on ∼ ravastars and boson stars. ting boson stars when we set r discriminating rotating BH s and BH bomb 5 of an event horizon in the for- ckers (for example electrically y are unstable. However typi- r” that scatters the superradiant − ompact rotating objects without presented. Section 4 contains a ergoregion instability in the near-horizon limit [22]. o the horizon and are not easily to measure its mass. If the latter 10 of their broad mass spectrum. The to the absorption by the event hori- c ultra-compact object is the same ructure of ultra-compact objects is is of this instability can be traced back rzschild BH and a static neutral BH eir instability timescale. In Section horizon has been proposed [18, 19]. ∼ n of stars. This conclusion changes s. Moreover there are evidences that bble time [32]. Thus the ergoregion allowed to repeat itself ad infinitum. oughout the paper geometrized units 4]. For some of the rotating BH mim- s in the rr BHs are stable aganist small scalar, form it is expected to detect the pres- to an inverted BH bomb. Some super- r made of matter non-interacting with ed to be a BH. However, this method tion of energy from the scattering body ausing the energy inside the ergoregion may occur for any rotating star with an operties of orbiting objects occur. Since een ure of ultra-compact objects it is thus im- radiance occurs when scattered waves have ing BH mimickers from ordinary Kerr BHs. 3 1) are used, except during the discussion of results for rota = c Rapidly rotating stars do possess an ergoregion and thus the This paper is organized as follows. In Section 2 we deal with g Here, we describe a method originally proposed in [23, 24] fo = G [26, 27, 28]. InstabilityThis may happens, arise for whenever example, this when processwave a back is BH to is the surrounded horizon, by amplifying a it “mirro at each scattering, a amplitudes larger than incident waves. This leads to extrac any system with ergoregions and noto horizons superradiant [25]. scattering. The In origin a scattering process, super cal instability timescales are showninstability to is be too larger weak thandrastically to the for produce Hu BH any mimickers due effect to onickers their compactness described the [23, above, evolutio 2 instability timescales range betw ergoregion: the mirror can be either its surface or, for a sta the mirror is inside the ergoregion,radiant superradiance waves escape may to lead infinity carryingto positive decrease energy, and c eventually generating an instability. This the wave, its center. On thezon other hand being BHs larger could than be superradiant stableelectromagnetic amplification. due and Indeed gravitational Ke perturbations [31]. detectable electromagnetically. To ascertain theportant true to nat devise observational tests to distinguishThe rotat traditional way to distinguishis a larger BH than from the a Chandrasekharcannot neutron be limit, used star the for is object the BH ismain mimickers difference believ discussed between above, a because BHmer. and Some a indirect BH experimental mimicker methods isAnother to the detect very presence the promising event observational methodgravitational wave to astronomy. probe From the theence st gravitational of wave an event horizoncharged in quasi-BHs the [21]) source are [20]. alreadysome ruled Some model out other for by BH BH experiment mimickers mimi is plagued by a singular behavior mimickers from ordinary BHs.event This horizon method are uses unstable the when fact an that ergoregion is c present. Th the object, its mass and its angular momentum. We describe rotating models for3 these a objects toy and model discussbrief for th discussion both of rotating the wormholes results( and and superspinars concludes is the paper. Thr Ergoregion instability of black hole mimickers of orbiting objects both around a static BH and around a stati BH mimickers are very compact these deviations occur close t and it makes virtually impossiblemimicker. to Instead discern for between rotating a objects Schwa deviations in the pr PoS(BHs, GR and Strings)027 , ) ] 2 t r p )= , 1 (2.1) (2.3) (2.2) t < r p ( r ′ , r ρ < p Paolo Pani , 1 r ρ )= 2 − r d Rezzolla the [ ( ρ . , diag 0 dr spherical symmetric ρ r = p ν 3 )) r )= r µ 1 ( π , T r 8 m ( ) nt intermediate ( 2 r ρ sotropic pressure to avoid the ( r )+ − m r 2 2 own, they can be studied in the r ( 2 )= intermediate exterior interior ( zolla [10, 11]. Ω 0 r m d boson stars as well as the method ( d − 2 for a typical gravastar are shown in 2 ρ 1 r r rotating stars with uniform density. In 0 , completely determine the structure of shell model by Mazur and Mottola [9] ons [34]. This procedure was used in the thickness of the intermediate region ) = Z + 2 r a more detailed discussion see [23]. 1 1 r ρ 2 r 1 nergy tensor is r B ( r < dr )= − ≤ p ) r r r 2 r ( , ) described by a de Sitter metric, an exterior r r ( ) 1 = Γ 2 < ≤ r B r ≤ r ( = 1 2 4 p Γ r 0 r < + δ − , 2 r ) r d ( dt Γ ) as in Ref. [33]. + e r ( ) f cr π 4 2 − + M ( r , 2 2 / = dr br − 2 05 . ρ 1 2 + 0 ds , r 3 , 0 π = 0 ρ ar = 4 G r f 0      Z )= the compactness of the gravastar. In the model by Chirenti an )= r are found imposing continuity conditions r ( 2 ( is found fixing the total mass, M. The metric coefficients are r are the radial and tangential pressures, respectively. The ρ d / 0 m t ρ p M and = c , and µ 0 and b r , p ) described by the Schwarzschild metric and a model-depende a 2 )= r This section discusses the main properties of gravastars an Although exact solutions for spinning gravastars are not kn The model assumes a thick shell with continuous profile of ani 2 r > ( ′ r where metric is ( and with region. In the following we shall indicate with and it consists of three regions: an interior ( ρ Ergoregion instability of black hole mimickers the Newton constant to be where the gravastar [10]. TheFig. behaviors 1. of the metric coefficients 2. Gravastars and boson stars to compute the ergoregion instability for these objects. Fo limit of slow rotation byRef. [35] perturbing to the study nonrotating the existencethe soluti of following, ergoregions we for omit ordinary theand discussion we for focus the on original the anisotropic thin- fluid model by Chirenti and Rez introduction of an infinitesimally thin shell. The stress-e 2.1 Nonrotating Gravastars The above equations and some closure relation, density function is where PoS(BHs, GR and Strings)027 ) ω (2.5) (2.6) (2.4) − 1). Ω ( Paolo Pani = M 0 and s the results the . 8 and = . 2 ′ 1 ) ) = ω dt ) 1 r r − , ( vanishes [35]. An approx- 2, ) Ω . ω t tt ( 2 p

− are uniquely determined. The . arison with the results for stars = eveloped in Ref. [34]. A rota- is the azimuthal coordinate and + φ J 2 Ω d component of Einstein equations r φ ρ quation for isotropic ﬂuids [34]. ( mputed from the above equations. ) )( 4 θ ditions φ and . 2 , Ω are given in terms of the nonrotating oregion can be located by drawing an t mpact gravastars is θ Ω ( t − sin 2 , p order in , where 2 ) is the angular momentum of the gravastar. ω , r 3 sin ω J φφ ρ )( 2 + g r r − ( 2 2 ω 2 B

θ Ω ω r/r ( π d 5 2 2 r ≡− , where in the diagonal coefﬁcients of the metric (2.1) and 16 )+ 3 φ r 2 and r + t = ( . The minimum of the curve is the minimum values of / ′ 2 g Ω f 2 ) J , 2 dr 1 ′ − ω j M Ω j ) constant, from the [23] / r = − = 10f(r) B(r) ) ( J

= + r B 0 ω ( Ω r 4 ( Ω ω + 1 2 10 ′ 0.1 dt

ω tric et M ) r + ( ′′ f ω − depends on the initial condition at the origin. Figure 2 show = is evaluated at zeroth order and 2 2 Ω / 1 ds gives corrections of order − ) Metric coefficients for the anisotropic pressure model ( Ω fB ( is the angular velocity of frame dragging. The full metric is ) ≡ r j ( which are required for the existence of the ergoregion. Comp Figure 1: ω The ergoregion can be found by computing the surface on which Slowly rotating solutions can be obtained using the method d 2 = M / where imated relation for the location of the ergoregion in very co ω Ergoregion instability of black hole mimickers 2.1.1 Slowly rotating gravastars and ergoregions tion of order The existence and the boundariesWe of integrate the equation ergoregions can (2.5)finite. be from The co the exterior solution origin satisfies with initial con gravastar model described in thehorizontal previous line sections. at The the erg desired value of introduces a non-diagonal term of order rotation parameter geometry. The aboveSolutions equation of reduces Eq. (2.5) to describe the rotating gravastars corresponding to e first J If the gravastar rotates rigidly, i.e. Demanding the continuity of both we find a differential equation for PoS(BHs, GR and Strings)027 1 and 8 and . 1 = 60. The . 1 where = M Paolo Pani < 1 r 3, , a spinning . K 2 δ 2, Ω . = 65 and 0 2 / . 2 Ω r = 2 r 70, 0 . , for 1 r − 75, 0

. 2 r = 1.2 80, 0 . δ , ion of an ergoregion for rotating , and the thickness, sily around gravastars due to their J 1.0 nsidered valid for 85, 0 1.0 . s space in Ref. [36]. A delicate issue . δ 90, 0 . 0.8 0.8 M 95, 0 . ∆ 2 0.6

0 6 r/r = 0.6 , which cannot be directly measured by experiments. 2 δ 0.4 M / J for the anisotropic pressure model with 5M Ω 0.4 2 0.2 , the angular momentum, J/M µ 0. 0.0 → 0.2 0.0 2.0 1.5 1.0 0.5 δ 1.4 1.2 1.0 0.8 0.6 0.4 0.2 and angular frequency 2 is the Keplerian frequency. values. From top to bottom: M 2 J / / 3 J Ergoregion width (in units of M) as function of the thickness µ = 1. K = Ω Figure 3: Figure 2: Ergoregion instability of black hole mimickers Depending on the compactness, Figure 3 shows how the ergoregion width is sensitive to gravastar does or doesgravastar not is develop exhaustively an discussed ergoregion. inis the the The whole strong format dependence parameter on the thickness, ergoregion width decreases as for different M of uniform density [35], shows that ergoregions form more ea higher compactness. The slow-rotation approximation is co M PoS(BHs, GR and Strings)027 and (2.7) (2.8) r . The b one can λ [37]. The Paolo Pani tt √

nQ = B = m J 10 y is invariant under 2 ) depend only on 0 and different values . dt φ 2 ) , r mentum ) ( = g/g | 1 . The solution has spherical ζ z, List and Schaffer (KKLS) a Φ | − (

0, f/f ,... . uting the coefﬁcient ϕ U 2 , is conserved and it is associated 1 r/(GM) 770. A more complete discussion d / . . l/l ( − c interacting complex scalar ﬁeld ± . = ng the equatorial plane, with parameters , . [33]. 858. In Fig. 4 the metric functions e star. For this particular choice of c . θ 1 µ 0.1 λ 2 , d is described in Ref. [33]. Throughout + ± , Φ 1, sin Φ 0 ν . ∗ , µ 2 1 0471 to 0 r = Φ ∂ . ∗ = 0 n + + Φ b 731 and 0 i ν . ∼ 0.01 , 2 7 ) − 1.2 1.0 0.8 0.6 0.4 0.2 0.0 2, θ Φ 566. Right panel: Fractional difference of the metric . d µ = 0 ∗ , = 2 GM r µ Φ ( 566, 0 n j ∼ .

/ . The mass of the boson is given by

+ 0 ) ) r 2 2 b µν ∼ g dr + ) 4 for the same star. 2 1 GM 2 2 ( 10 / | g / − (r) π J Φ | f k = = GM a ( 0, . θ / + − 2 J 2 is single-valued implies 4 | KKLS = 1 Φ Φ a f dt L 2 and | ( /

− 2 0, π | . r/(GM) 1 , where the metric components and the real function = Φ = | ϕ 2 0 and axial symmetry otherwise. Since the Lagrangian densit = θ λ in g(r) ds + = λ t 0.1 s transformation, the current, n ω , satisfying the quantization condition with the angular mo )= 1, i ) | . e Q 1 1 f(r) ( Φ corresponding to | l(r) φ Left panel: Metric coefficients for a rotating boson star alo ( = ) U b = U M , A example of rotating boson star is the model by Kleihaus, Kun Φ 2, 1.0 0.8 0.6 0.4 0.2 0.0 J ( = . The requirement that of numerical procedure to extract the metricthe and the paper scalar we fiel will consider solutions with for a boson star along the equatorial plane are shown. By comp n prove that boson starsparameters, develop the ergoregions ergoregion extends deeply from inside th Figure 4: θ Ergoregion instability of black hole mimickers 2.2 Rotating boson stars [33]. The KKLS solution is based on the Lagrangian for a self- where ansatz for the axisymmetric spacetime is and to a charge potentials between on the ergoregions of rotating boson stars can be found in Ref symmetry for a global PoS(BHs, GR and Strings)027 δ (2.9) [38]. and (2.11) (2.10) (2.12) (2.13) m µ , Paolo Pani J t the timescale . g terms of order . Σ . ) ) r = φ ( , r f − θ V ( p del for different values of lm ± Y . (2.10) is in excellent agree- , t ω ω ,... i nd of a gravastar. The metric of 2 − − values, these results still provide Tdr , e vitational perturbations of perfect 1 = √ m , scale decreases as the star becomes Σ l numerical integration of the Klein- dr 0 on stars , ± d he strength of the instability. d gravitational perturbations for rotating rs: WKB approach able modes are determined by rturbatively by considering small devi- V 0 = b , but it crucially depends on cial for the star evolution. Gravitational r ′ a B alar perturbations. However the equation r n = le of scalar perturbations for low B 2 ntical to the equation for scalar perturba- M Z lm + , ¯ χ ′ f ) dr π limit, which is appropriate for a WKB analysis | f n Σ 2 , T | m 8 r + + ( p = 2 r π 2 , T c l 2 r ) b is determined by the condition r = m − Z c V Z r + dr make the star more unstable. The maximum growth time m 1 2 ) − 2 2 ′′ r lm Σ ( − ¯ and χ M T )( / Σ J + p V 4exp = exp b r a ) r − + r = Z ( V Σ τ ( m lm ¯ ) ) χ r r limit [23]. There are also generic arguments suggesting tha ( ( lm f ∑ B m = = are determined by the Klein-Gordon equation which, droppin = and l Φ T m lm ¯ / χ ω are solutions of b r , yields ≡− , 2 a r Σ m . Although the WKB approximation breaks down at low / 2 Table 2 shows the WKB results for the anisotropic pressure mo The WKB method [32] for computing the eigenfrequencies of Eq Consider now a minimally coupled scalar field in the backgrou 1 M / where J reliable estimates [32]. ThisGordon claim has equation. be verified The withmore results a compact. show ful that Larger the values of instability time of the instability can be of the order of a few thousand [36]. For a large range of parameters this instability is cru Ergoregion instability of black hole mimickers 2.3 Ergoregion instability for rotating gravastars and bos for axial gravitational perturbations oftions in gravastars the is large ide The stability of gravastars and bosonations stars around can equilibrium. be studied Due pe to theobjects, difficulty the of calculations handling below are mostly restricted to sc The functions ment with full numerical results [39]. The quasi-bound unst Equation (2.10) can befluid shown stars to [24]. be identical for the axial gra O where and have an instability timescale of gravitational perturbations is smaller than the timesca gravastars is given by Eq. (2.4). In the large Thus, scalar perturbations should provide a lower bound on t 2.3.1 Scalar field instability for slowly rotating gravasta [32, 39], the scalar field can be expanded as PoS(BHs, GR and Strings)027 M 6 1. = 0 82 3 4 5 6 7 . . Paolo Pani 1 0 M 10 10 10 10 10 = = 0 and different × × × × × . 2 K 2 8 and 34 33 45 73 07 M Ω ...... = / / 1 2 2 2 2 3 6 8 J a 10 12 1 (see Table 1). The Ω = 857658 10 10 0, 10 10 1 . . < r rotating boson stars. 1 0 × × × × − µ 2, 74 3 4 5 7 8 = . 90 = . . 2 0 2 < λ 0 478 815 5; (ii) the stability analysis 10 10 10 10 10 815 717 . . . . . = 5 = 1 2 0 . 1, = 2 1 × × × × × . 2 GM r 1 K 2 > / 58 81 82 02 52 Ω J . . . . . M star model is of the order of 10 = µ / 3 4 6 1 1 / 05. b . J Ω 0 , we expect that instability timescales ghtforwardly computed following the or 2, is worth to notice that the slowly rotat- s computed. For most of the BH mim- e reader to [23] and we only summarize µ = his case. Possible future developments = ity for these models is a non-trivial task. 5 r solutions of rotating wormholes and su- G th 7 8 3 4 1. It would be interesting to study whether n 65 3 5 6 7 9 80 . π . ) ars; (iii) a gravavastar model which is not 10 0 730677 10 10 10 10 ∼ 0 . 10 10 10 10 10 0 × M µ × × × × = = × × × × × . GM / ( K 2 = τ / 9 99 11 25 81 02 2 Ω 603 915 303 839 . . . . . M τ 274 . . . . . / 5 1 2 4 1 / 1 and 1 2 6 5 9 J Ω GM ∼ / 2 J 5, while for rotating BHs 0 . M 0 49 4 6 7 9 60 . 11 . / . 0 J δ 0 < 10 10 10 10 10 = µ = × × × × × 5 6 2 3 4 K 2 78 14 65 95 566139 10 10 10 10 10 59 Ω . . . . M . . / 2 1 5 2 / 0 1 × × × × × J Ω = 2 824 554 847 057 274 . . . . . 5 5 8 7 6 33 7 7 40 . 10 12 14 . GM 0 0 / 10 10 10 10 10 J = = × × × × × K 2 857658. Thus the instability seems to be truly effective for . 33 25 Ω 31 50 06 . . 0 4 5 1 2 3 M m . . . WKB results for the instability of rotating gravastars with / 1 8 / 1 2 5 J = Ω (from [23]). The Newton constant is defined as 4 2 Instability for rotating boson stars with parameters J 4 5 1 2 3 GM m / The ergoregion instability of a rotating boson star is strai Table 2: J values of 3. A toy model for Kerr-like objects This section discusses Kerr-like objects suchperspinars. as particula A rigorous analysis of the ergoregion instabil Table 3: Ergoregion instability of black hole mimickers perturbations are expected to be more unstable. Moreover it ing approximation allows only for method described above for spinning gravastars. We referthe th results in Table 3. The maximum growth time for this boson for realistic gravastars should be muchickers models shorter to than be the viable one we require strongly dependent on the thickness, against gravitational perturbations for rotating gravast ergoregion instability being monotonically increasing wi the ergoregion instability is orinclude: is (i) not always a effective full in rotating t gravastar model, which allows f for PoS(BHs, GR and Strings)027 , 2 θ (3.3) (3.2) (3.4) (3.1) d Σ and no + M Paolo Pani dt > φ a , d 0 θ , 2 0 = sin = lm a S s lm Σ R Mr 2 4 ) less object with a excision 2 λ x − sx . Using the Kinnersley tetrad sufficiently close to the Kerr 0 2 − r − + φ r 1 , d m ω field are reduced to the radial and ( b” [29, 30], i.e. a rotating BH sur- b θ can be modeled by the Kerr geom- is 4 s will be modeled by the Kerr metric 4 laced by a reflecting surface. For a utions of the gravitational field equa- dx − es the essential features of most Kerr- a of perturbed Kerr BHs [24]. Thus the sin + aM -like boundary conditions are imposed. lm sidering Kerr geometries with arbitrary e the angular variables from the radial pin- rotating wormholes will be modeled by dx A nown. To overcome these difficulties, the K 2 s = ected. ab ) iff matter is assumed close to the would-be a J alysis of the stress-energy tensor. Moreover g + M δ , the space-time possesses naked singularities , which excludes the pathological region. s Σ 0 Mr ∞ − 0 [40]. High energy modifications (i.e. stringy r + 2 + r + ( ∆ < sx + 10 is 2 Kerr < . Unlike Kerr BHs, superspinars have ) ω 2 φφ r 2 a ds g θ a 2 − Mr < 2 2 2 = + − ∞ K 2 − sin 2 r ) − 2 ( x a + ω from the would-be horizon [15]. Wormholes require exotic + a + 2 2 wormhole ( ε 2 r lm ds dr = dr + and angular momentum ∆ Σ dR x ∆ , 1 M + + x , s 2 and ∆ lm dt S θ s 2 ) and a “mirror” at some Boyer-Lindquist radius d 2 dr x a s Σ Mr cos − − 2 2 ∆ a 1 ( − + 1 is infinitesimal. In general, Eq. (3.2) describes an horizon 2 r ab − = g δ Σ = A superspinar of mass If the background geometry of superspinars and wormholes is Kerr-like wormholes are described by metrics of the form 2 Kerr horizon. Since the domain of interest is angular master equations [41] and Boyer-Lindquist coordinates, it is possible to separat ones, decoupling all quantities. Small perturbations of a s etry [16] ds at some small distance of order Ergoregion instability of black hole mimickers Indeed known wormhole solutions are special non-vacuum sol tions, thus their investigation requires a case-by-caseexact an solutions of four-dimensional superspinars are notfollowing k analysis will focus on a simplelike model horizonless which ultra-compact captur objects. Superspinars and This problem is very similar torounded Press by and a Teukolsky’s “BH perfectly bom reflectingmore detailed mirror discussion with see [24]. its horizon rep 3.0.2 Superspinars and Kerr-like wormholes the exterior Kerr metric down to their surface, where mirror matter and/or divergent stress tensors, thus somehorizon. ultra-st In the following, both superspinars and wormholes with a rigid “wall” at finite Boyer-Lindquist radius 3.1 Instability analysis geometry, its perturbations is determinedinstability by of the superspinars and equations wormholes isrotation studied parameter by con and closed timelike curves in regions where where corrections) in the vicinity of the singularity are also exp where PoS(BHs, GR and Strings)027 − is = = M r 0 a and λ r (3.6) (3.7) (3.5) ∼ 998 λ . + 0 r Paolo Pani = , a ∗ dG

. The eigenvalues − 2 ω 0.0010 G am − 2 ) 0.0008 − λ 2 − a=0.998Ms=2 , ω . The separation constant s 2 2 e canonical form of Eq. (3.3) a ω l=m=4

0.0006 − nalytically solved in the slowly- l=m=3 ) ir l=m=2 + 2 al frequencies for an object with . Scalar, electromagnetic and grav- 4 a ( lm , in agreement with the assumptions used 2 ∆ nsidered. The instability timescale for am + ) A . ng objects. The mirror location is at c solutions for a star with 0.0004 Ω s 2 2 de smaller than the instability timescale k + − r a m ) ( ≡ , K ω ω + ∆ ) 0 ) λ r a 2 2 ( 0.0002 r M = a ) ( k 2 respectively. The separation constants ( − )+ slm + f 0.07 0.06 0.05 0.04 0.03 2 r ± 2 VY a (

m / m - ) e( R r = is 1, + ∑ + k 2 . ± 2 1, and in the rapidly-spinning regime, where 2 ∗ = ( Y r = ω 2 − ( dr 0, d 2 K as [42] ≪ / lm is the angular velocity at the horizon. The details of the am ) K = A

) ω M s M s a and + = − ω 0.0010 − 2 2 V a r Mr ω ( 2 2 s + ( a / = 0.0008 a + , Mr G a=0.998Ms=2 , 2 R ≡ lm , 2 / h A − s 1 l=m=4 2

) 0.0006 Ω am r 2 ≡ a − = l=m=2 λ + ω ∆ ) 2 0.0004 2 , r ( a θ , where 2 h / + s Ω 2 cos ∆ Imaginary and real parts of the characteristic gravitation 0.0002 r m . The real part is approximately constant and close to ≡ = + , according to the analytic calculation for rapidly-spinni ∼ x = ( r 0.085 0.080 0.075 Y ) M

are expanded in power series of are related by

m( ( Im ) The oscillation frequencies of the modes can be found from th Following Starobinsky [27], equations (3.3)-(3.4) can be a ω ε K lm lm + 998 . 1 A A and 0 Figure 5: Ergoregion instability of black hole mimickers where analytic approximation are described in Ref. [24]. Analyti related to the eigenvalues of the angular equation by s s 3.1.1 Analytic results and itational perturbations correspond to where ( are shown in Fig. 5 wheregravitational gravitational perturbations perturbations is are about co five ordersfor scalar of perturbations. magnitu 3.2 Instability analysis: numerical results in the analytic approach. rotating and low-frequency regime, PoS(BHs, GR and Strings)027 M 0 is < (3.8) . The . Thus a M infinity )= 0 M r Paolo Pani 6 998 , . The inte- . 0 ∞ ω ÷ r ( 2 = ts suggest that 400, where the Y a ∼ = 2. The minimum τ ∞ . ± until M ω Mr 6 ) ) ) , where the value of the . Thus the compactness . 0 0 + r r 1 and = ± 2900 2977 3035 > . ues of . . . a 2 ) 0 0 0 0, , , , ε ic results [24]. The instability − = Kerr starting at µ s = + 9 ) . 1 s ow as ω ( 0 4342 7803 1336 perturbations of objects with , extremal Kerr BHs are marginally regory-Laflamme instability [44, 45]. . . . M + t initial point does not affect the final ) 0 0 1 ∼ r a Kerr-like object with r ard from a large distance and ω stability. Secondly, fast-spinning objects µ s of the mirror re 6 shows that the ergoregion instability ( = ve orders of magnitude stronger than the lly special perturbations [40]. For objects m seem to be unstable against a certain class 0 . ) ) ( ) ( ) ( r Im ∗ 5 5 5 6 , r 6= escales are of the order of − − − − ω l i M e 10 10 10 10 ) s 0, it is infinitesimally close to the compactness − ω 12 vanishes, the field satisfies the boundary condition × × × × r ( ) → 0 0 ∼ r Re ε , ( = 6244 5373 1928 5927 for a wide range of mirror locations. Figure 6 shows the Y . . . . ω s 0 0 0 0 ( , , , , M Y 5 10 1120 4440 7902 1436 is the oscillation frequency of the mode. . . . . ∼ 0 0 0 1 0 τ ( ( ( ( ω = are included in the calculation. Absence of ingoing waves at m and, in the limit could potentially describe superspinars. Several argumen ω 2 M M 1 2 3 4 requires a surface or mirror at = ) 1, corresponding to the compactness + . . This result holds also for M r l > < 0 ω / M m a a a > ( = M < a ) ε a ε − is extracted. The integration is repeated for different val 1 ) ( 0 r , Characteristic frequencies and instability timescales fo ∼ ω 0 ( r Y / Objects with The regime M are summarized in Tableis 4 weaker and for are larger in agreement with the analyt obtained with the desired precision. If asymptotic behavior (3.8) is imposed.results.) (Choosing The numerical a integration differen is stopped at the radiu of a Kerr BH. Numerical results for scalar and gravitational field instability timescale is of order 3.2.2 Objects with Table 4: Ergoregion instability of black hole mimickers Terms up to order Numerical results are obtained by integrating Eq. (3.5) inw 3.2.1 Objects with gravitational perturbations lead to an instability about fi instability due to scalar perturbations (see Table 4). Figu remains relevant even for values of the angular momentum as l results for gravitational perturbations. Instability tim mirror is located at for perfect reflection and implies is gration is performed with the Runge-Kutta method with fixed objects rotating above thestable. Kerr Thus bound the addition are of extra unstable. rotation should lead Firstly to in usually take a pancake-like form [43] and are subject to the G of gravitational perturbations [46, 47] called algebraica Finally, Kerr-like geometries, like naked singularities, PoS(BHs, GR and Strings)027 = m M = 001. . / l 0 a = Paolo Pani M /

modes and 0.35 m = a=0.9M 0.30 l 0.01 l=m=2, l=m=2, s=2 a=0.998Ms=2 , a=0.8M 0.25 0. In general the instability is = 0.20

ion instabilities. Analytical and r some of the most viable BH mim- l=m=4 ompact, horizonless objects which eir intermediate region. In a recent rs can be many orders of magnitude 0.15 1E-3 a=0.6M s, for different elop an ergoregion depending on their may not develop an ergoregion. In the l=m=3 odels. ltra-compact objects rotating above the rtant role is played by the thickness (see tigations are needed to better understand 0.10 s. with uniform density. In the large l=m=2 against scalar ﬁeld perturbations for a large 1. 0.05 1E-4 < 0.1 0.01 0.00

) M e( R 13 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 /

(M M e( R )

0.8

0.7 l=m=4 0.4 l=m=3 2 and different l=m=2 0.6 = a=0.9M a=0.998Ms=2 , l=m=2, l=m=2, s=2 m 0.5 regime. An example in shown in Fig. 7 for the surface at = l

0.4 a=0.8M 0.2 < a 0.3 a=0.6M 0.2 the surface or mirror can be placed anywhere outside 0.1 Details of the instability for gravitational perturbation M 0.0 0.1

) M ( Im a

We investigated the ergoregion instability of some ultra-c If rotating, boson stars and gravastars may develop ergoreg The instability timescale for both boson stars and gravasta 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

) M ( Im 998 (top panels) and for . This result conﬁrms other investigationsKerr suggesting bound that are unstable u [48]. 4. Conclusion can mimick the spacetime ofickers: a rotating gravastars, boson black stars, hole. wormhole We and studied superspinar numerical results indicate that these objects arerange unstable of the parameters. Slowlyangular rotating momentum, gravastars their can dev compactness andwork the [36] thickness it of has th pointed outformation of that the slowly ergoregion rotating for gravastars rotating gravastarsFigure an 3) impo which is not easily detectable. Thus further inves the ergoregion formation in physical resonable gravastar m stronger than the instability timescale for ordinary stars as strong as in the Figure 6: Ergoregion instability of black hole mimickers with 0 PoS(BHs, GR and Strings)027 , of ⊙ en M 6 10 seconds 5 Paolo Pani = − M s=2 a/M , 433 (1977). , 312 (2004). een captured by a simple model ussions and for sharing some of 179 602 gion instability of these objects is tes for astrophysical ultra-compact ience Foundation through LIGO th timescales of order 10 strophysical observations of rapidly on suggests that exotic objects with- a e Tecnologia (FCT) - Portugal c. J. dynamical details of the gravitational l=m=4 l and scalar perturbations have similar l=m=3 l=m=2 compactness. 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 . Instability timescales can be as low as 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

object. Therefore, high rotation is an indirect

) (M e R ⊙ M 14 6 10

, 152 (1972).

= 25 001. . M 0 4 modes of an object spinning above the Kerr bound as function s=2 regime gravitational perturbations are expected to have ev , = 3 , m M 2 objects and about a week for supermassive BHs, / 0 = r ⊙ , 199 (2005). m 7 M 1 = l

= a/M M l=m=2 l=m=3 l=m=4 object and 10 seconds for a 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 The fundamental ⊙

0.100 0.096 0.092 0.088 0.084 0.080 0.076 0.072 0.068 0.064

) (M Im Although further studies are needed, the above investigati The authors warmly thank Matteo Losito for interesting disc The essential features of wormholes and superspinars have b 1 seconds for a . 0 [2] R. Narayan, New J. Phys. [1] S. W. Hawking, Commun. Math. Phys. [3] R. D. Blandford and R. L. Znajek, Mon. Not. Roy. Astron. So [4] C. F. Gammie, S. L. Shapiro and J. C. McKinney, Astrophys. evidence for horizons. out event horizon are likelyobjects. to This be strengthens ruled the outspinning role compact as of objects. viable BHs candida as candidates for a Acknowledgements his results. This work isthrough supported project by PTDC/FIS/64175/2006 Fundação and para by aResearch the Ciênci Support National grant Sc NSF PHY-0757937. 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