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Home , Accretion disk, Gravastar, Rotating black hole, Supermassive black hole, Supernova

Can disk properties distinguish from Title holes?

Author(s) Harko, T; Kovacs, Z; Lobo, FSN

Classical and Quantum , 2009, v. 26 n. 21, article no. Citation 215006

Issued Date 2009

URL http://hdl.handle.net/10722/125293

Rights Creative Commons: Attribution 3.0 Hong Kong License arXiv:0905.1355v2 [gr-qc] 4 Sep 2009 edrt 1 o ealdrve) nti otx,re- context, been has as this collapse In denoted gravitational of review). proposed, state final detailed the new a refer a for cently ambigu- (we [1] solution certain Schwarzschild to ex- a the reader as the in well literature inherent as the ity misconceptions, in of encounters istence still . curved one in However, theory field quantum of in- singularitie aspects for and beyond, horizons, event and regarding relativity, stance, general in role conceptual nitro eSte odnae oendb nequa- by an given by governed state with condensate, of object Sitter compact tion a de of inves- interior consist extensively an models been These have properties tigated. their and holes, ‡ † ∗ lcrncades fl[email protected] address: Electronic [email protected] address: Electronic lcrncades [email protected] address: Electronic h cwrshl ouinhspae fundamental a played has solution Schwarzschild The 2,wihrpeetaval lentv oblack to alternative viable a represent which [2], ) a crto ikpoete itnus rvsasfrom gravastars distinguish properties disk accretion Can ASnmes 45.d 47.w 97.10.Gz 04.70.Bw, 04.50.Kd, numbers: PACS h sign black observational Kerr-type these effi from gravastars Thus, less rotating a holes. distinguishing provide black gravastars radiatio than i.e., into holes, black accreting for the efficiency of diffe efficiency conversion signatu are observational the Due clear spectrum) giving velocity. consequently radiation angular objects, equilibrium the and in properti distribution electromagnetic order and second thermodynamic the the gravastargeometry, to rotating up slowly holes around estimated th black disks through Kerr-type accretion is and consider gravastars holes axiall rotating black and around stationary from geometr disks of gravastars exterior context the distinguishing the expec reg distinguish of in the stability However, to difficult to large hole. black be close gravastars would sufficiently static it are in that Ob that that exist fact relativity. challen configurations) major the general h a of to matter, is gravastars picture due anisotropic and hole holes of black black shell astrophysical standard thin the correlated to strongly a by eateto xeietlPyis nvriyo Szeged, of University , Experimental of Department etod ´sc eoiaeCmuainl audd eCi de Faculdade Computacional, F´ısica Te´orica e de Centro rvsas yohtcatohsclojcs consistin objects, astrophysical hypothetic Gravastars, a-lnkIsiuefu aiatooi,AfdmH¨uge dem Auf f¨ur Radioastronomie, Max-Planck-Institute .INTRODUCTION I. eateto hsc n etrfrTertcladComput and Theoretical for Center and Physics of Department gravastars p = vnd rfso aaPno2 -6903Lso,Portuga Lisboa, P-1649-003 2, Pinto Gama Professor Avenida h nvriyo ogKn,PkF a od ogKong Hong Road, Lam Fu Pok Kong, Hong of University The − ρ ace oaselof shell a to matched , ( grav itational Dtd etme ,2009) 4, September (Dated: rnic .N Lobo N. S. Francisco va iei Harko Tiberiu ota Kov´acsZolt´an cuum s, nt hcns iha qaino state of equation an with thickness finite we w nt ai.I oeohrcsste collapse they cases other be- some oscillating In is radii. shell finite thin two tween the excursion” “bounded where represent gravastars, they stable cases gravastars, other stable in represent models while the prototype found cases was of some It in [5]. that models studied and dynamical constructed Recently, were gravastars Reissner-Nordstr¨om and or interior [4]. Sitter Sitter exterior de (anti)-de or Schwarzschild Sitter a anti-de furth an was physically analysis to stability some generalized latter that are This layer transition stability. the there to the of for that state features found of equations key was reasonable the It constructing shares by scenario. that [3], model in considered a was layer perturbations symmet transition ric spherically the against of shell) thin stability infinitesimally slightly view. dynamic (an of the radius point of a thermodynamic issue config- a The at These from stable located [2]. are radius is urations Schwarzschild surface the hori- event than rigid no greater its and origin as the at zon, singularity gravastar no the Sitter has de therefore, model the horizons, Schwarzschild both replaces vac- the Schwarzschild shell and thick exterior The an solution. to uum matched then is latter e.I diint hs ti losonthat shown also is it this, to addition In res. fadr nrycnest surrounded condensate dark a of g ∗ † oles. ,wt l h erctno components tensor metric the all with s, efrti atrtertclmdl This model. theoretical latter this for ge in ehns o ovrigms to mass converting for mechanism cient so h ik eeg u, flux, (energy disks the of es trspoietepsiiiyo clearly of possibility the provide atures epciey ntepeetppr we paper, present the In respectively. , sawy mle hnteconversion the than smaller always is n ymtia emtis possibility a geometries, symmetrical y etfrteetocasso compact of classes two these for rent oprtv td fti accretion thin of study comparative e os(ftetasto ae fthese of layer transition the (of ions ‡ e oiino h vn oio,so horizon, event the of position ted fgaatr rma astrophysical an from gravastars of y v enpooe sa alternative an as proposed been ave 9 32 on emn and Germany Bonn, 53121 69, l ´ mTe ,See 70 Hungary 6720, Szeged D´om T´er 9, evtoal itnusigbetween distinguishing servationally e ca aUiesdd eLisboa, de Universidade ˆencias da otedffrne nteexterior the in differences the to toa Physics, ational l lc holes? black p = ρ The . er - 2 until the formation of black holes. without the presence of exotic fluids or modifications of In addition to this, gravastar models that exhibit the Friedmann equations. continuous pressure without the presence of infinitesi- However, observationally distinguishing between astro- mally thin shells were introduced in [6] and further an- physical black holes and gravastars is a major challenge alyzed in [7]. By considering the usual TOV equation for this latter theoretical model, as in static gravastars for static solutions with negative central pressure, it large stability regions exist that are sufficiently close was found that gravastars cannot be perfect fluids and to the expected position of the . The anisotropic pressures in the ‘crust’ of a gravastar-like constraints that present-day observations of well-known object are unavoidable. The anisotropic TOV equation candidates place on the gravastar model were can then be used to bound the pressure anisotropy, and discussed in [15]. The heating of stars via ac- the transverse stresses that support a gravastar permit cretion is well documented by astronomical observations. a higher compactness than the Buchdahl-Bondi bound A gravastar would acquire most of its mass via accre- for perfect-fluid stars. A wide variety of gravastar mod- tion, either as part of its birth (e.g., during core collapse els within the context of nonlinear electrodynamics were in a followed by the rapid accretion also constructed in [8]. Using the F representation, spe- of a fallback disk), or over an extended period of time cific forms of Lagrangians were considered describing after birth. Similar heating processes have not been ob- magnetic gravastars, which may be interpreted as self- served in the case of black hole candidates. However, gravitating magnetic monopoles with charge g. Using the the absence of a detectable heating may also be consis- dual P formulation of nonlinear electrodynamics, electric tent with a gravastar, if the heat capacity is large enough gravastar models were constructed by considering specific that it requires large amounts of heat to produce small structural functions, and the characteristics and physical changes in temperature. Nevertheless, some level of ac- properties of the solutions were further explored. Gravas- cretion heating is unavoidable, and provides some strong tar solutions with a Born-Infeld phantom replacing the constraints on the gravastar model. The large surface de Sitter interior were also analyzed in [9]. employed in gravastar models implies that the It has also been recently proposed by Chapline that internal energy generated per unit rest mass accreted is 2 this new emerging picture consisting of a compact ob- very nearly c , and that the radiation emitted from the ject resembling ordinary spacetime, in which the vacuum surface of the object should be an almost perfect black energy is much larger than the cosmological vacuum en- body. The energy evolution of an accreting gravastar is determined by the equation dU/dt = Mc˙ 2 L, where ergy, has been denoted as a ‘ ’ [10]. In- − deed, a generalization of the gravastar picture was con- U is the internal energy, M˙ is the mass accretion rate, sidered in [11] by considering a matching of an interior and L is the . Assuming that the gravastar solution governed by the dark energy equation of state, is the result of a BEC-like induced by ω = p/ρ < 1/3, to an exterior Schwarzschild vacuum strong gravity [10], the gravastar which starts at zero − solution at a junction interface. Several relativistic dark temperature and rapidly accretes a mass ∆mMSun will energy stellar configurations were analyzed by imposing be heated to a temperature as observed at infinity of specific choices for the mass function, by assuming a con- T 3.1 106m−2/3ξ1/3 (∆m/m)1/3 K, where m is the h ≈ × stant energy , and a monotonic decreasing energy mass of the gravastar in units, and ξ = l/lPl is density in the star’s interior, respectively. The dynamical the length scale in at which general relativ- stability of the transition layer of these dark energy stars ity fails to adequately describe gravity. In the case of slow to linearized spherically symmetric radial perturbations accretion, the temperature is given by the equilibrium about static equilibrium solutions was also considered, 1/4 value T 1.7 107 M/˙ M˙ m−1/4 K, where and it was found that large stability regions exist that eq ≈ × Edd −9 are sufficiently close to where the event horizon is ex- M˙ Edd =2.3 10 mM⊙/yr is the Eddington mass accre- pected to form. Evolving dark energy stars were explored tion rate. By× comparing these theoretical predictions on in [12], where a time-dependent dark energy parameter the temperature with the upper limits of the temperature was considered. The general properties of a spherically of the black hole candidates, obtained through observa- symmetric body described through the generalized Chap- tions, one can find some limits on ξ. Two black hole lygin equation of state were also extensively analyzed in candidates, known to have extraordinarily low luminosi- [13]. In the context of cosmological equations of state, ties, namely, the in the galactic in [14] the construction of gravastars supported by a van center, *, and the stellar-mass black hole, der Waals equation of state was studied and their re- XTE J1118 + 480, respectively, were considered in the spective characteristics and physical properties were fur- analysis [15]. For XTE J1118+480, due to the low mass ther analyzed. It was argued that these van der Waals and low heat capacity, the value of ξ is constrained to quintessence stars may possibly originate from density ξ 5 103, while for Sagittarius A* values for ξ in the fluctuations in the cosmological background. Note that range≈ × 104 and 1011 are excluded. Therefore the length the van der Waals quintessence equation of state is an scale for the considered gravastar models must be sub- interesting scenario that describes the late , and Planckian. Thus, if any significant fraction of the mass of seems to provide a solution to the puzzle of dark energy, the gravastars is due to accretion, we should see the ther- 3 mal emission associated with that phase, which is ruled the equation of state, the speed of sound and the surface out for Sagittarius A* and XTE J1118 + 480. were calculated for both models. The dipolar The question of whether gravastars can be distin- magnetic field configuration for gravastars was studied guished from black holes at all was also considered, from in [21], and solutions of Maxwell equations in the inter- a theoretical point of view, in [16], where two basic ques- nal background spacetime of a slowly rotating gravastar tions analyzed were: (i) Is a gravastar stable against were obtained. The shell of the gravastar where the mag- generic perturbations and, (ii) if it is stable, can an ob- netic field penetrated was modeled as a sphere consisting server distinguish it from a black hole of the same mass? of a highly magnetized perfect fluid, with infinite con- A general class of gravastars was constructed, and the ductivity. It was assumed that the dipolar magnetic field equilibrium conditions in order to exist as solutions of of the gravastar is produced by a circular current loop the Einstein equations were obtained. It was found that symmetrically placed at radius a at the equatorial plane. gravastars are stable to axial perturbations, and that The mass accretion around rotating black holes was their quasi-normal modes differ from those of a black hole studied in for the first time in [22]. By of the same mass. Thus, these modes can be used to dis- using an equatorial approximation to the stationary and cern, beyond dispute, a gravastar from a black hole. The axisymmetric spacetime of rotating black holes, steady- formation hysteresis effects were ignored in this study. state models were constructed, extending the In addition to this, sharp analytic bounds on the surface theory of non-relativistic accretion [23]. In these mod- compactness 2m/r that follow from the requirement that els hydrodynamical equilibrium is maintained by efficient the dominant energy condition (DEC) holds at the shell cooling mechanisms via radiation transport, and the ac- were derived in [17]. In the case of a Schwarzschild exte- creting matter has a Keplerian rotation. The radiation rior, the highest surface compactness is achieved with the emitted by the disk surface was also studied under the stiff shell in the limit of vanishing (dark) energy density in assumption that black body radiation would emerge from the interior. In the case of a Schwarzschild-de Sitter ex- the disk in thermodynamical equilibrium. The radiation terior, it was shown that gravastar configurations with a properties of thin accretion disks were further analyzed in surface pressure and with a vanishing shell pressure (dust [24, 25], where the effects of capture by the hole shells), are allowed by the DEC. The causality require- on the spin evolution were presented as well. In these ment (sound speed not exceeding that of ) further works the efficiency with which black holes convert rest restricts the space of allowed gravastar configurations. mass into outgoing radiation in the accretion process was The ergoregion instability is known to affect very com- also computed. More recently, the emissivity properties pact objects that rotate very rapidly, and that do not of the accretion disks were investigated for exotic cen- possess an horizon. A detailed analysis on the relevance tral objects, such as [26], and non-rotating or of the ergoregion instability for the viability of gravas- rotating quark, boson or fermion stars and brane-world tars was presented in [18, 19]. In [18], it was shown that black holes [27, 28, 29, 30, 31, 32]. The radiation power ultra-compact objects with high redshift at their surface per unit area, the temperature of the disk and the spec- are unstable when rapidly spinning, which strengthens trum of the emitted radiation were given, and compared the role of black holes as candidates for astrophysical with the case of a Schwarzschild black hole of an equal observations of rapidly spinning compact objects. In mass. The physical properties of matter forming a thin particular, analytical and numerical results indicate that in the static and spherically symmetric gravastars are unstable against scalar field perturbations. spacetime metric of vacuum f(R) modified gravity mod- Their instability timescale is many orders of els were also analyzed [33], and it was shown that partic- stronger than the instability timescale for ordinary stars ular signatures can appear in the electromagnetic spec- with uniform density. In the large l = m approximation, trum, thus leading to the possibility of directly testing suitable for a WKB treatment, gravitational and scalar modified gravity models by using astrophysical observa- perturbations have similar instability timescales. In the tions of the emission spectra from accretion disks. low-m regime gravitational perturbations are expected to It is the purpose of the present paper to consider have shorter instability timescales than scalar perturba- another observational possibility that may distinguish tions. In [19], the analysis shows that not all rotating gravastars from black holes, namely, the study of the gravastars are unstable, and stable models can be con- properties of the thin accretion disks around rotating structed also with J/M 2 1, where J and M are the an- gravastars and black holes, respectively. Thus, we con- gular and mass∼ of the gravastar, respectively. sider a comparative study of the thin accretion disks Therefore, the existence of gravastars cannot be ruled around slowly rotating gravastars and black holes, respec- out by invoking the ergoregion instability. The gravas- tively. In particular, we consider the basic physical pa- tar model was extended by introducing an electrically rameters describing the disks, such as the emitted energy charged component in [20], where the Einstein–Maxwell flux, the temperature distribution on the surface of the field equations were solved in the asymptotically de Sit- disk, as well as the spectrum of the emitted equilibrium ter interior, and a source of the electric field was coupled radiation. Due to the differences in the exterior geome- to the fluid energy density. Two different solutions that try, the thermodynamic and electromagnetic properties satisfy the dominant energy condition were given, and of the disks (energy flux, temperature distribution and 4 equilibrium radiation spectrum) are different for these ∆ < 0 indicates that pr >p⊥. Note that ∆/r represents two classes of compact objects, thus giving clear obser- a force due to the anisotropic of the stellar model, vational signatures, which may allow to discriminate, at which is repulsive, i.e., being outward directed if p⊥ >pr, least in principle, gravastars from black holes. We would and attractive if p⊥ rior solution, governed by an equation of state, pr = ρ, 0, r = 0 the body is tangential pressure dominated while to an exterior Schwarzschild vacuum solution with p−= ∀ 6 5

ρ = 0, at a junction interface Σ, with junction radius a. where b0 is a non-negative constant, which was exten- The is given by sively analyzed in [11] in the context of dark energy stars. The latter may be determined from the regularity con- − 2M 2M 1 ditions and the finite character of the energy density at ds2 = 1 dt2 + 1 dr2 + r2 dΩ2 , − − r − r the origin r = 0, and is given by b0 = 8πρc/3, where ρc     (8) is the energy density at r = 0. 2 which possesses an event horizon at rb =2M, and dΩ = This choice of the mass function represents a mono- dθ2 +sin2 θ dϕ2. To avoid the event horizon, the junction tonic decreasing energy density in the star interior, and radius lies outside 2M, i.e., a > 2M. We show below was used previously in the analysis of isotropic fluid that M, in this context, may be interpreted as the total spheres by Matese and Whitman [35] as a specific case mass of the gravastar. of the Tolman type IV solution [36], and later by Finch Using the Darmois-Israel formalism [34], the surface and Skea [37]. Anisotropic− stellar models, with the re- stresses of the thin shell are given by [11] spective astrophysical applications, were also extensively analyzed in Refs. [38], by considering a specific case of σ = 1 1 2M +a ˙ 2 1 2m +a ˙ 2 , (9) the Matese-Whitman mass function. The numerical re- − 4πa − a − − a sults outlined show that the basic physical parameters, q 2 q′ 2  1− M +˙a +aa¨ 1−m − m +˙a +aa¨ such as the mass and radius, of the model can describe = 1 a a , (10) 8πa 1− 2M +˙a2 1− 2m +˙a2 realistic astrophysical objects such as neutron stars [38]. P √ a − √ a ! The spacetime metric for this solution is provided by where the overdot denotes a derivative with respect to 2 2 2 1+b0r 2 1+2b0r 2 2 2 proper time, τ. σ and are the surface energy density 2 2 ds = 1+2b0r dt + 1+b0r dr + r dΩ .(13) and the tangential pressure,P respectively [11]. The dy- −     namical stability of the transition layer of these compact The stress-energy tensor components are given by spheres to linearized spherically symmetric radial pertur- bations about static equilibrium solutions was explored 2 b 3+2b0r using Eqs. (9)-(10) (see [11] for details). Large stability p = ρ = 0 r − − 8π (1+2b r2)2 regions were found that exist sufficiently close to where   0  2 2 2 2 the event horizon is expected to form, so that it would b0 (3+2b0r ) b0r (5+2b0r ) be difficult to distinguish the exterior geometry of these p⊥ = 2 2 + 2 3 . − 8π (1+2b0r ) 4π (1+2b0r ) gravastars from astrophysical black holes.     The surface mass of the thin shell is given by ms = The anisotropy factor takes the following form 4πa2σ. By rearranging Eq. (9), evaluated at a static solution a , i.e.,a ˙ =¨a = 0, one obtains the total mass of 0 b2r2 (5+2b r2) the gravastar, given by ∆= 0 0 , (14) 4π (1+2b r2)3   0 2m(a0) ms(a0) M = m(a0)+ms(a0) 1 . (11) which is always positive, implying that p⊥ > pr, and s − a0 − 2a0  ∆ = 0 at the center, i.e., p⊥(0) = p (0), as expected. |r=0 r  

3. Specific model: Tolman-Matese-Whitman mass function B. Slowly rotating gravastars

To gain insight into the problem, it is interesting to When the equilibrium configuration described in Sec. present a specific example. For instance, consider the II A 1 is set into slow rotation, the geometry of spacetime following choice for the mass function, given by around it and its interior distribution of stress-energy are 3 changed. With an appropriate change of coordinates, the b0r m(r)= 2 , (12) perturbed geometry is described by [39, 40] 2(1+2b0r )

1+2[m + m P (cos θ)] / (r 2M) ds2 = eνrot(r) 1+2[h + h P (cos θ)] dt2 + 0 2 2 − dr2 + − { 0 2 2 } 1 2M/r − r2 [1 + 2 (v h ) P (cos θ)] dθ2 + sin2 θ (dϕ ωdt)2 + O Ω3 . (15) 2 − 2 2 − S h i  Here, P (cos θ) = 3cos2 θ 1 /2 is the Legendre polynomial of order two; the angular velocity, ω, of the local 2 − inertial frame, is a function of the radial coordinate r, and is proportional to the star’s angular velocity ΩS; and h0,  6

2 h2, m0, m2, v2 are functions of r that are proportional to ΩS. Outside the star the metric can be written as [40] 2M J 2 J 2 M 5 Q J 2/M ds2 = 1 +2 1+2 1+ + − Q2 (χ) P (cos θ) dt2 − − r r4 Mr3 r 8 M 3 2 2         − 2M J 2 1 J 2 5M 5 Q J 2/M + 1 +2 1 2 1 + − Q2 (χ) P (cos θ) dr2 − r r4 − Mr3 − r 8 M 3 2 2         2 2 2 J 2M 5 Q J /M 2M 1 2 +r 1+2 3 1+ + − 3 Q2 (χ) Q2 (χ) P2 (cos θ) ( "−Mr r 8 M r (r 2M) − !# )   − 2J 2 p dθ2 + sin2 θ dϕ dt . (16) × − r3 (   )

In Eq. (16) the variable χ = r/M 1, and the quantities The , describing a , is given 1 2 − Q2 and Q2 denote associated Legendre polynomials of the in the Boyer-Lyndquist coordinate system by second time, so that 2Mr 2Mr Σ ds2 = 1 dt2 +2 a sin2 θdtdφ + K dr2 3χ2 2 3 χ +1 − − ΣK ΣK ∆K Q1(χ)= χ2 1 − χ ln , (17)   2 2 2Mr − " χ 1  − 2 χ 1# +Σ dθ2 + r2 + a2 + a2 sin2 θ sin2 θdφ2, (19) p − − K Σ  K  and 2 2 2 2 2 where ΣK = r + a cos θ and ∆K = r + a 2mr, 5χ 3χ2 3 χ +1 respectively. In the equatorial plane, the metric− compo- Q2(χ)= − + χ2 1 ln , (18) 2 (χ2 1) 2 − χ 1 nents reduce to − −  2Mr 2M respectively. The line element outside the star is deter- g = 1 = 1 , tt − − Σ − − r mined by three constants: the total mass of the rotating  K    star M, the star’s total J and the 2Mr 2 Ma gtφ = a sin θ =2 , star’s mass quadrupole moment Q. The quadrupole mo- ΣK r 2 ment Q can be expressed in terms of the eccentricity e = ΣK r 2 ∗2 ∗2 grr = = , (re/rp) 1 of the star as e = 3Q/Mr +O 1/r , ∆K ∆K ∗ − qwhere r is a large distance from the origin, and re and 2 2 2Mr 2 2 2 p  gφφ = r + a + a sin θ sin θ rp are the equatorial and radius of the star, re- ΣK spectively [40]. The metric given by Eq. (16) is valid   2M in the case of slow rotation, that is, the angular veloc- = r2 + a2 1+ , r ity of the star ΩS must be small enough so that the   fractional changes in pressure, energy density, and grav- respectively. For the Kerr metric J = Ma and itational field, due to rotation, are all much less than Q = J 2/M, respectively. The latter relationship,− i.e., unity. The condition of slow rotation can be formulated Q = J 2/M, between the quadrupole momentum and the 2 as Ω2 (c/R) GM/Rc2 , where R is the radius of angular momentum shows the very special nature of the S ≪ the static stellar configuration [39]. The critical angu- Kerr solution. lar velocity at which mass shedding occurs is given by 2 3 ΩK = GM/R [40], and therefore the condition of slow rotation implies ΩS ΩK . All models considered in the III. ELECTROMAGNETIC RADIATION present paper satisfy≪ this condition. PROPERTIES OF THIN ACCRETION DISKS IN The metric given by Eq. (16) is used to determine STATIONARY AXISYMMETRIC SPACETIMES the electromagnetic signatures of accretion disks around slowly rotating gravastars, which is analyzed in detail To set the stage, we present the general formalism below. of electromagnetic radiation properties of thin accretion disks in stationary axisymmetric spacetimes.

C. The Kerr black hole A. Stationary and axially symmetric spacetimes For self-completeness and self-consistency, we present the Kerr metric, as it will be compared to metric (16) in In this work we analyze the physical properties and the electromagnetic signature analysis of accretion disks. characteristics of particles moving in circular 7 around general relativistic compact spheres in a station- For a Kerr black hole the geodesic equation (23) for r ary and axially symmetric geometry given by the follow- becomes ing metric r2 dr 2 2 2 2 2 2 = Veff (r) (29) ds = gtt dt +2gtφ dtdφ + grr dr + gθθ dθ + gφφ dφ . ∆K dτ (20)   Note that the metric functions gtt, gtφ, grr, gθθ and gφφ with the effective potential given by only depend on the radial coordinate r in the equatorial approximation, i.e., θ π 1, which is the case of V (r)= 1+ E2 r2(r2 + a2)+2ma2r | − | ≪ eff − interest here. In the following we denote the square root n of the determinant of the metric tensor by √ g. +4ELmar L2 r2 2mr  r2 r2 2mr + a2 . − − e − To compute the flux integral given by Eq. (39),− we de-  o (30) termine the radial dependence of the angular velocity Ω, e e e of the specific energy E and of the specific angular mo- Note that these relationships may be rewritten in the mentum L of particles moving in circular orbits around following manner compact spheres in a stationarye and axially symmetric geometrye through the geodesic equations. The latter take dr 2 r4 = V (r) (31) the following form dτ   dt Egφφ + Lgtφ with V (r) given by = 2 , (21) dτ g gttgφφ tφ − 2 2 2 2 e e V (r)= r ∆K Veff (r)= r (r 2mr + a )Veff (r) . (32) dφ Egtφ + Lgtt − = 2 , (22) 2 2 2 dτ −gtφ gttgφφ where the relationship ∆K = gtφ gttgφφ = r 2mr+a e − e along the equatorial plane has been− used. − dr 2 E2g +2ELg + L2g g = 1+ φφ tφ tt . (23) rr dτ − g2 g g   tφ − tt φφ e ee e B. Properties of thin accretion disks One may define an effective potential term defined as For the thin accretion disk, it is assumed that its verti- E2g +2ELg + L2g V (r)= 1+ φφ tφ tt . (24) cal size is negligible, as compared to its horizontal exten- eff − g2 g g tφ − tt φφ sion, i.e, the disk height H, defined by the maximum half e ee e thickness of the disk in the vertical direction, is always For stable circular orbits in the equatorial plane much smaller than the characteristic radius r of the disk, the following conditions must hold: Veff (r) = 0 and defined along the horizontal direction, H r. The thin ≪ Veff, r(r) = 0. These conditions provide the specific en- disk is in hydrodynamical equilibrium, and the pressure ergy, the specific angular momentum and the angular gradient and a vertical gradient in the accret- velocity of particles moving in circular orbits for the case ing matter are negligible. The efficient cooling via the of spinning general relativistic compact spheres, given by radiation over the disk surface prevents the disk from cu- mulating the heat generated by stresses and dynamical g + g Ω E = tt tφ , (25) . In turn, this equilibrium causes the disk to sta- 2 − gtt 2gtφΩ gφφΩ bilize its thin vertical size. The thin disk has an inner − − − g + g Ω edge at the marginally stable of the compact ob- Le = p tφ φφ , (26) g 2g Ω g Ω2 ject potential, and the accreting has a Keplerian − tt − tφ − φφ motion in higher orbits. 2 e dφ gtφ,r + (gtφ,r) gtt,rgφφ,r In steady state accretion disk models, the mass ac- Ω = p = − − . (27) cretion rate M˙ is assumed to be a constant that does dt p gφφ,r 0 not change with time. The physical quantities describ- The marginally stable orbit around the central object can ing the orbiting plasma are averaged over a characteristic be determined from the condition Veff, rr(r) = 0, which time scale, e.g. ∆t, over the azimuthal angle ∆φ = 2π provides the following important relationship for a total period of the orbits, and over the height H [22, 23, 24]. E2g +2ELg + L2g (g2 g g ) =0. The particles moving in Keplerian orbits around the φφ,rr tφ,rr tt,rr − tφ − tt φφ ,rr (28) compact object with a rotational velocity Ω = dφ/dt have e ee e a specific energy E and a specific angular momentum L, By inserting Eqs. (25)-(27) into Eq. (28) and solving this which, in the steady state thin disk model, depend only equation for r, we obtain the marginally stable orbit for on the radii of thee orbits. These particles, orbiting withe µ the explicitly given metric coefficients gtt, gtφ and gφφ. the four-velocity u , form a disk of an averaged surface 8 density Σ, the vertically integrated average of the rest can be expressed in terms of the specific energy, angu- mass density ρ0 of the plasma. The accreting matter in lar momentum and of the angular velocity of the central the disk is modeled by an anisotropic fluid source, where compact object [22, 24], µ the density ρ0 , the energy flow vector q and the stress tensor tµν are measured in the averaged rest-frame (the ˙ r specific heat was neglected). Then, the disk structure M0 Ω,r F (r)= (E ΩL)L,rdr . (39) can be characterized by the surface density of the disk −4π√ g 2 − (E ΩL) Zrms [22, 24], − − e e e e e H Another important characteristics of the mass accre- Σ(r)= ρ0 dz, (33) −H h i tion process is the efficiency with which the central object Z converts rest mass into outgoing radiation. This quan- with averaged rest mass density ρ0 over ∆t and 2π and tity is defined as the ratio of the rate of the radiation of the torque h i energy of escaping from the disk surface to infin- ity, and the rate at which mass-energy is transported to H r r the central compact general relativistic object, both mea- Wφ = tφ dz, (34) sured at infinity [22, 24]. If all the emitted photons can −H h i Z escape to infinity, the efficiency is given in terms of the r specific energy measured at the marginally stable orbit with the averaged component tφ over ∆t and 2π. The time and orbital average of theh energyi flux vector gives rms, the radiation flux (r) over the disk surface as (r) = qz . F F h i ǫ =1 Ems. (40) The stress-energy tensor is decomposed according to −

µν µ ν (µ ν) µν e T = ρ0u u +2u q + t , (35) For Schwarzschild black holes the efficiency ǫ is about 6%, whether the photon capture by the black hole is con- µ µν where uµq = 0, uµt = 0. The four-vectors of sidered, or not. Ignoring the capture of radiation by the the energy and angular momentum flux are defined by hole, ǫ is found to be 42% for rapidly rotating black holes, Eµ T µ(∂/∂t)ν and J µ T µ(∂/∂φ)ν , respectively. − ≡ ν ≡ ν whereas the efficiency is 40% with photon capture in the The structure equations of the thin disk can be derived Kerr potential [25]. by integrating the conservation laws of the rest mass, of the energy, and of the angular momentum of the plasma, The accreting matter in the steady-state thin disk respectively [22, 24]. From the equation of the rest mass model is supposed to be in thermodynamical equilibrium. conservation, (ρ uµ) = 0, it follows that the time aver- Therefore the radiation emitted by the disk surface can ∇µ 0 be considered as a perfect black body radiation, where aged rate of the accretion of the rest mass is independent 4 of the disk radius, the energy flux is given by F (r)= σSB T (r) (σSB is the Stefan-Boltzmann constant), and the observed luminos- M˙ 2π√ gΣur = constant. (36) ity L (ν) has a redshifted black body spectrum [28]: 0 ≡− − µ The conservation law µE = 0 of the energy has the 8 rf 2π ν3rdφdr integral form ∇ L (ν)=4πd2I (ν)= cos γ e . π exp(ν /T ) 1 Zri Z0 e − r (41) [M˙ 0E 2π√ gΩWφ ],r =4π√ gF E , (37) − − − Here d is the distance to the source, I(ν) is the Planck which statese that the energy transported bye the rest mass distribution function, γ is the disk inclination angle, and flow, M˙ 0E, and the energy transported by the dynami- ri and rf indicate the position of the inner and outer r cal stresses in the disk, 2π√ gΩW , is in balance with edge of the disk, respectively. We take ri = rms and − φ the energy radiated away from the surface of the disk, rf , since we expect the flux over the disk surface e → ∞ 4π gF E. The law of the angular momentum conser- vanishes at r for any kind of general relativistic √ → ∞ vation,− J µ = 0, also states the balance of these three compact object geometry. The emitted frequency is given ∇µ forms of thee angular momentum transport, by νe = ν(1+z), where the redshift factor can be written as [M˙ L 2πrW r] =4π√ gF L . (38) 0 − φ ,r − 1+Ωr sin φ sin γ r 1+ z = (42) By eliminatinge Wφ from Eqs. (37) ande (38), and g 2Ωg Ω2g applying the universal energy-angular momentum rela- − tt − tφ − φφ tion dE = ΩdJ for circular geodesic orbits in the form p E,r =ΩL,r, the flux F of the radiant energy over the disk where we have neglected the light bending [41, 42]. e e 9

IV. ELECTROMAGNETIC AND frequency for black holes increases moderately, whereas THERMODYNAMIC SIGNATURES OF it has only a negligible increment for gravastars. This ACCRETION DISKS AROUND SLOWLY makes the differences in the spectral properties more ROTATING GRAVASTARS AND BLACK HOLES acute for higher values of the spin parameter (a∗ > 0.3). The increase in the quadrupole moment somewhat∼ low- A. Electromagnetic and thermodynamic properties ers the amplitude of the spectra but causes a stronger of the disks decrease in the cut-off frequencies.

In this section we compare the radiation properties of thin accretion disks around gravastars and black holes in B. Conversion efficiency of the accreting mass the slowly rotating case when the spin parameter a∗ = 2 J/M has the maximal value of 0.5. In Fig. 1 we present We also present the conversion efficiency ǫ of the ac- the time averaged energy flux F (r) radiated by the disk creting mass into radiation, measured at infinity, which for both types of central objects with the total mass M 6 is given by Eq. (40), for the case where the photon cap- of 10 M⊙ and increasing spin parameter from 0.1 to 0.5. ture by the slowly rotating central object is ignored. The The quadrupole moment Q of the gravastar models runs value of ǫ measures the efficiency of energy generating between 0.1M 3 and 2M 3. Here the mass accretion rate −5 mechanism by mass accretion. The amount of energy M˙ is set to 2.5 10 M⊙/yr, which is in the typical 0 × released by matter leaving the marginally stable orbit, range for super massive cental objects. and falling down the black hole, is the binding energy If we compare the flux emerging from the surface of the Ems of the black hole potential. In Tabs. I and II, the thin accretion disk around black holes and gravastars, marginally stable orbits rms and ǫ are given for black we find that its maximal value is systematically lower holese and gravastars with the parameters a∗ and Q in the for gravastars, independently of the values of the spin range used for the plots presenting the radiation proper- parameter or the quadrupole momentum. For very slow ties of the accretion disks. rotation, (a∗ = 0.1), and a relative small value of the 3 quadrupole moment (Q =0.1M ), the radial distribution −2 a∗ rms [M] ǫ [10 ] of the disk radiation is close to each other for the two 0.1 5.67 6.06 types of compact central objects. The maximal flux for gravastars is roughly 90% of the black hole’s flux, and 0.2 5.33 6.46 the maximum of the inner edge of the accretion disk is 0.3 4.98 6.94 located at somewhat higher radii for gravastars. With 0.4 4.62 7.51 increasing rotational frequency of the central object, the 0.5 4.24 8.21 flux values also increase, but the increment is higher for black holes than for gravastars. For a∗ = 0.5 the flux maximum for black holes is almost twice the maximal flux TABLE I: The marginally stable orbit and the efficiency for value for gravastars. The more rapid rotation does not slowly rotating Kerr black holes with different spin parame- cause strong effect on the location of the inner disk edge ters. for gravastars, as we find a slight decrease in the value of rms as the rotation of central object increases (see in These values demonstrate the variation in the location Tab II). For black holes, the effect of the rotation is also of the inner disk edge with the changing spin parame- stronger here. Although the error of the approximation ter and quadrupole momentum, as we have seen in the applied for the slow rotation is rapidly increasing in the discussion on the radial distribution of the flux. The regime around a∗ 0.5, this picture is still definitely ∼ higher these values are, the closer the marginally sta- adequate for lower values of a∗. ble orbits are to the center in the dimensionless radial The variation of the quadrupole moment causes con- scale. For slowly rotating black holes, the conversion ef- siderable changes in both the maximal value of disk ra- ficiency is still close to 6%, which is the value obtained diation and the location of the inner edge of the disk. for Schwarzschild black holes. Up to a∗ =0.5 it increases As we increase Q, the maximal flux decreases and rms to 8%, which is still much lower than the one for extreme increases. These effects are presented in Fig. 2, show- Kerr black holes. For gravastars, ǫ has a smaller vari- ing the disk , although the differences are ation, and always remains smaller than the conversion somewhat less striking. efficiency for black holes. For very slow rotation, the ef- The disk spectra, presented in Fig. 3, have similar fea- ficiency is approximately 5.8%, which is close to the one tures in the dependence of the disk radiation on the ro- for the static black hole, and it decreases to 5% as the tation parameter and the quadrupole momentum. The quadrupole moment increases to 2M 3. For a higher spin amplitudes and the cut-off frequencies of the spectra for parameter (a∗ 0.5), ǫ is still around 6.5% but it be- gravastars are always lower than those for black holes. comes smaller than∼ 6% as Q increases. We conclude that For higher rotational velocity, the amplitudes are some- the conversation efficiency is higher for more rapidly ro- what higher but do not exhibit much change. The cut-off tating gravastars, but this is moderated by the increment 10

12 18 BH BH Q=0.1M3 16 Q=0.1M3 10 Q=0.5M3 Q=0.5M3 ] 3 ] 14 3 -2 Q=1.0M -2 Q=1.0M Q=2.0M3 Q=2.0M3

cm 8 cm 12 -1 -1 10 6 erg s erg s 8 13 13 4 6 (r) [10 (r) [10

F F 4 2 2 0 0 5 10 15 20 30 40 60 5 10 15 20 30 40 60 r/M r/M

25 30 BH BH Q=0.1M3 Q=0.1M3 Q=0.5M3 25 Q=0.5M3 ] 20 3 ] 3

-2 Q=1.0M -2 Q=1.0M Q=2.0M3 Q=2.0M3

cm cm 20 -1 15 -1 15 erg s erg s

13 10 13 10 (r) [10 (r) [10

F 5 F 5

0 0 5 10 15 20 30 40 60 5 10 15 20 30 40 60 r/M r/M

FIG. 1: The energy flux emerging from the accretion disk of slowly rotating gravastars and black holes for the spin parameter a∗ = 0.1 (upper left hand plot), a∗ = 0.3 (upper right hand plot), a∗ = 0.4 (lower left hand plot), and a∗ = 0.5 for (lower right 6 3 hand plot). All the plots are given for the total mass M = 10 M⊙, the quadrupole moments Q = 0.1, 0.5, 1.0, 2.0 times M , −5 and the mass accretion rate 2.5 × 10 M⊙/yr.

5 5 BH BH Q=0.1M3 Q=0.1M3 Q=0.5M3 Q=0.5M3 Q=1.0M3 Q=1.0M3 Q=2.0M3 Q=2.0M3 4 4 K] K] 4 4

T(r) [10 3 T(r) [10 3

2 2 5 10 15 20 30 40 60 5 10 15 20 30 40 60 r/M r/M

5 5 BH BH Q=0.1M3 Q=0.1M3 Q=0.5M3 Q=0.5M3 Q=1.0M3 Q=1.0M3 Q=2.0M3 Q=2.0M3 4 4 K] K] 4 4

T(r) [10 3 T(r) [10 3

2 2 5 10 15 20 30 40 60 5 10 15 20 30 40 60 r/M r/M

FIG. 2: The disk temperature of slowly rotating gravastars and black holes for the spin parameter a∗ = 0.1 (upper left hand plot), a∗ = 0.3 (upper right hand plot), a∗ = 0.4 (lower left hand plot), and a∗ = 0.5 for (lower right hand plot). All the plots 6 3 are given for the total mass M = 10 M⊙, the quadrupole moments Q = 0.1, 0.5, 1.0, 2.0 times M , and the mass accretion rate −5 2.5 × 10 M⊙/yr. 11

1040 1040

1039 1039

1038 1038 ] ] -1 1037 -1 1037

1036 1036 ) [erg s ) [erg s ν ν 1035 1035 L( L( ν ν 34 BH 34 BH 10 Q=0.1M3 10 Q=0.1M3 3 3 33 Q=0.5M 33 Q=0.5M 10 Q=1.0M3 10 Q=1.0M3 Q=2.0M3 Q=2.0M3 1032 1032 1015 1016 1015 1016 ν [Hz] ν [Hz]

1040 1040

1039 1039

1038 1038 ] ] -1 1037 -1 1037

1036 1036 ) [erg s ) [erg s ν ν 1035 1035 L( L( ν ν 34 BH 34 BH 10 Q=0.1M3 10 Q=0.1M3 3 3 33 Q=0.5M 33 Q=0.5M 10 Q=1.0M3 10 Q=1.0M3 Q=2.0M3 Q=2.0M3 1032 1032 1015 1016 1015 1016 ν [Hz] ν [Hz]

FIG. 3: The disk spectra of slowly rotating gravastars and black holes for the spin parameter a∗ = 0.1 (upper left hand plot), a∗ = 0.3 (upper right hand plot), a∗ = 0.4 (lower left hand plot), and a∗ = 0.5 for (lower right hand plot). All the plots are 6 3 given for the total mass M = 10 M⊙, the quadrupole moments Q = 0.1, 0.5, 1.0, 2.0 times M , and the mass accretion rate −5 2.5 × 10 M⊙/yr.

3 −2 a∗ Q [M ] rms [M] ǫ [10 ] in its quadrupole moment. In addition to this, it is al- 0.1 0.1 5.91 5.82 ways smaller than the ǫ for black holes, i.e., gravastars 0.1 0.5 6.20 5.60 provide a less efficient mechanism for converting mass to 0.1 1.0 6.50 5.37 radiation than black holes. 0.1 2.0 7.00 5.02 In order that our proposal for discriminating gravastars 0.2 0.1 5.75 6.00 from black holes by using the electromagnetic emission 0.2 0.5 6.05 5.75 properties of accretion disks could be effectively applied 0.2 1.0 6.37 5.50 in concrete observational cases, it is necessary to know at least the value of the mass and of the spin parameter 0.2 2.0 6.77 5.23 of the central rotating compact object. To estimate the 0.3 0.1 5.58 6.19 spins of stellar-mass black holes in X-ray binaries, one 0.3 0.5 5.89 5.90 has to fit the continuum X-ray spectrum of the radia- 0.3 1.0 6.23 5.63 tion from the accretion disk, by using the standard thin 0.3 2.0 6.89 5.12 disk model, and extract the dimensionless spin parame- 0.4 0.1 5.39 6.40 ter a = a/M of the black hole as a parameter of the fit ∗ 0.4 0.5 5.74 6.08 [44]. Recently, a number of precise black hole spin de- 0.4 1.0 6.09 5.77 terminations have been reported. By using Chandra and Gemini-North observations of the eclipsing X-ray binary 0.4 2.0 6.65 5.33 M33 X-7, precise values of the mass of its black hole pri- 0.5 0.1 5.20 6.64 mary and of the system’s angle have 0.5 0.5 5.58 6.27 been obtained. The distance to the binary is also known 0.5 1.0 5.95 5.93 to a few percent. By using these precise results, from the 0.5 2.0 6.52 5.45 analysis of 15 Chandra and XMM-Newton X-ray spectra, and a fully relativistic accretion disk model, one can find that the dimensionless spin parameter of the black hole TABLE II: The marginally stable orbit and the efficiency for is a = 0.77 0.05 [45]. Therefore, even that slowly rotating gravastars with different spin parameters and presently there∗ are severe± observational limitations for quadrupole moments. the application of our proposal, with the future improve- 12 ments of the observational techniques, the observation expansion bursts from a given source remain constant of the emission spectra of accretion disks could be effec- between bursts to within a few percent, giving empirical tively used to discriminate between gravastars and black verification to the theoretical expectation that the emerg- holes. ing luminosity is approximately equal to the Eddington critical luminosity. The at infinity of a gravastar is given by [48] V. DISCUSSIONS AND FINAL REMARKS 4πm R2 deν/2 L∞ = 0 e−λ , (44) E σ dr If gravastars are surrounded by a thin shell of mat- r=R ter, the presence of a turning point for matter (the point where the motion of the infalling matter suddenly stops) where m0 is the mass of the particle and σ is the inter- at the surface of the gravastars may have important as- action cross section. For the gravastar we obtain trophysical and observational implications. Since the velocity of the matter at the gravastar surface is zero, ∞ 4πm0M 2M LE = 1 . (45) matter can be captured and deposited on the surface of σ − r r r=R gravastar.Therefore gravastars may have a gaseous sur- face, formed from a a thin layer of dense and hot gas. The ratio of the Eddington at infinity for 6 12 Moreover, because matter is accreted continuously, the a gravastar with mass of 4 10 M⊙ and radius 1.4 10 × × increase in the size and density of the surface will ignite cm and a with mass of 2 solar and some thermonuclear reactions [29]. The ignited reactions radius of 10 km is L∞ /L∞ = 1.22 106. Finally, Egrav ENS × are usually unstable, causing the accreted layer of gas we consider the apparent surface area during burst cool- to burn explosively within a very short period of time. ing. Observations of the cooling tails of multiple type After the nuclear fuel is consumed, the gravastar reverts I bursts from a single source have shown that the ap- to its accretion phase, until the next thermonuclear in- parent surface area of the emitting region, defined as ∞ 2 4 stability is triggered. Thus, gravastars may undergo a S =4πD Fc,∞/σSBTc,∞, where Fc,∞ is the measured semi-regular series of , called type I thermonu- flux of the source during the cooling tail of the burst, clear bursts, discovered first for X-ray binaries [46, 47]. Tc,∞ is the measured color temperature of the burst spec- The observational signatures indicating the presence trum, D is the distance to the source and σSB is the of X-ray bursts from gravastars are similar to those of Stefan-Boltzmann constant, remains approximately con- standard neutron stars, and are the stant during each burst, and between bursts from the of a surface atomic line, the touchdown luminosity of a same source. The color temperature on the surface of radius-expansion burst, and the apparent surface area the compact object Tc,h is related to the color temper- −ν(R)/2 during the cooling phases of the burst [48]. ature measured at infinity by Tc,h = Tc,∞e [48]. If the with wavelength λe emitted by By introducing the color correction factor fc = Tc/Teff , the matter at the surface of the gravastar has absorption where Teff is the effective temperature at the surface, we or emission features characteristic of atomic transitions, obtain these features will be detected at infinity at a wavelength 2 ∞ R 2 λo, gravitationally redshifted with a value S =4π 4 [z(R)+1] . (46) fc λ λ z = o − e = e−ν(R)/2 1, (43) grav λ − Since the radius of the gravastar as well as its surface e redshift may be very large quantities, the apparent area where we have assumed, for simplicity, that the gravas- of the emitting region as measured at infinity may be also tar is static, and that the exterior metric can be approx- very large. Hence all the astrophysical quantities related imated by the standard Schwarzschild metric, given by to the observable properties of the X-ray bursts, origi- 6 Eq. (8). By assuming a gravastar of mass M =4 10 M⊙ nating at the surface of the gravastar can be calculated, and radius R = 1.4 1012 cm, we obtain a surface× red- and have finite values on the surface of the gravastar and × shift of zgrav = 1.55. The corresponding value of the at infinity. redshift for a neutron star with mass M =2M⊙ and ra- It was argued in [29] that any neutron star, composed 6 dius R = 10 cm is zNS =0.56. Therefore the radiation by matter described by a more or less general equation coming from the surface of a gravastar may be highly red- of state, should experience thermonuclear type I bursts shifted (in standard general relativity the redshift obeys at appropriate mass accretion rates. The question asked the constrain z 2). in [29] is whether an “abnormal” surface may allow such Type I X -ray≤ bursts show strong spectroscopic evi- a behavior. The gravastars may have such a zero veloc- dence for a rapid expansion of the radius of the X-ray ity, particle trapping, abnormal surface. The presence . The luminosities of these bursts reach the of a material surface located at the “event horizon” im- Eddington critical luminosity at which the outward radi- plies that energy can be radiated, once matter collides ation force balances gravity, causing the expansion lay- with that surface. Thus, gravastar models, characterized ers of the star. The touchdown luminosity of radius- by high mass, normal matter crusts/surfaces and type 13

−10 I thermonuclear bursts can be theoretically constructed. A*: M˙ > 10 M⊙/yr. Thus it follows that Sgr A* Moreover, some of the so-called SXT’s (soft X-ray tran- does not have a surface, i.e., that it must have an event sients), having a relatively low mass function (e.g. SXT horizon. This argument could be made more restrictive A0620-00, with a mass function f(M) 3M⊙ [29]), but by an order of magnitude with microarcsecond resolution ≥ still exceeding the equilibrium limit of 3M⊙, or very mas- imaging, e.g., with submillimeter very long baseline inter- sive neutron stars showing the presence of a crust, may ferometry. Submilliarcsecond and imaging of in fact be gravastars. the black hole Sgr A* at the Galactic Centre may become It is generally expected that most of the astrophysical possible in the near future at and submillimetre objects grow substantially in mass via accretion. Re- wavelengths [55]. The expected images and light curves, cent observations suggest that around most of the active including polarization, associated with a compact emis- galactic nuclei (AGN’s) or black hole candidates there sion region orbiting the central black hole were computed exist gas clouds surrounding the central far object, and in [56]. From spot images and light curves of the observed an associated accretion disk, on a variety of scales from flux and polarization it is possible to extract the black a tenth of a to a few hundred [43]. These hole mass and spin. At radio wavelengths, disc opacity clouds are assumed to form a geometrically and optically produces significant departures from the infrared behav- thick torus (or warped disk), which absorbs most of the ior, but there are still generic signatures of the black hole ultraviolet radiation and the soft x-rays. The gas exists in properties. Detailed comparison of these results with fu- either the molecular or the atomic phase. Evidence for ture data can be used to test general relativity, and to the existence of super massive black holes comes from improve existing models for the accretion flow in the im- the very long baseline interferometry (VLBI) imaging of mediate vicinity of the black hole. molecular H2O in active , like NGC 4258 With the improvement of the imaging observational [49], and from the astrometric and mea- techniques, it will also be possible to provide clear obser- surements of the fully unconstrained Keplerian orbits for vational evidence for the existence of gravastars, and to short period stars around the supermassive black hole at differentiate them from other types of compact general the center of our [50, 51]. The VLBI imaging, relativistic objects. Indeed, in this work we have shown produced by Doppler shift measurements assuming Kep- that the thermodynamic and electromagnetic properties lerian motion of the masering source, has allowed a quite of the disks (energy flux, temperature distribution and accurate estimation of the central mass, which has been equilibrium radiation spectrum) are different for these 7 found to be a 3.6 10 M⊙ super massive dark object, × two classes of compact objects, consequently giving clear within 0.13 parsecs. Hence, important astrophysical in- observational signatures. More specifically, comparing formation can be obtained from the observation of the the energy flux emerging from the surface of the thin ac- motion of the gas streams in the gravitational field of cretion disk around black holes and gravastars of similar compact objects. masses, it was found that its maximal value is system- Therefore the study of the accretion processes by com- atically lower for gravastars, independently of the val- pact objects is a powerful indicator of their physical na- ues of the spin parameter or the quadrupole momentum. ture. However, up to now, the observational results have These effects are confirmed from the analysis of the disk confirmed the predictions of general relativity mainly in temperatures and disk spectra. In addition to this, it is a qualitative way. With the present observational preci- also shown that the conversion efficiency of the accreting sion one cannot distinguish between the different classes mass into radiation is always smaller than the conversion of compact/exotic objects that appear in the theoreti- efficiency for black holes, i.e., gravastars provide a less ef- cal framework of general relativity [29]. However, with ficient mechanism for converting mass to radiation than important technological developments one may allow to black holes. Thus, these observational signatures may image black holes and other compact objects directly [52]. provide the possibility of clearly distinguishing rotating Recent observations at a wavelength of 1.3 mm have set a gravastars from Kerr-type black holes. size of microarcseconds on the intrinsic diameter of SgrA* [53]. This is less than the expected apparent size of the event horizon of the presumed black hole, thus suggest- ing that the bulk of SgrA* emission may not be centered Acknowledgments on the black hole, but arises in the surrounding accre- tion flow. A model in which Sgr A* is a compact object with a thermally emitting surface was considered in [54]. We would like to thank the two anonymous referees Given the very low quiescent luminosity of Sgr A* in for suggestions and comments that helped us to signif- the near-infrared, the existence of a hard surface, even icantly improve the manuscript. The work of TH was in the limit in which the radius approaches the hori- supported by the General Research Fund grant number zon, places a severe constraint on the steady mass ac- HKU 701808P of the government of the Hong Kong Spe- −12 cretion rate onto the source: M˙ 10 M⊙/yr. This cial Administrative Region. ZK was supported by the limit is well below the minimum≤ accretion rate needed Hungarian Scientific Research Fund (OTKA) grant No. to power the observed submillimeter luminosity of Sgr 69036. 14

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