Accretion process onto super-spinning objects
Cosimo Bambi, Katherine Freese, Tomohiro Harada, Rohta Takahashi, Naoki Yoshida
aa r X i v : . [ g r- q c ] O c t IPMU09-0113
Accretion process onto super-spinning objects
Cosimo Bambi , ∗ Katherine Freese , † Tomohiro Harada , ‡ Rohta Takahashi , § and Naoki Yoshida ¶ Institute for the Physics and Mathematics of the Universe,The University of Tokyo, Kashiwa, Chiba 277-8568, Japan The Michigan Center for Theoretical Physics, Department of Physics,University of Michigan, Ann Arbor, Michigan 48109, USA Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan Cosmic Radiation Laboratory, the Institute of Physical and Chemical Research, Wako, Saitama 351-0198, Japan (Dated: October 7, 2018)The accretion process onto spinning objects in Kerr spacetimes is studied with numerical simu-lations. Our results show that accretion onto compact objects with Kerr parameter (characterizingthe spin) | a | < M and | a | > M is very different. In the super-spinning case, for | a | moderatelylarger than M , the accretion onto the central object is extremely suppressed due to a repulsiveforce at short distance. The accreting matter cannot reach the central object, but instead is ac-cumulated around it, forming a high density cloud that continues to grow. The radiation emittedin the accretion process will be harder and more intense than the one coming from standard blackholes; e.g. γ -rays could be produced as seen in some observations. Gravitational collapse of thiscloud might even give rise to violent bursts. As | a | increases, a larger amount of accreting matterreaches the central object and the growth of the cloud becomes less efficient. Our simulations findthat a quasi-steady state of the accretion process exists for | a | /M > ∼ .
4, independently of the massaccretion rate at large radii. For such high values of the Kerr parameter, the accreting matter formsa thin disk at very small radii. We provide some analytical arguments to strengthen the numericalresults; in particular, we estimate the radius where the gravitational force changes from attractiveto repulsive and the critical value | a | /M ≈ . PACS numbers: 04.20.Dw, 97.60.-s, 95.30.Lz, 97.10.Gz
I. INTRODUCTION
It is widely believed that the final product of the gravitational collapse of matter is a black hole (BH). In classicalgeneral relativity (GR), astrophysical BHs should be completely characterized by just three parameters: the mass M , the charge Q , and the spin J . In this paper we focus on chargeless BHs. The spin is often replaced by the Kerrparameter a = J/M . In classical GR, the values of M and a cannot be completely arbitrary, as they must satisfythe relation | a | < M , which is the condition for the existence of the horizon. To see this we can examine the 3+1dimensional Kerr solution. The position of the horizon is given by the expression [1, 2] r H = M + p M − a . (1)It is clear that in (3+1)D spacetime the horizon cannot be formed if M < | a | . (2)In the absence of a horizon, there would be naked singularities which are not allowed in GR. Indeed, if condition (2)is fulfilled, the Kerr metric makes it possible to reach the physical singularity at r = 0 from some large r in finitetime without crossing any horizon. One could thus consider closed time-like curves and violate causality (see e.g.section 66c of [3] or ref. [4]). For this reason, usually some kind of cosmic censorship is assumed and naked singularitiesare forbidden [5]. In particular, it is believed that naked singularities cannot be created by any physical process andtherefore that the end-state of the gravitational collapse of matter is a Kerr BH with | a | < M [5]. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected]
However, in this paper we consider objects which do violate the Kerr bound, i.e. with | a | > M . We call them“super-spinars”, as proposed in [6]: since they have no event horizon, by the standard definition they are not BHs.Our main motivation is simple. The singularity can be viewed as the place where new physics should be expected:here observer-independent quantities like the scalar curvature diverge, while GR presumably breaks down above thePlanck scale. It is therefore not unreasonable to expect that causality is conserved, not because the collapsing mattercan form only objects with | a | < M , but because actually the central singularity is replaced by some high curvatureregion due to some quantum gravity effects, see e.g. [6, 7]. In this case, there is apparently no reason to believethat the final product of the gravitational collapse of matter cannot have | a | > M . Another possibility is that thecollapsing matter forms a super-compact star with | a | > M and exotic equations of state: now there is no centralsingularity, since the Kerr metric is a solution of Einstein equations only in vacuum; matter could have very exoticequation of state once it reaches densities so high that our knowledge of physics becomes inadequate. Actually, ingeneral the metric at very small radii may deviate from the Kerr solution (the uniqueness theorem does not hold inabsence of a regular horizon [8]), but in our discussion we will neglect such a possibility.In this paper, we extend the studies started in [9, 10]. The goal is to examine the main differences between the cases | a | < M and | a | > M . Previous papers [9, 10] discussed implications on the apparent shape. There it was found that,even if the bound is violated by a small amount, the shadow cast by the super-spinar (i.e. how it blocks light comingto us from an object behind it) changes significantly from the case with | a | < M : the shadow for the super-spinar isabout an order of magnitude smaller as well as distorted. This distinction can be used as an observational signaturein the search for these objects. Based on recent observations at mm wavelength of the super-massive BH candidateat the Center of the Galaxy [11], the authors speculated on the possibility that it might violate the Kerr bound.In this paper we discuss the process of accretion in a Kerr background with arbitrary value of the Kerr parameter.For | a | moderately larger than M , we find that the accreting matter cannot reach the central object, but is accumulatedaround it, forming a high density cloud. That may have interesting observational consequences. First, because ofthe high density and the high temperature of the plasma, the radiation produced in the accretion process can bemuch harder and more intense than the one coming from BHs. Second, there might be violent phenomena like bursts,when the amount of accumulated gas is large enough to gravitationally collapse. For higher values of | a | , the cloudevolves into a sort of disk, which is however very different from the usual disk of accretion around a BH: here the diskextends from r ≈ M to the center, leading to rapid accretion, increasing efficiently the mass of the central object,and producing hard radiation at very small radii.Unlike the case of Kerr BHs, we do not know if super-spinars are stable under small perturbations. Previous workhas found that some very rapidly rotating objects in 3+1 and higher dimensions can be unstable [12, 13]. To addressthis point regarding the super-spinars studied in this paper, one should do a linearized analysis of the perturbationsof these objects. However, the conclusion would be determined by the boundary conditions at the surface of super-spinars, which presumably depend on the quantum theory of gravity and are therefore unknown. Such a questioncannot thus be addressed at present: here we just assume that super-spinars are stable and we study the accretionprocess onto these objects. Conventions : We use natural units G N = c = k B = 1. The metric has signature ( − + ++). II. MODEL AND ASSUMPTIONSA. Equations
In this subsection we briefly review the basic ingredients of the formalism used in our study. For more details, seee.g. [14, 15, 16] and references therein.We are going to simulate the accretion process of a test fluid in a background curved spacetime; that is, we neglectthe back-reaction of the fluid to the geometry of the spacetime, as well as the increase in mass and the variation inspin of the central object due to accretion. Such an approximation is surely reasonable if we want to study a stellarmass compact object in a binary system, because in this case the matter captured from the stellar companion istypically small in comparison with the total mass of the compact object. The results of our simulations should notbe applied to long-term accretion onto a super-massive object at the center of a galaxy, where accretion makes themass of the compact object increase by a few orders of magnitude from its original value.Our master formulas are the equations of conservation of baryon number and of the fluid energy-momentum tensor ∇ µ J µ = 0 , (3) ∇ µ T µν = 0 , (4)where J µ and T µν are respectively the current of matter and the fluid energy-momentum tensor J µ = ρu µ , (5) T µν = ρhu µ u ν + pg µν . (6)Here ρ is the rest-mass energy density (for example, in the case of hydrogen plasma, ρ = n ( m p + m e ), where n isthe number density of protons/electrons and m p ( m e ) the proton (electron) mass), p is the pressure, u µ is the fluidfour-velocity, h = 1 + ǫ + p/ρ is the specific enthalpy, and ǫ is the specific internal energy density. In other words, ρǫ is the thermal energy density ( ρǫ = 3 nT for non-relativistic hydrogen plasma), while ρ (1 + ǫ ) is the total energydensity of the fluid. In order to solve the system, an equation of state p = p ( ρ, ǫ ) must be specified.In the 3+1 formalism, the line element of the spacetime is written in the form ds = − (cid:0) α − β i β i (cid:1) dt + 2 β i dtdx i + γ ij dx i dx j , (7)where α is the lapse function, β i the shift vector and γ ij the 3-metric induced on each space-like slice. Greek indicesrun from 0 to 3 and are lowered (uppered) by the 4-metric g µν (inverse 4-metric g µν ). Latin indices run from 1 to3 and are lowered (uppered) by the 3-metric γ ij (inverse 3-metric γ ij ). For Eulerian observers, the 3-velocity of thefluid is given by v i = u i αu + β i α , (8)while the covariant components can be obtained by using γ ij , i.e. v i = γ ij v j . For what follows, it is convenient to usetwo sets of variables. The primitive variables are V = (cid:0) ρ, v i , p (cid:1) T (9)and are the quantities whose evolution and quasi-steady state (if any) we want to determine. The hydrodynamicalequations are instead solved in term of the conserved variables U = ( D, S i , τ ) T , (10)which can be written in term of the primitive ones as D = ρW , (11) S i = ρhW v i , (12) τ = ρhW − D − p , (13)where W = αu = (1 − v ) − / is the Lorentz factor and v = v i v i . The equations of conservation (3) and (4) cannow be written as 1 √− g (cid:20) ∂∂t ( √ γ U ) + ∂∂x i (cid:0) √− g F i (cid:1)(cid:21) = S , (14)where F i and S are defined by [14] F i = (cid:16) D (cid:0) v i − β i /α (cid:1) , S j (cid:0) v i − β i /α (cid:1) + pδ ij , τ (cid:0) v i − β i /α (cid:1) + pv i (cid:17) T , (15) S = (cid:16) , T µν (cid:0) ∂ µ g νj − Γ λµν g λj (cid:1) , α (cid:0) T µ ∂ µ ln α − T µν Γ µν (cid:1) (cid:17) T . (16)These equations are solved numerically by integrating over the computational cells of the discretized spacetime. B. Background metric and test fluid
In our study, the spacetime is described by the Kerr metric. Using the Boyer-Lindquist coordinates, the lineelement (7) becomes ds = − α dt + Σ sin θ̺ ( dφ − ωdt ) + ̺ ∆ dr + ̺ dθ , (17)where α = ̺ ∆ / Σ , ω = 2 aM r/ Σ , and ∆, ̺ and Σ are defined by∆ = r − M r + a , (18) ̺ = r + a cos θ , (19)Σ = (cid:0) r + a (cid:1) − a ∆ sin θ . (20)The equation of state of the accreting matter is the one of an ideal gas with constant polytropic index Γ: p = (Γ − ǫρ . (21)In the simulations, we take Γ = 5 / C. Calculation method
Our calculations are made with the relativistic hydrodynamics (RHD) module of the public available codePLUTO [17, 18], properly modified for the case of curved spacetime, as described in subsection II A. We do notsolve the Riemann problem to compute fluxes, but we use a Lax-Friedrichs scheme; flux contributions are evalu-ated from all directions simultaneously (no dimensional splitting); time evolution uses a second order Runge-Kuttaalgorithm.The computational domain is the 2D axysimmetric space r in < r < M and 0 < θ < π , where r in is set accordingto the case under study. In this paper we present the results for six different values of the Kerr parameter: a = 0 . M , a = 0 . M , a = 1 . M , a = 1 . M , a = 2 . M , and a = 3 . M . In the first two cases, we have a BH with event horizonat radial coordinate r H = 1 . M and r H = 1 . M respectively, and the inner boundary is set just outside the eventhorizon: r in = 2 . M for a = 0 . M and r in = 1 . M for a = 0 . M . We adopt free-outflow boundary conditions,i.e. we set zero gradient across the inner boundary:. ∂ρ∂n = ∂p∂n = ∂v i ∂n = 0 , (22)where n is the coordinate orthogonal to the boundary plane.For | a | > M , there is no event horizon but a naked singularity at r = 0. In the simulations presented in this paper,we took r in = 0 . M and we imposed no flow from the boundary. The choice of the value of r in can appear arbitrary.However, it does not significantly alter the final result for any value of | a | /M , while for smaller r in the computationaltime increases considerably. As for the choice of the boundary condition, we have imposed no flow from the boundary,in order to prevent an unphysical injection of gas from the center.Both for BHs and super-spinars, at the beginning of the simulations the density of the gas (plasma) around thecompact objects is constant (and equal to ρ , see below) and we start injecting gas from the outer boundary at aconstant rate. The gas is injected with radial velocity v r = − . v θ = v φ = 0. We found that, for a givenaccretion rate, the final density profile is independent of the injection velocity for v r < ∼ − . M : M = 1 . · (cid:18) M M ⊙ (cid:19) cm , (23)while the unit of density, ρ , is unspecified. The mass accretion rate of the system in the simulations is˙ M = 3 . · (cid:18) M M ⊙ (cid:19) (cid:18) ρ − g / cm (cid:19) g / s , (24)
0 1 2 3 4 5-2-1 0 1 2 1 10 100 1000 10000 0 1 2 3 4 5-2-1 0 1 2 1 10 100 1000 10000
FIG. 1: Density of the accretion flow around a Kerr BH with a = 0 . M (left panel) and a = 0 . M (right panel). The innerboundary is at r in = 2 . M for the case a = 0 . M , and at r in = 1 . M for the case a = 0 . M . The unit of length along the x and y axes is M . The density scale (shown on the right side of each figure) ranges from ρ to 10 ρ , where ρ is our unit ofdensity (see discussion in subsection II C). or, in Eddington units , ˙ M ˙ M E ! = 0 . (cid:18) M M ⊙ (cid:19) (cid:18) ρ − g / cm (cid:19) . (25)If we want to consider the accretion process onto an object with a certain mass at a particular accretion rate, we haveto adjust the unit of density ρ . For example, in the case of a super-massive object with M = 10 M ⊙ and accreting at10 − its Eddington limit, probably like the BH candidate at the Galactic Center, one has to take ρ = 10 − g / cm .Because of our simple treatment of the accreting matter, the gas temperature becomes extremely high. Here wesimply impose a maximum temperature: this work does not aspire to get an accurate description of the accretionprocess, but only to figure out the main differences between the accretion process onto objects with | a | < M and | a | > M . We have checked that the choice of T max does not change significantly the final results: for T max = 10 KeV,100 KeV and 1 MeV, the density changes by at most a few percent, while we do not expect that our simulations areas accurate. III. RESULTSA. Numerical study
The results of the simulations are summarized in figs. 1, 2, 3, 4, and 5. In fig. 1, we present the density, ρ , of theaccretion flow onto BH with Kerr parameter a = 0 . M (left panel) and a = 0 . M (right panel). The density scale inthese and the other pictures ranges from ρ to 10 ρ , where ρ is our unit of density, as discussed in subsection II C.The white area is out of the domain of computation. For both choices of a , the accretion flow reaches a quasi-steadystate, that is, the density does not depend on time.Fig. 2 (left panel) shows the density of the accretion flow around a super-spinar with a = 1 . M . We use the samecolor intensity scale of fig. 1, in order to facilitate the comparison between the cases | a | < M and | a | > M . Forthe cells with density higher (lower) than 10 ρ ( ρ ), we use the same color scheme as though their density were10 ρ ( ρ ). For example, a cell with ρ = 10 ρ will have the same color as a cell with ρ = 10 ρ , and a cell with ρ = 10 − ρ will have the same color as a cell with ρ . As in the BH case, the white area is the region out of the Since we assume that the process is adiabatic, the unit of density ρ can be left unspecified and our results can be applied (in principle)to any accretion rate, even orders of magnitude different. Such an assumption holds only for small accretion rates and optically thingas. For example, our results would predict no qualitative differences between sub-Eddington and super-Eddington accretions, which isdefinitively not true because for high accretion rates the pressure of radiation cannot be neglected.
0 1 2 3 4 5-2-1 0 1 2 1 10 100 1000 10000 D en s i t y Radius
FIG. 2: Left panel: density plot of the accretion flow around a super-spinar with a = 1 . M (color scheme as described infig. 1). Right panel: radial density profile on the equatorial plane of the same object. The inner boundary is at r in = 0 . M .
0 1 2 3 4 5-2-1 0 1 2 1 10 100 1000 10000 D en s i t y Radius
FIG. 3: Left panel: density plot of the accretion flow around a super-spinar with a = 1 . M (color scheme as described infig. 1). Right panel: radial density profile on the equatorial plane of the same object. The inner boundary is at r in = 0 . M . domain of computation (now r < . M ). In the right panel, we present the radial density profile on the equatorialplane (i.e. the plane with vertical axis y = 0 in the previous plot).The peculiar feature of the case a = 1 . M is that the space around the central object is almost empty (the blackregion in the picture, where actually ρ ≪ ρ ). This is not the result of very high inflow velocity of the gas, butthe effect of the repulsive force at short distances from the center . The gas around the center is pushed away tolarger radii, while the gas far from the object is attracted by the usual gravitational force. The object does not reallyaccrete; instead matter accumulates around it, thus forming a high density cloud. Actually, on the equatorial planethere is some amount of gas falling into the super-spinar, but the fraction of matter with respect to the injected massis so small that it is negligible. It is therefore clear that we cannot find any quasi-steady state in this case: matter iscontinuously accumulated into the cloud (fig. 2 shows the density profile at the time t = 10 M of the simulation) . Here the repulsive force is due to the singularity; it is not the centrifugal force due to the rotation of the accreting gas. The fact that thegravitational force of a time-like singularity might be repulsive in its neighborhood has been already noticed in the literature, see e.g.refs. [19, 20]. The simplest example is the Schwarzschild spacetime with negative mass: there is no horizon, but a time-like singularityat r = 0, and the gravitational force is repulsive, as one can easily see by considering the Newtonian limit. We notice that a similar result was obtained in ref. [21] for the case of Reissner-Nordstr¨om naked singularity: even there, no quasi-steadyaccretion is possible and an “atmosphere” of fluid is formed around the center.
100 1000 10000 100000 1e+06 0 2000 4000 6000 8000 10000 M a ss Time 100 1000 10000 100000 1e+06 0 2000 4000 6000 8000 10000 M a ss Time
FIG. 4: Rest mass of the gas (in units M ρ ) as a function of time (in units M ) inside the region 0 . M < r < . M in thecases of super-spinars with a = 1 . M (left panel) and a = 1 . M (right panel). For values of the Kerr parameter just above theKerr bound, we do not find any quasi-steady state and the gas is continuously accumulated around the object. For a = 1 . M ,the flow reaches a quasi-steady state at t ≈ M . Such a statement is checked by plotting the rest mass of the gas, defined by M gas = Z ρ √ γ d x , (26)in the region 0 . M < r < . M , as a function of time (fig. 4, left panel): as the time goes on, the total mass of thegas increases and there is no quasi-steady state.As | a | /M increases, more and more matter falls to the center and leaves the domain of computation. The radialcoordinate, say r max , of the point of the cloud with the highest density decreases (e.g. r max ≈ . M for | a | /M = 1 . r max ≈ . M for | a | /M = 1 .
2, and r max ≈ . M for | a | /M = 1 .
3) and the growth of the cloud slows down. For | a | /M = 1 . r max approaches r in and the process reaches aquasi-steady state; that is, the flux leaving the simulation at r in (moving towards the center) equals the flux of matterentering the simulation at r out = 20 M . In fig. 3 we can see the density plot of a super-spinar with a = 1 . M . Stillthere is an empty region (black in the picture) in the neighborhood of the center. However, that is not true near theequatorial plane, where the density increases as r decreases and r max ≈ r in , see the right panel of fig. 3. The rightpanel of fig. 4 shows that in this case the accreting flow reaches a quasi-steady state.Since for | a | /M > ∼ . r max goes to r in , it is natural to wonder whether the threshold a = 1 . M dividing two qualitatively different cases is determined by the value of r in . We have therefore run the codewith smaller values of r in , finding however the same result: r max → r in and for | a | /M > ∼ . | a | /M = 1 . a = 2 . M (left panel) and a = 3 . M (rightpanel). The pictures show that the volume of the empty region around the compact object increases for higher valuesof the Kerr parameter. Yet the accreting matter reaches the inner boundary at r in more and more easily, thus leavingthe computational domain and forming a structure similar to a disk. Such a disk is very different from the standardaccretion disk around BHs. In very rapid super-spinars, the disk extends from r ∼ | a | / ∼ M , inward to r ∼ r isco , where r isco is the innermoststable circular orbit (ISCO). The latter is 6 M for a Schwarzschild BH and as small as M for a BH with | a | /M = 1.In addition to this, it is important to notice that here the disk appears in the case of initially spherically symmetricaccretion, as the gas initially does not have angular momentum. It is the result of the peculiar gravitational force atsmall radii around super-spinars. B. Qualitative Explanation
The results of our simulations may seem too exotic and unexpected. In this subsection, we present a few argumentsto show that the emerging picture is physically reasonable.
0 1 2 3 4 5-2-1 0 1 2 1 10 100 1000 10000 0 1 2 3 4 5-2-1 0 1 2 1 10 100 1000 10000
FIG. 5: Density plot of the accretion flow around a super-spinar with a = 2 . M (left panel) and a = 3 . M (right panel). Colorscheme as described in fig. 1. The inner boundary is at r in = 0 . M . First, let us see that the effective force can be repulsive at small radii. Assuming negligible velocity (i.e. ˙ r ≈ ˙ θ ≈ ˙ φ ≈ r ≈ − Γ rtt ˙ t = − ∆ M (cid:0) r − a cos θ (cid:1) ̺ ˙ t , (27)where ’dot’ stands for a derivative with respect to the affine parameter and ∆ is defined in Eq.(18). Since ∆ > r when | a | > M , the force is attractive or repulsive according to the sign of ( r − a cos θ ). The same conclusioncan be deduced from the master equations (14): if the fluid is non-relativistic, i.e. v ≪ h ≈
1, the radialcomponent of eq. (14) becomes ∂S r ∂t ≈ − √− g √ γ Γ rtt T tt g rr = − √− g √ γ M ρ (cid:0) r − a cos θ (cid:1) Σ ̺ ∆ , (28)and the direction of the force still depends on the sign of ( r − a cos θ ). In the case of Kerr BH with | a | < M , theforce is always attractive because ( r − a cos θ ) > r > r H > M , while for r < r H the Kerr metric doesnot hold. We note also that, for the case of r ≫ | a | and r ≫ M , eq. (28) reduces to the usual Newtonian case, with − M ρ/r on the right hand side.Eqs. (27) and (28) suggest that the force is repulsive inside the two spherical regions of radius | a | / r = | a | / θ = 0, π . This is basically the shape and the size of the empty region around thecenter in the cases a = 1 . M (fig. 3), a = 2 . M , and 3 . M (fig. 5), while it is less clear for a = 1 . M . According tothis interpretation, the origin of the repulsive force at small radii is a pure geometrical effect due to the spin a , andnot the result of an effective force due to the angular momentum of the accreting plasma. We checked such a guessby performing some simulations injecting gas with different angular momenta. We found that the shape and the sizeof the empty regions around the center are essentially unaffected by the choice of the initial angular momentum ofthe gas, whose effect is instead to make the density profile asymmetric with respect to the equatorial plane.The origin of the critical value | a | /M ≈ . φ ≈ r ≈ − Γ rtt ˙ t − rtφ ˙ t ˙ φ − Γ rφφ ˙ φ , (29)where ˙ t = Σ E − aM rL z ̺ ∆ , (30)˙ φ = 2 aM rE + ( ̺ − M r ) L z csc θ̺ ∆ . (31)In order to study the sign of ¨ r , we need to specify the constant of motion E and L z , which are respectively the energyand the angular momentum along the spin (per unit mass) at infinity. For a simple estimate, we can take E = 1(marginally bound orbits) and L z = 0. Generally speaking, the first term on the right hand side of eq. (29) producesan attractive force at large radii and a repulsive force at small radii, while the second and third terms behave in theopposite way; that is, they give a force which is attractive and repulsive respectively for small and large r . It is easyto see that, on the equatorial plane, for | a | /M slightly larger than 1, the force turns out to be attractive at very smalland large radii, and repulsive around r = 0 . M . The two values of r , say r and r , for which the force is zero aregiven by the equation 4 a M r Σ + 4 a M (cid:0) r − a M r (cid:1) − Σ = 0 , (32)where Σ = ( r + a ) − a ( r − M r + a ). Eq. (32) reduces to the following simple equation a + 2 a r ( r − M ) + r = 0 , (33)which can be seen as a quadratic equation for a , with solution a = r (2 M − r ) ± r √ M − r . (34)As | a | /M increases, the r and r converge to r = 0 . M and eventually coincide for a crit /M = 3 √ ≈ . , (35)which is the maximum of eq (34). For larger values of the Kerr parameter, eq. (32) has no solutions for r > | a | /M = 3 √ / r = 0 . M and only when | a | /M is slightly larger than 3 √ / IV. ASTROPHYSICAL IMPLICATIONS
Our simulations show that the accretion process onto objects with | a | < M and | a | > M is very different. Inthe first case, the injected matter is swallowed by the object and the density profile reaches always a quasi-steadyconfiguration: the BH grows at the same rate of the injection of matter. For super-spinars, accretion is more difficultdue to a repulsive force in the neighborhood of the center, except near the equatorial plane. Here we can distinguishthe cases | a | /M < . | a | /M > ∼ .
4. For | a | moderately larger than M , the repulsive force prevents that theaccreting gas reaches the central object: the mass of the super-spinar does not increase and the gas is accumulatedaround the object. A cloud forms and grows; the matter density of this cloud becomes higher and higher. As shownin the left panel of fig. 4, we do not find a quasi-steady state. As the quantity | a | /M increases, more and more of thematter is able to fall all the way into the center. The highest density region of the cloud moves to smaller radii. Then,for | a | /M > ∼ .
4, the repulsive force in the neighborhood of the center is no longer able to prevent a regular accretionof the super-spinar. The accreting flow can reach a quasi-steady state (see the right panel of fig. 4) by forming apeculiar high density disk on the equatorial plane at very small radii (figs. 3 and 5).Limiting our considerations to a qualitative level, super-spinars might explain observations like the ones reportedin ref. [22]: the blazar PKS 0537-441 produces most of its flux in gamma rays, while for standard accretion onto usualKerr BH it is not natural to produce gammas more than a few percent in bolometric luminosity [23]. Indeed, if thecentral object is a BH, one would expect an accretion luminosity proportional to ST , where S ∝ M is the emittingsurface area and T the temperature: a BH does not produce a significant amount of hard radiation. In the case ofsuper-spinars, the temperature of the accreting gas could become higher, because of much higher plasma densities,thus producing harder radiation.It is however important to bear in mind that i ) some radiative processes might become important and ii ) here theaccreting matter is modeled as a test fluid. Actually, the particles of the accreting gas can lose energy and angularmomentum through some dissipative mechanisms, and that can have some effect on the accretion process, even if weexpect that our basic conclusions still hold, especially for small accretion rates. Secondly, the test fluid approximation0breaks down when the mass of the accreting gas is non-negligible in comparison to the original mass of the object,e.g. that is true for super-massive objects at the center of galaxies. For | a | /M < .
4, as the cloud grows, eventuallyits gravitational field becomes non-negligible, invalidating the test fluid approximation. Here it is difficult to predictwhat happens only from the present results, but we suggest a few possibilities. For example, the cloud could collapseonto the object, as soon as it wins against the repulsive force around the center. If the accretion rate at large radiiis at 10% of the Eddington limit, the mass of the cloud becomes comparable to the mass of the super-spinar after10 yr. The gravitational collapse of the cloud would be likely a violent event, and with two possible results: theformation of a BH with event horizon, i.e. an object with | a | < M , or the formation of a heavier super-spinar.The fate of the system presumably depends on the details of the accretion process and of the release of energy andangular momentum during the collapse of the cloud. Another possibility is that an event horizon just forms due tothe accumulated matter, hides the mass from outside and no striking radiation is emitted. In the second scenario, thesuper-spinar would be the first product of the collapse, and then it would evolve into a Kerr BH, presumably withoutleaving any signature of its previous super-spinning stage. V. CONCLUSIONS
In this paper we have studied the accretion process onto a spinning object. In particular, we have considered aKerr spacetime with absolute value of the Kerr parameter a (ratio of spin to mass) either smaller or larger than M ,the mass of the object. In the first case, the spacetime contains a BH, in the second one a naked singularity. Ourmain motivation for considering the possibility of a naked singularity is based on the observation that the singularityis more likely a pathology of classical GR and that in the full theory it must be replaced by something else. We donot know how the central singularity is resolved, but our results are probably not significantly affected by the detailsof the correct theory. The only relevant quantity astrophysically is likely to be | a | /M .We can distinguish three cases:1. BHs with | a | /M <
1. We find the usual accretion picture: injected matter always reaches a quasi-steadystate configuration, in which matter is lost behind the event horizon at the same rate as it enters into thecomputational domain.2. Super-spinars with 1 < | a | /M < .
4. Here the gas cannot reach the central object, because of a repulsive forcein the neighborhood of the center. As a result, the gas is efficiently accumulated around the super-spinar. Thatleads to the formation and growth of a high density cloud. However, the accumulation process will stop atsome point. One possibility is that it is interrupted by violent events due to the gravitational collapse of thecloud onto the object. This could be associated with the formation of a new object, either a BH or a heaviersuper-spinar. Another possibility is that the accumulated matter creates an event horizon, hiding the objectfrom the outside, and with no abundant release of energy.3. Super-spinars with | a | /M > ∼ .
4. Now the repulsive force around the center is no longer capable of preventinga regular accretion of the object. Our simulations find that the flow forms a high density thin disk on theequatorial plane and reaches a quasi-steady state, i.e. matter enters and leaves the computational domain atthe same rate. This disk is much closer to the center of the object than in the case of a standard BH.
Acknowledgments
We would like to thank Sergei Blinnikov, Hideki Ishihara, Ken’ichi Nakao, and Lev Titarchuk for useful discussionsat different stages of this work. C.B. and N.Y. were supported by World Premier International Research CenterInitiative (WPI Initiative), MEXT, Japan. K.F. thanks the DOE and the MCTP at the University of Michigan forsupport. T.H. was partly supported by the Grant-in-Aid for Scientific Research Fund of the Ministry of Education,Culture, Sports, Science and Technology, Japan (Young Scientists (B) 18740144 and 21740190). R.T. was supported bythe Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Culture, Sports, Science and Technology,Japan (Young Scientists (B) 18740144). [1] Ch. W. Misner, K. S. Thorne and J. A. Wheeler,
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