Testing the Kerr-nature of stellar-mass black hole candidates by combining the continuum-fitting method and the power estimate of transient ballistic jets Cosimo Bambi∗ Arnold Sommerfeld Center for Theoretical Physics Ludwig-Maximilians-Universita¨t Mu¨nchen, 80333 Munich, Germany (Dated: January 26, 2012) Astrophysical black hole candidates are thought to be the Kerr black holes predicted by General Relativity, as these objects cannot be explained otherwise without introducing new physics. How- ever, there is no observational evidence that the space-time around them is really described by the Kerr solution. The Kerr black hole hypothesis can be tested with the already available X-ray data byextendingthecontinuum-fittingmethod,atechniquecurrentlyusedbyastronomerstoestimate the spins of stellar-mass black hole candidates. In general, we cannot put a constraint on possible deviationsfromtheKerrgeometry,butonlyonsomecombinationbetweenthesedeviationsandthe spin. The measurement of the radio power of transient jets in black hole binaries can potentially 2 break this degeneracy, thus allowing for testing the Kerr-nature of these objects. 1 0 PACSnumbers: 97.60.Lf,97.80.Jp,04.50.Kd,97.10.Gz,98.38.Fs 2 n I. INTRODUCTION ThepossibilityoftestingtheKerrnatureofastrophys- a ical BH candidates with present and near future exper- J The 5−20 M compact objects in X-ray binary sys- iments is becoming an active research field [10–16]. In 5 (cid:12) 2 tems and the 105 −109 M(cid:12) dark bodies at the center particular, one canextend the continuum-fitting method of every normal galaxy are thought to be the Kerr black to constrain possible deviations from the Kerr geome- ] holes (BHs) predicted by General Relativity [1]. There try [12]. That can be achieved by considering a more c q is no evidence that the space-time around these objects general background, which includes the Kerr solution as - is really described by the Kerr metric, but, at the same special case. The compact object will be thus character- gr time, there is no other explanation in the framework of izedbyM,a∗,andatleastone“deformationparameter”, [ conventional physics. A Kerr BH is completely speci- measuring deviations from the Kerr geometry. If obser- fied by two parameters: its mass, M, and its spin an- vationaldatarequireavanishingdeformationparameter, 2 gular momentum, J. A fundamental limit for a BH in the Kerr BH hypothesis is verified. However, the fit of v 8 4-dimensional General Relativity is the bound |a∗| ≤ 1, the X-ray spectrum cannot be used to measure a∗ and 3 where a∗ = J/M2 is the dimensionless spin parameter1. the deformation parameter at the same time, but it is 6 This is just the condition for the existence of the event only possible to constrain a combination of them. This 1 horizon: for |a | > 1, there is no horizon and the Kerr is not a problem of the continuum-fitting method, but ∗ . metric describes a naked singularity, which is forbidden of any approach (see e.g. Ref. [13] for the case of the 1 0 by the weak cosmic censorship conjecture [2]. analysis of the Kα iron line). 2 In the case of the stellar-mass BH candidates in X-ray In what follows, I will apply the recent finding of 1 binary systems, the mass M can be deduced by study- Ref. [17] to show that one can potentially break the de- v: ing the orbital motion of the stellar companion. This generacy between a∗ and the deformation parameter by i measurement is reliable, because the system can be de- combining the continuum-fitting method with the power X scribed in the framework of Newtonian mechanics, with estimate of transient ballistic jets. r no assumptions about the nature of the compact object. a The situation changes when we want to get an estimate of the spin parameter a . The most reliable approach is II. TRANSIENT BALLISTIC JETS ∗ currently the continuum-fitting method [3–8]. Basically, one fits the X-ray continuum spectrum of the BH candi- Observationally, BH binaries can emit two kinds of date using the standard accretion disk model of Novikov jets[18]. Steadyjetsoccurinthehardspectralstate,over and Thorne [9]. Under the assumption that the back- a wide range of luminosity of the source, and they seem ground geometry is described by the Kerr metric, it is to be not very relativistic. Transient ballistic jets are in- possible to infer the spin parameter, a∗, and the mass stead launched when a BH binary with a low-mass com- accretion rate, M˙ , if the mass of the BH candidate, its panion undergoes a transient outburst: the jet appears distance from us, and the inclination angle of the disk whenthesourceswitchesfromthehardtosoftstateand are known independently. its luminosity is close to the Eddington limit. Transient jets are observed as blobs of plasma moving ballistically outward at relativistic velocities. The common interpre- ∗ [email protected] tationisthatsteadyjetsareproducedrelativelyfarfrom 1 Throughoutthepaper,IuseunitsinwhichGN=c=1. the compact object, say at about 10 to 100 gravitational 2 BH Binary a η P (kpc2 GHz Jy/M ) Reference ∗ jet (cid:12) GRS 1915+105 0.975, a >0.95 0.224, η>0.190 39.4 [5] ∗ GRO J1655-40 0.7±0.1 0.104+0.018 19.7 [6] −0.013 XTE J1550-564 0.34±0.24 0.072+0.017 2.79 [7] −0.011 A0620-00 0.12±0.19 0.061+0.009 0.173 [8] −0.007 TABLEI.Thefourstellar-massBHcandidatesofwhichthespinparametera hasbeenestimatedwiththecontinuum-fitting ∗ method and we have radio data of their outbursts. The accretion efficiency η in the third column has been deduced from the corresponding a for a Kerr background. The mass-normalized jet power P in the fourth column has been inferred from the ∗ jet data reported in Ref. [17], using Eq. (1). radii [19], while transient jets are launched within a few the log adopted in Ref. [17]. Despite there being only gravitational radii [20]. As discussed in Ref. [17], it is four objects, there is evidence for a correlation between thereforeplausiblethattransientjetsarepoweredbythe jet power and Ω , and one finds the behavior expected H rotational energy of the BH and, since they occur at a in the case of a jet powered by the rotational energy well defined luminosity, they may be used as “standard of the BH. For more details about the systematics, the candles”. interpretation of the finding, and the comparison with In Ref. [17], the authors show there is a correla- previous results, see Ref. [17]. The conclusions of the tion between the spin parameter a , as inferred by the authorsarethereforethat: i)theyhaveprovidedthefirst ∗ continuum-fitting method, and the radio power of tran- evidence that some jets may be powered by the BH spin sient ballistic jets. Moreover, the behavior is close to energy, and ii) the observed correlation also provides an what should be expected if these jets were powered by additionalconfirmationofthecontinuum-fittingmethod. the BH spin via the Blandford-Znajek mechanism [21]. So far, the continuum-fitting method has provided the estimate of the spin parameter of nine stellar-mass BH candidates [4]. Five of these objects have a low-mass companion and undergo mass transfer via Roche lobe outflow: during their outbursts, they produce ballistic III. NON-KERR SPACE-TIMES jets. For three of them (GRS 1915+105, GRO J1655-40, andXTEJ1550-564), wehavegoodradiodataduringat least one of their outbursts. For A0620-00, the data are I this section, I will show that the jet power of a BH not so good. 4U 1543-47 has never been monitored well candidate can provide additional information about the atradiowavelengthduringanyofitsoutbursts. ForGRS natureofthecompactobjectandpotentiallycanbeused 1915+105, GRO J1655-40, XTE J1550-564, and A0620- to break the degeneracy between the spin and the defor- 00,theauthorsofRef.[17]computethemass-normalized mation parameter. I will outline the basic idea, without jet radio power: following a rigorous study: the latter would require a D2(νS ) complete reanalysis of the X-ray continuum spectrum of P = ν max,5GHz , (1) jet M the four objects and new GRMHD simulations in a par- ticularnon-Kerrbackground,bothbeyondthepurposeof where D is the distance of the binary system from us this work, as well as more observational data, which we and (νS ) is the estimate of the maximum of ν max,5GHz do not have yet. I will consider two specific non-Kerr the radio power at 5 GHz (see Tab. I). Then, they plot space-times: the braneworld-inspired BHs of Ref. [23] the jet power P against the BH spin parameter a , as jet ∗ and the Johannsen-Psaltis (JP) BHs of Ref. [24]. These inferred from the continuum-fitting method, and against space-times can be seen as the two prototypes of non- the corresponding BH angular frequency Kerr background, or at least of the ones proposed in the Ω =−gtφ(cid:12)(cid:12) = a , (2) literature [25]. H gφφ(cid:12)r=rH rH2 +a2 where r is the radius of the BH outer event horizon H and a = a M. The scaling P ∼ a2 was derived in ∗ jet ∗ Ref. [21], under the assumption |a | (cid:28) 1. P ∼ Ω2 ∗ jet H wasinsteadobtainedinRef.[22]andworksevenforspin parameters quite close to 1. The top left panel of Fig. 3 A. Example 1: braneworld black holes shows the plot P vs Ω , which is basically the plot in jet H Fig.3ofRef.[17]. Theblue-dashedlinehasslopeof2,as expected from the theoretical scaling. The uncertainty A braneworld-inspired BH solution was found in in P is the somehow arbitrary uncertainty of 0.3 in Ref. [23]. In Boyer-Lindquist coordinates, the non-zero jet 3 (cid:96)/M2 = 0.5 (cid:96)/M2 = 0.5 0.5 (cid:96)/M2 = 0.0 1 (cid:96)/M2 = 0.0 (cid:96)/M2 = -0.5 (cid:96)/M2 = -0.5 0.4 (cid:96)/M2 = -1.0 (cid:96)/M2 = -1.0 (cid:96)/M2 = -1.5 0.5 (cid:96)/M2 = -1.5 (cid:96)/M2 = -2.0 (cid:96)/M2 = -2.0 0.3 (cid:100) (cid:49)H 0 0.2 -0.5 0.1 -1 0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 a a * * FIG. 1. Braneworld-inspired black holes of Eq. (3). Accretion efficiency η=1−E (left panel) and BH angular frequency ISCO Ω (right panel) as a function of the spin parameter a for different values of β/M2. H ∗ Naked Singularities Naked Singularities 1 1 GRS 1915+105 GRO J1655-40 0 0 2 2 M M /(cid:96) /(cid:96) -1 -1 A0620-00 XTE J1550-564 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 a a * * FIG. 2. Braneworld-inspired black holes of Eq. (3). Allowed regions in the parameter space (a ,β/M2) for the BH candidates ∗ GRS1915+105andXTEJ1550-564(leftpanel)andGROJ1655-40andA0620-00(rightpanel). Theredsolidcurveseparates BHs from naked singularities. See text for details. components of the induced 4D metric are by ∆=0; the radius of the outer event horizon is (cid:18) 2Mr−β(cid:19) r =M +(cid:112)M2−a2−β. (5) g =− 1− , H tt ρ2 (cid:112) The event horizon exists only for M ≥ a2−β. When 2a(2Mr−β) (cid:112) g =− sin2θ, M < a2−β, there is no horizon and the space-time tφ ρ2 has a naked singularity2. For the metric in Eq. (3) it is (cid:20) (cid:21) 2Mr−β straightforwardtorepeattheanalyticalderivationofthe g = r2+a2+ a2sin2θ sin2θ, φφ ρ2 jet power (see Ref. [27] and Appendix A of [22]) and one still finds P ∼Ω2 , as in Kerr. ρ2 jet H g = , TheanalysisoftheX-raycontinuumspectraofthefour rr ∆ objects in Tab. I would provide a constraint on a and gθθ =ρ2, (3) β/M23. Thecorrectprocedurewouldbetoreanalyz∗ethe where ρ2 =r2+a2cos2θ, 2 LetusnoticethatthesebraneworldBHsmayviolatethefamil- ∆=r2−2Mr+a2+β, (4) iarbound|a∗|≤1,withoutviolatingtheweakcosmiccensorship conjecture. Itisalsopossibletocheckthatthereexistastrophysi- and β is the tidal charge parameter, encoding the im- calprocessescapableofproducingsuchfast-rotatingobjects[26]. prints of the non-local effects from the extra dimension. 3 IftheBirkhoff’sTheoremholds,SolarSystemexperimentswould require|β/M2|<4.6·10−4. Whileitisnotclearifthisisthecase ThemetriclooksliketheusualKerr-Newmansolutionof in braneworld models, the aim of this paper is not to constrain General Relativity, which describes a rotating BH with thesetheories,buttoshowhowtwoindependentmeasurements electric charge Q, with β =Q2. However, here β can be canbreakthedegeneracybetweenthespinandthedeformation either positive or negative. The event horizon is defined parameter. 4 )]n )]n MSu 2 (cid:96)/M2 = 0.0 MSu 2 (cid:96)/M2 = 0.5 y / y / J J z 1 z 1 H H G G 2 2 c c p p k 0 k 0 / (et / (et Pj Pj [0 -1 [0 -1 1 1 g g o o l l -1.5 -1 -0.5 -1.5 -1 -0.5 log ((cid:49) M) log ((cid:49) M) 10 H 10 H )]n )]n MSu 2 (cid:96)/M2 = -0.5 MSu 2 (cid:96)/M2 = -1.0 y / y / J J z 1 z 1 H H G G 2 2 c c p p k 0 k 0 / (et / (et Pj Pj [0 -1 [0 -1 1 1 g g o o l l -1.5 -1 -0.5 -1.5 -1 -0.5 log ((cid:49) M) log ((cid:49) M) 10 H 10 H )]n )]n MSu 2 (cid:96)/M2 = -1.5 MSu 2 (cid:96)/M2 = -2.0 y / y / J J z 1 z 1 H H G G 2 2 c c p p k 0 k 0 / (et / (et Pj Pj [0 -1 [0 -1 1 1 g g o o l l -1.5 -1 -0.5 -1.5 -1 -0.5 log ((cid:49) M) log ((cid:49) M) 10 H 10 H FIG.3. Braneworld-inspiredblackholesofEq.(3). PlotsofthejetpowerP againsttheBHangularfrequencyΩ . Thetop jet H left panel shows the data in the case of the familiar Kerr background and the blue dotted line corresponds to P ∼Ω2 , the jet H theoretical scaling derived in Ref. [22]. X-ray data of these objects in the background (3); how- energy of the gas at the innermost stable circular orbit ever, that would take a long time and is beyond the pur- (ISCO), which is supposed to be the inner edge of the pose of the present paper. A simple estimate can be ob- accretion disk. The common statement in the literature tained from the following consideration. In the standard thatthecontinuum-fittingmethodmeasurestheinnerra- case of Kerr background, the continuum-fitting method diusofthedisk,r ,iscorrectbecauseintheKerrmetric in provides the BH spin parameter a and its mass accre- there is a one-to-one correspondence between η and r . ∗ in tionrateM˙ ,whentheBHmass,itsdistancefromus,and However, in a non-Kerr background one can see that the theinclinationangleofthediskareknown. Actually,the actualkey-parameterisη. Wecanthenwritethepresent low frequency region of the spectrum constrains M˙ [28], estimates of a of the four objects in terms of the accre- ∗ while the position of the peak constrains the accretion tion efficiency η (see the third column in Tab. I), and efficiency η =1−E [12], where E is the specific then get the allowed regions in the space (a ,β/M2) for ISCO ISCO ∗ 5 (cid:161) = -0.5 0.5 3(cid:161) = 0 3 (cid:161) = 1 3 0.4 (cid:161)3 = 5 (cid:161) = 10 3 (cid:161) = 15 3 0.3 (cid:100) 0.2 0.1 0 -1.5 -1 -0.5 0 0.5 1 1.5 a * FIG. 4. JP black holes of Eq. (6) with deformation parameter (cid:15) and (cid:15) =0 for i(cid:54)=3. Accretion efficiency η=1−E as a 3 i ISCO function of the spin parameter a for different values of (cid:15) . ∗ 3 25 25 GRO J1655-40 20 XTE J1550-564 20 15 15 3 10 GRS 1915+105 3 10 (cid:161) (cid:161) 5 5 0 0 A0620-00 -5 -5 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 a a * * FIG. 5. JP black holes of Eq. (6) with deformation parameter (cid:15) and (cid:15) =0 for i(cid:54)=3. Allowed regions in the parameter space 3 i (a ,(cid:15) )fortheBHcandidatesGRS1915+105andXTEJ1550-564(leftpanel)andGROJ1655-40andA0620-00(rightpanel). ∗ 3 See text for details. every BH candidate (see App. A for more details). The B. Example 2: JP black holes accretion efficiency and the BH angular frequency as a function of the spin parameter are shown in Fig. 1. The final results are reported in Fig. 2, where the red solid TheJPBHshavebeenproposedin[24]explicitlytobe curveseparatesBHsfromnakedsingularities. Theregion used to test the Kerr BH hypothesis. The non-vanishing with naked singularities can be excluded for at least two metric coefficients in Boyer-Lindquist coordinates are: reasons: these space-times have equatorial stable circu- larorbitswithnegativeenergy,whichwouldimplyη >1, andtheyarepresumablyunstable, duetotheergoregion instability [29]. As we can see in Fig. 2, we cannot esti- mate a∗ and β/M2 independently, but we can only con- (cid:18) 2Mr(cid:19) strain a combination of the these two parameters. This g =− 1− (1+h), tt ρ2 is the usual situation we find when we want to test the Kerr BH hypothesis. 2aMrsin2θ g =− (1+h), tφ ρ2 (cid:20) 2a2Mrsin2θ(cid:21) alsFooirmbpraonrteawnotrtldoBnoHtsic,eΩtHhaits sPtill gisivpernopboyrEtiqon.a(2l)t.oItthies gφφ =sin2θ r2+a2+ ρ2 + jet secondpowerofΩH; thatis,Pjet doesnotdependonthe a2(ρ2+2Mr)sin4θ sense of BH rotation with respect to the one of the disk. + h, ρ2 InFig.3,IplotthepowerjetagainstΩ forsomevalues H ρ2(1+h) ofβ/M2. HereIassumethatalltheBHcandidateshave g = , thesamevalueofβ/M2. Thisassumptioncanberelaxed rr ∆+a2hsin2θ and tested when more data will be available. g =ρ2, (6) θθ 6 )]n )]n Su 2 Su 2 M (cid:161) = 0 M (cid:161) = -0.5 3 3 y / y / J J z 1 z 1 H H G G 2 2 c c p p k 0 k 0 / (et / (et Pj Pj [0 -1 [0 -1 1 1 g g o o l l -2 -1 0 -2 -1 0 log a log a 10 * 10 * )]n )]n Su 2 Su 2 M (cid:161) = 1 M (cid:161) = 5 3 3 y / y / J J z 1 z 1 H H G G 2 2 c c p p k 0 k 0 / (et / (et Pj Pj [0 -1 [0 -1 1 1 g g o o l l -2 -1 0 -2 -1 0 log a log a 10 * 10 * )]n )]n Su 2 Su 2 M (cid:161) = 10 M (cid:161) = 15 3 3 y / y / J J z 1 z 1 H H G G 2 2 c c p p k 0 k 0 / (et / (et Pj Pj [0 -1 [0 -1 1 1 g g o o l l -2 -1 0 -2 -1 0 log a log a 10 * 10 * FIG. 6. JP black holes of Eq. (6) with deformation parameter (cid:15) and (cid:15) =0 for i(cid:54)=3. Plots of the jet power P against the 3 i jet BHspinparametera . ThetopleftpanelshowsthedatainthecaseofthefamiliarKerrbackgroundandthebluedottedcurve ∗ corresponds to P ∼Ω2 , the theoretical scaling derived in Ref. [22]. jet H where while (cid:15) is constrained at the level of 10−4 from current 2 tests in the Solar System [24]. For the sake of simplic- ρ2 =r2+a2cos2θ, ity, in what follows I will consider only the case with the ∆=r2−2Mr+a2, deformation parameter (cid:15) and (cid:15) =0 for i(cid:54)=3. 3 i (cid:88)∞ (cid:18) Mr (cid:19)(cid:18)M2(cid:19)k For some values of the deformation parameters, the h= (cid:15) + (cid:15) . (7) 2k ρ2 2k+1 ρ2 JPBHshaveafewpropertiescommontoothernon-Kerr k=0 metrics,butabsentintheKerrsolution(existenceofver- The metric has an infinite number of free parameters (cid:15) tically unstable circular orbits on the equatorial plane, i andtheKerrsolutionisrecoveredwhenalltheseparam- topologically non-trivial event horizons, etc.). In partic- eters are set to zero. However, in order to recover the ular, herewecannotdefinetheBHangularfrequency, at correct Newtonian limit we have to impose (cid:15) = (cid:15) = 0, least in the usual way, as from Eq. (2) we would obtain 0 1 7 something that depends on the polar angle θ. Anyway, can potentially be used to constrain the deformation pa- if we want to check the Kerr-nature of astrophysical BH rameter. As it is particularly clear in Fig. 6, where (cid:15) 3 candidates, wecanstillplot P againstthespinparam- is the deformation parameter and (cid:15) = 0 corresponds to jet 3 eter a and see if the correlation if the one expected for the Kerr metric, the expected correlation (the blue dot- ∗ Kerr BHs. ted curve in the topleft panel of Fig. 6)is not consistent Theaccretionefficiencyη =1−E asafunctionof with observations when the space-time has large devia- ISCO thespinparametera forsomevaluesofthedeformation tions from the Kerr solution (the cases (cid:15) = 10 and 15 ∗ 3 parameter(cid:15) isshowninFig.4. Togettheconstraintson in Fig. 6). The interpretation of the authors of Ref. [17] 3 a and(cid:15) forthefourobjectsinTab.I,wecanstillapply needs to be confirmed and the study of a larger num- ∗ 3 the simplified analysis of the previous subsection. The ber of objects is compulsory. However, as shown in this resultsareshowninFig.5. Fig.6showstheplotsP vs work through a simplified analysis, the combination of jet a intheJPspace-timewith(cid:15) . Theblue-dottedcurvein the continuum-fitting method and the estimate of jet ∗ 3 the top left panel is the theoretical scaling P ∼Ω2 in powermaybeabletotesttheKerr-natureofstellar-mass jet H Kerr background. Let us notice that the cases with (cid:15) = BH candidates in the near future. 3 10and15areallowedwiththesoleuseofthecontinuum- fitting method, while they seem to be at least strongly disfavoredwhenweaddtheinformationcomingfromthe ACKNOWLEDGMENTS jet power. Indeed, when (cid:15) =10 and 15, the continuum- 3 fitting method would predict a counterrotating disk (i.e. This work was supported by the Humboldt Founda- a < 0) for some sources, while the jet power should be ∗ tion. independent of the sense of BH rotation with respect to the accreting matter. Appendix A: Accretion efficiency in the Novikov-Thorne model IV. CONCLUSIONS The Novikov-Thorne model is the standard model for Astrophysical BH candidates are thought to be the accretion disks [9]. It describes geometrically thin and Kerr BHs predicted in General Relativity, but direct ob- optically thick disks and it is the relativistic general- servational evidence for this identification is still lack- ization of the Shakura-Sunyaev model [30]. Accretion ing. In order to test and verify the Kerr BH hypoth- is possible because viscous magnetic/turbulent stresses esis, we have to probe the geometry of the space-time and radiation transport energy and angular momentum around these objects. The current most robust ap- outwards. The model assumes that the disk is on the proach to do that with already available data seems to equatorial plane and that the disk’s gas moves on nearly be the continuum-fitting method, a technique used by geodesic circular orbits. The model can be applied for astronomers to measure the spin of the stellar-mass BH a generic stationary, axisymmetric, and asymptotically candidates. Thephysicsinvolvedisrelativelysimpleand space-time. Here, thelineelementcanalwaysbewritten there are both astrophysical observations and numerical as calculationssupportingthecrucialingredientsofthisap- proach. However, the continuum-fitting method cannot ds2 =g dt2+2g dtdφ+g dr2+g dθ2+g dφ2. tt tφ rr θθ φφ provide at the same time an estimate of the spin and of (A1) some deformation parameter measuring the deviations from the Kerr geometry. The problem is that there is a Since the metric is independent of the t and φ coordi- degeneracy between these two parameters and therefore nates, we have the conserved specific energy at infinity, it is only possible to get a constraint on some combina- E, andtheconservedaxial-componentofthespecifican- tion of them. The reason is that the continuum-fitting gularmomentumatinfinity,L. Fromtheconservationof method is sensitive to the accretion efficiency, which de- the rest-mass, g uµuν =−1, we can write µν pends on the spin and on the deformation parameter. Inthispaper,Iexploredawaytobreakthisdegeneracy g r˙2+g θ˙2 =V (r,θ), (A2) rr θθ eff and get an estimate of the spin and on the deformation parameter separately. If transient ballistic jets in BH where the effective potential V is given by eff binaries are powered by the BH spin via the Blandford- Znajekmechanism,thejetpowerandtheBHspinshould V = E2gφφ+2ELgtφ+L2gtt −1. (A3) be correlated in a specific way. In Ref. [17], the authors eff g2 −g g tφ tt φφ showed for the first time evidence for such a correlation. Here, I showed that, if this interpretation is correct, the Circular orbits in the equatorial plane are located at the estimate of jet power provides an additional information zeros and the turning points of the effective potential: about the nature of the stellar-mass BH candidates and, r˙ = θ˙ = 0, which implies V = 0, and r¨ = θ¨ = 0, eff when combined with the continuum-fitting method, it requiring respectively ∂ V = 0 and ∂ V = 0. From r eff θ eff 8 these conditions, one can obtain the angular velocity, E, The orbits are stable under small perturbations if and L: ∂2V ≤ 0 and ∂2V ≤ 0. In Kerr space-time, the sec- r eff θ eff ond condition is always satisfied, so one can deduce the radiusoftheinnermoststablecircularorbit(ISCO)from ∂2V = 0. 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