Table Of ContentTesting 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. In general, however, that may not be true.
r eff
(cid:113) Forinstance,intheJPspace-times,theISCOradiusmay
−∂ g + (∂ g )2−(∂ g )(∂ g ) be determined by the orbital stability along the vertical
r tφ r tφ r tt r φφ
Ω= , (A4)
direction. When we know the ISCO radius, we can com-
∂ g
r φφ
pute the corresponding specific energy E and then
g +g Ω ISCO
E =− tt tφ , (A5) the accretion efficiency:
(cid:112)
−g −2g Ω−g Ω2
tt tφ φφ
η =1−E . (A7)
g +g Ω ISCO
L= tφ φφ . (A6)
(cid:112)
−g −2g Ω−g Ω2
tt tφ φφ
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