Towards the use of the most massive black hole candidates in AGN to test the Kerr paradigm

0.6 MB·English

by  Cosimo Bambi

Towards the use of the most massive black hole candidates in AGN
to test the Kerr paradigm
Cosimo Bambi∗
Arnold Sommerfeld Center for Theoretical Physics
Ludwig-Maximilians-Universita¨t Mu¨nchen, 80333 Munich, Germany
(Dated: January 9, 2012)
The super-massive objects in galactic nuclei are thought to be the Kerr black holes predicted by
GeneralRelativity,althoughadefiniteproofoftheiractualnatureisstilllacking. Themostmassive
objects in AGN (M ∼109 M ) seem to have a high radiative efficiency (η ∼0.4) and a moderate
(cid:12)
mass accretion rate (L /L ∼ 0.3). The high radiative efficiency could suggest they are very
bol Edd
rapidly-rotating black holes. The moderate luminosity could indicate that their accretion disk is
geometrically thin. If so, these objects could be excellent candidates to test the Kerr black hole
hypothesis. An accurate measurement of the radiative efficiency of an individual AGN may probe
thegeometryofthespace-timearoundtheblackholecandidatewithaprecisioncomparabletothe
one achievable with future space-based gravitational-wave detectors like LISA. A robust evidence
2
of the existence of a black hole candidate with η > 0.32 and accreting from a thin disk may be
1
interpreted as an indication of new physics. For the time being, there are several issues to address
0
before using AGN to test the Kerr paradigm, but the approach seems to be promising and capable
2
of providing interesting results before the advent of gravitational wave astronomy.

n
a PACSnumbers: 98.54.Cm,04.50.Kd,98.62.Mw
J

6
I. INTRODUCTION state [6], while the super-massive objects at the centers

] ofgalaxiesaretooheavy,compact,andoldtobeclusters
c
q Gravity has been tested and verified for distances in of non-luminous bodies, as the cluster lifetime would be
- therange∼1mmto∼1pc(mainlywithinitsNewtonian shorter than the age of these systems [7]. All these ob-
gr limit) and for weak gravitational fields [1, 2]. The re- jects are therefore thought to be the BHs predicted by
[ search is now moving to check the validity of the theory GR, as they cannot be explained otherwise without in-
at cosmological scales, sub-millimeter distances, and for troducingnewphysics. Therearealsosomeobservations
2 strong gravitational fields. One of the most intriguing interpretedasanindirectevidencefortheexistenceofthe
v
3 predictions of General Relativity (GR) is that the col- event horizon [8] (but see [9]). On the contrary, there is
6 lapsing matter produces singularities in the space-time. no indication that the geometry around these objects is
6 Accordingtotheweakcosmiccensorshipconjecture,sin- described by the Kerr metric.
4 gularitiesofgravitationalcollapsemustbehiddenwithin
2. black holes (BHs) [3]. In 4-dimensional GR, uncharged TestingtheKerrBHhypothesisisthusthenextstepto
progressinthisresearchfieldandseveralauthorshavein-
1 BHs are described by the Kerr solution, which is com-
deed suggested possible ways to do it using present and
1 pletelyspecifiedbytwoparameters,themass,M,andthe
future data (for a review, see e.g. Ref. [10]). A very
1 spinangularmomentum,J [4]. Theconditionfortheex-
v: istenceoftheeventhorizonisa∗ ≤1,wherea∗ =|J/M2| ptiroominisspinirgalasp(pErMoaRchIsi,si.teh.esdysetteemctsiocnonofsiestxitnrgemofeamsatsesllarar--
i isthespinparameter1. Whena >1,thereisnohorizon
X ∗ mass compact object orbiting a super-massive BH can-
and the central singularity is naked, violating the weak
r cosmic censorship conjecture. didate) with future space-based gravitational-wave an-
a tennas. Missions like LISA will be able to follow the
Astronomers have discovered at least two classes of
stellar-mass compact object for millions of orbits around
BH candidates (for a review, see e.g. Ref. [5]): stellar-
the central super-massive BH candidate, and therefore
mass objects in X-ray binary systems (M ∼5−20 M )
(cid:12) deviations from the Kerr geometry will lead to a phase
and super-massive objects in galactic nuclei (M ∼105−
differenceinthegravitationalwaveformsthatgrowswith
109 M ). The estimates of the masses of these objects
(cid:12) the number of observed cycles [11]. However, these data
are robust, because determined via dynamical measure-
willnotbeavailableshortly,asthefirstmissionwillbeat
ments and without any assumption about the geometry
bestintheearly2020s. ThenatureofBHcandidatescan
of the space-time. The key-point is that the stellar-mass
alsobetestedbyextendingthemethodscurrentlyusedto
objectsinX-raybinarysystemsaretooheavytobeneu-
estimate the spin of these objects, such as X-ray contin-
tronorquarkstarsforanyreasonablematterequationof
uum[12]andKα-ironmeasurements[13],observationsof
quasi-periodic oscillations [14], and measurements of the
cosmic X-ray background [15, 16]. These methods can
∗ [email protected] in principle be applied even with present data, provided
1 Throughout the paper I use units in which GN =c=1, unless that the systematic errors are properly understood. Fu-
statedotherwise. ture observations of the shadow of nearby super-massive
2
BHcandidatesareanotherexcitingpossibilitytotestthe sarily requires a very rigid compact object, much stiffer
Kerr BH paradigm [17]. than a self-gravitating fluid with “normal” equations of
state. The confirmation of the existence of individual
Previous studies have clearly pointed out that
AGN with high radiative efficiency (η >0.3) can poten-
“rapidly-rotating”objectsarethebestcandidatestotest
tiallyeitherbeusedtoputstrongconstraintsontheKerr
the Kerr BH hypothesis: if the object rotates fast, even
nature of astrophysical BH candidates and to discover
a small deviation from the Kerr background can cause
new physics, as in the framework of the standard accre-
significantdifferencesinthepropertiesoftheelectromag-
tion disk model η cannot exceed 0.32. In Section VI, I
netic radiation emitted by the gas of the accretion disk
discussthefindingsofthispaperinrelationwithcurrent
andpeculiarfeatures,otherwiseabsentintheKerrgeom-
estimates of the radiative efficiency of AGN. The con-
etry, may show up [18, 19]. The aim of this paper is to
clusions and the issues that need to be addressed before
investigatethepotentialitiesofthemostmassiveBHcan-
usingthemostmassiveobjectsinAGNtoreallytestthe
didatesinAGN,throughthemeasurementoftheirradia-
tiveefficiencyη,definedbyL =ηM˙ ,whereL isthe Kerr BH hypothesis are reported in Section VII. In Ap-
bol bol
bolometric luminosity of the source and M˙ is the mass pendices A and B, the reader can find the non-vanishing
metric coefficients respectively of the JP and of the MN
accretion rate of the BH candidate. The estimate of the
metric.
mean radiative efficiency of AGN through the Soltan’s
argument [20] already suggests the presence of rapidly-
rotating BHs [21, 22]. Recently, Davis and Laor have
II. ACCRETION DISKS
proposed a way to measure the radiative efficiency of in-
dividual AGN [23]. The authors found that the most
massive BH candidates (with a mass M ∼ 109 M ) A. Novikov-Thorne model
(cid:12)
would have a high radiative efficiency, up to η ∼ 0.4,
and a moderate mass accretion rate, L /L ∼ 0.3, The Novikov-Thorne (NT) model is the standard
bol Edd
where L is the Eddington luminosity of the source. model for accretion disks [26]. It describes geometrically
Edd
ThestandardaccretiondiskmodelinaKerrbackground thin and optically thick disks and it is the relativistic
would predict a high value of the spin parameter a for generalization of the Shakura-Sunyaev (SS) model [27].
∗
theseobjects,extremelycloseto1. Atthesametime,the The disk is thin in the sense that the disk opening angle
moderate luminosity (in Eddington units) may indicate is h = H/r (cid:28) 1, where H is the thickness of the disk
a thin accretion disk and the applicability of the stan- at the radius r. Magnetic fields are ignored. In the Kerr
dard accretion disk model. If these estimates and these background,therearefourparameters(BHmassM,BH
considerationsarecorrect,themostmassivecompactob- spin parameter a , mass accretion rate M˙ , and viscos-
∗
jects in AGN would be excellent candidates to test the ity parameter α), but the model can be easily extended
Kerr paradigm. The sole measurement of η can poten- to any (quasi-)stationary, axisymmetric, and asymptoti-
tially constrain either a∗ and a deviation from the Kerr cally flat space-time. Accretion is possible because vis-
geometry. cousmagnetic/turbulentstressesandradiationtransport
The paper is organized as follows. In Section II, I re- energy and angular momentum outwards. The model
view the standard accretion disk model, its assumptions assumes that the disk is on the equatorial plane and
and properties, its effects on the evolution of the spin that the disk’s gas moves on nearly geodesic circular or-
parameter of the central object, and its applicability. In bits. For long-term accretions, the disk is adjusted on
SectionIII,IconsideranaccretiondiskintheKerrback- theequatorialplaneasaresultoftheBardeen-Petterson
ground, summarizing well-known results that should be effect [28]. That remains true even in non-Kerr back-
expected if the BH candidates are the BHs of GR. In grounds [29]. The assumption of nearly geodesic circular
Sections IV and V, I discuss accretion disks respectively orbits requires that the radial pressure is negligible com-
in the Johannsen-Psaltis (JP) [24] and in the Manko- paredtothegravitationalforceoftheBH.Heatadvection
Novikov (MN) [25] space-times. These are two metrics is ignored (it scales as ∼h2) and energy is radiated from
that can be conveniently used to describe a background the disk surface.
deviatingfromtheKerrgeometry. Thenatureofthetwo The key-ingredient of the NT model is that the inner
metrics is definitively different: the JP metric describes edge of the disk is at the innermost stable circular orbit
non-Kerr BHs in a putative alternative theory of grav- (ISCO), where viscous stresses are assumed to vanish.
ity, while the MN one is an exact solution of the Ein- When the gas’s particles reach the ISCO, they quickly
stein’s vacuum equations and can describe the exterior plunge into the BH, without emitting additional radia-
gravitational field of generic compact objects. In both tion. At first approximation, the total efficiency of the
cases, the body is characterized by a mass, a spin angu- accretion process is
lar momentum, and an infinite number of “deformation
η =1−E , (2.1)
parameters”, even if here, for the sake of simplicity, I tot ISCO
will consider only a single deformation parameter at a whereE isthespecificenergyofthegasattheISCO
ISCO
time. I will show that the two metrics present common radius and depends uniquely on the background geome-
features. In particular, a high radiative efficiency neces- try. Ingeneral,thetotalpoweroftheaccretionprocessis
3
converted into radiation and kinetic energy of jet/wind where g (g ≥0) and Ω are respectively the
ISCO ISCO ISCO
outflows, so we can write η = η + η . η is the ra- torque and the angular velocity of the gas at the ISCO.
tot k
diative efficiency and can be inferred from the bolomet- Another crucial point of the NT model is the α viscosity
ric luminosity L if the mass accretion rate is known: parameter: magnetic fields, which are thought to drive
bol
L = ηM˙ . In this paper, I will assume η = 0, i.e. turbulence in disks, may not behave like a local scalar
bol k
no gravitational energy of the gas is converted to kinetic viscosity. If magnetohydrodynamics torques are present,
energyofoutflows. Thisisaconservativeassumptionfor the radiative efficiency η can even exceed 1, as the flow
what follows. taps the spin energy of the BH [32]. In this case, it is
Asaconsequenceoftheaccretionprocess,theBHspin not easy to recover E from the estimate of η and
ISCO
parameter evolves. Since the gas particles arriving at even the equilibrium spin parameter aeq is not given by
∗
the ISCO plunge quickly onto the central object, with- Eq.(2.2),butitturnsouttobelower[33]. GRMHDsim-
out emission of additional radiation, the BH changes its ulationsshowalsothatthegas’sparticlesmaynotfollow
massbyδM =E δmanditsspinangularmomentum the geodesics of the space-time inside the ISCO [34] and
ISCO
by δJ = L δm, where L is the specific angular thatasignificantemissionofradiationfromtheplunging
ISCO ISCO
momentum of the gas at the ISCO, while δm is the gas region is possible [35].
rest-mass. The evolution of the spin parameter of the So, how good is the NT model to describe the ac-
BH turns out to be governed by the following equation: cretion disk around astrophysical BH candidates? For
non-magnetized and weakly-magnetized disks, there is a
da 1 L
∗ = ISCO −2a . (2.2) commonconsensusthattheNTmodeldescribescorrectly
dlnM M EISCO ∗ thin disks, h(cid:28)1, when the viscosity parameter is small,
α (cid:28) 1 [36]. A common criterion to select sources with
IftherighthandsideofEq.(2.2)ispositive,theaccretion
thin disks is that L /L does not exceed 0.3 [37].
processspinstheBHup. Ifitisnegative,theBHisspun bol Edd
In the case of magnetized disks (in Kerr background, as
down. The equilibrium spin parameter aeq is reached
∗ there are no simulations for other space-times), the issue
when the right hand side of Eq. (2.2) vanishes and its
isstillopenandcontroversial. TheGRMHDsimulations
value depends on the geometry of the space-time. As
in Ref. [38] (see also [39]) show that the stress at the in-
discussedin[16]andexplainedbrieflyalsoinSec.IV,the
ner edge of the disk scales as h and the authors conclude
valueofaeq wecaninferfromEq.(2.2)assumingthatall
∗ that the NT model with vanishing stress boundary con-
the accreting matter has its angular momentum in the
dition is recovered for h→0. The GRMHD simulations
same direction provides the maximum value for the spin
in [35] show instead large stress at the inner edge of the
parameterofthesuper-massiveBHcandidatesingalactic
disk even when h → 0; according to these authors, the
nuclei. This fact will be used to get the constraint a <
∗ NT model cannot describe magnetized disks, even when
aeq.
∗ the disk is very thin. It is not clear if the disagreement
Actually, not all the radiation emitted by the disk can
between the two groups can be attributed to different
escape to infinity. A part of it leaves the disk, but it is
configurations of the magnetic fields, different resolution
thencapturedbytheBH.Includingthiseffect,Eqs.(2.1)
of the simulations, or something else.
and (2.2) become [30]
η =1−E −ζ , (2.3)
ISCO E
III. KERR BLACK HOLES
da 1 L +ζ
∗ = ISCO L −2a , (2.4)
dlnM M E +ζ ∗
ISCO E A Kerr BH is completely specified by two parameters:
whereζE andζL takeintoaccounttheradiationcaptured the mass, M, and the spin parameter, a∗. In the NT
by the BH and their expression can be found in [30]. model with ηk = 0, the radiative efficiency of the accre-
There is also a part of radiation that is emitted by the tion process η is uniquely determined by the BH spin
diskandreturnstothediskasaconsequenceoflightde- parameter, as a∗ sets the ISCO radius. There is a one-
flection (returning radiation), interacting with the disk’s to-onecorrespondencebetweena∗ andη,andη increases
particlesandchangingthefluxprofileemittedbythedisk with increasing a∗2. At least in principle, an estimate of
(see e.g. Ref. [31]). η couldbeusedtoinferthespinparameteroftheBH.In
practice, assuming the validity of the NT model, we can
deduce a lower bound for a , as in general η (cid:54)=0.
∗ k
B. Validity of the model The radiative efficiency in Kerr background for differ-
ent values of the BH spin parameter is shown in Tab. I
and in the left panel of Fig. 1. If we use Eq. (2.1), for
If we relax the assumption of vanishing stresses at the
inner edge of the disk, the radiative efficiency increases.
The bolometric luminosity of the disk becomes
L =ηM˙ = 2 ThatistrueonlyforKerrBHs,i.e. whena∗ ≤1. Itisnottrue
bol in a Kerr background with arbitrary value of the spin parame-
=g Ω +(1−E −ζ )M˙ , (2.5) ter[40].
ISCO ISCO ISCO E
4
a E L η Ω M r /M
∗ ISCO √ISCO √ISCO ISCO
(cid:112) (cid:112)
0 8/9 12 1− 8/9≈0.057 1/ 63 ≈0.068 6
√ √ √
1 1/ 3 2/ 3 1−1/ 3≈0.423 1/2 1
0.99614 0.700 1.452 0.300 0.402 1.307
0.99793 0.680 1.395 0.320 0.420 1.240
0.99901 0.660 1.341 0.340 0.438 1.181
0.99960 0.640 1.292 0.360 0.455 1.129
0.99988 0.620 1.246 0.380 0.470 1.084
0.99998 0.600 1.205 0.400 0.484 1.045
0.9978(cid:63) 0.682 1.400 0.302 (0.318) 0.419 1.246
0.9983(cid:63) 0.674 1.379 0.308 (0.326) 0.426 1.223
0.9983† 0.674 1.379 0.309 (0.326) 0.426 1.223
0.9986† 0.669 1.365 0.315 (0.331) 0.430 1.207
TABLE I. Kerr space-time. E , L , maximum radiative efficiency in the NT model, angular frequency at the ISCO
ISCO ISCO
radius, and ISCO radius in Boyer-Lindquist coordinates for different value of the spin parameter. (cid:63) From Ref. [30], without
returning radiation. † From Ref. [31], including returning radiation. See text for details.
0.6 1
r /10
10 0.5 ISCO 0.8
/O
C 0.4
S
, rOI 0.3 a* 0.6
C
(cid:49)IS 0.2 0.4
, (cid:100) (cid:49)ISCO
0.2
0.1
(cid:100)
0 0
0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
a M/M
* 0
FIG. 1. Kerr space-time. Left panel: maximum radiative efficiency in the NT model, angular frequency at the ISCO radius,
and ISCO radius (divided by 10) in Boyer-Lindquist coordinates as a function of the spin parameter. Units M = 1. Right
panel: Evolution of the spin parameter for an initially non-rotating BH of mass M . See text for details.
0
corotatingaccretiondisksonefindsthatη isintherange equilibrium spin parameter is thus aeq = 1, and it is
∗
∼0.057 (Schwarzschild BH, a =0) to ∼0.423 (extreme reached after the object increased its mass by a factor
∗ √
Kerr BH, a = 1). However, η increases slowly with 6 ≈ 2.4. The evolution of the spin parameter as a
∗
a for low values of the spin parameters and it increases function of M/M is shown in Fig. 1, right panel.
∗ 0
muchfasterwhena approaches1. Forinstance,η >0.30
∗ For an astrophysical BH, the situation is slightly dif-
requires a > 0.996, η > 0.34 requires a > 0.999, and
∗ ∗ ferent, as there is no realistic mechanism to spin up the
η >0.40demandsa extremelycloseto1(seeTab.Iand
∗ object to a too close to 1. The accretion process from a
Fig. 1). ∗
thin disk is still a very efficient mechanism, but we have
In the Kerr background, we can analytically integrate
to include the effect of the radiation emitted by the disk
Eq. (2.2) and we obtain [41]
and captured by the BH, Eqs. (2.3) and (2.4). In this
(cid:114)2M (cid:34) (cid:114) M2 (cid:35) √ case,wefindslightlylowervaluesforηandae∗q. However,
a = 0 4− 18 0 −2 (forM/M ≤ 6), as η increases very quickly when a approaches 1, even a
∗ 3 M M2 0 tiny different in aeq causes a non-n∗egligible difference in
√ ∗
the maximum value of η. As the radiation with angular
a =1 (forM/M > 6), (3.1)
∗ 0
momentum anti-parallel to the BH spin has larger cap-
for an initially non-rotating BH with mass M . The ture cross section, from Eqs. (2.4) one finds aeq <1. For
0 ∗
5
instance, Thorne found aeq = 0.9978 when the emission circular orbits. When instead the ISCO is marginally
∗
of the disk is isotropic, and aeq =0.9983 when the emis- stable along the vertical direction, such a minimum does
∗
sion is limb-darkened [30]. The corresponding efficien- not exist in general. In the latter case, the curve of the
cies [Eq. (2.3)] are respectively 0.302 and 0.308. From energy of equatorial circular orbits around the ISCO is
Eq. (2.1), we would have found 0.318 (for a = 0.9978) clearly steeper, causing a lower value of E with re-
∗ ISCO
and 0.326 (for a = 0.9983). The effect of the return- specttothecaseofradiallyunstableISCOwiththesame
∗
ing radiation, which changes a little bit the emission of radius. The total efficiency η is therefore higher.
the disk, introduces an even smaller correction [31]. The As Eqs. (2.2) and (2.4) depend on the geometry of the
equilibrium spin parameter is now ae∗q = 0.9983 in the space-time,theequilibriumspinparameterinthesemet-
case ofisotropic emission and ae∗q =0.9986 inthe case of rics is determined by the deformation parameters (cid:15)i. We
limb-darkened emission. The radiative efficiencies turn can notice that in the case of the super-massive BH can-
out to be respectively 0.309 and 0.315 [0.326 and 0.331 didates, aeq should be the maximum value for the spin
∗
if we used Eq. (2.1)]. In conclusion, if the super-massive parameteroftheseobjects, independentlyoftheiractual
BHcandidatesareKerrBHs, arealisticupperboundfor nature [16, 29, 42]. Indeed, for the super-massive BH
η shouldbearound0.32,astheseobjectscanunlikelybe candidatestheinitialvalueofa ,i.e. theoneatthetime
∗
spun up to a∗ higher than roughly 0.998. of their birth, is thought to be completely irrelevant, as
Tab. I shows also the angular frequency at the ISCO, the mass of these objects has increased by several or-
ΩISCO, and the value of the ISCO radius in Boyer- dersofmagnitudefromtheoriginaloneandthespinhas
Lindquist coordinates, rISCO, for the same values of the evolved accordingly. Long term accretion from a thin
spinparameter. EvenforΩISCO andrISCO thereisaone- disk can efficiently spin the BH up to ae∗q (but it spins
to-onecorrespondence(foragivenmassM)withthespin the BH down if a >aeq), while other processes (chaotic
∗ ∗
parameter of the BH. In the case of corotating disks, the accretions, minor and major mergers) more likely spin
angular frequency at the ISCO increases with a∗, from the object down to a∗ ∼ 0. So, the value of ae∗q we can
∼0.068 M (a∗ =0) to 0.5 M (a∗ =1). The ISCO radius inferfromEqs.(2.2)and(2.4)canbeusedtoexcludethe
decreases with a∗, from 6 M (a∗ = 0) to M (a∗ = 1). region a∗ > ae∗q in the diagrams spin parameter vs de-
ΩISCO and rISCO as a function of the spin parameter are formation parameter. This point will be crucial for what
shown (for corotating disk) in the left panel of Fig. 1. follows.
Let us now start considering the properties of the JP
space-timewithdeformationparameter(cid:15) and(cid:15) =0for
3 i
IV. JOHANNSEN-PSALTIS SPACE-TIMES i (cid:54)= 3. Fig. 2 shows the efficiency η, as deduced from
Eq. (2.1). In particular, we are interested in the most
The JP space-times have been proposed in [24] explic- efficient systems and I report the curves on the a∗(cid:15)3-
itly to be used to test the Kerr geometry around astro- plane with η = 0.30, 0.32, 0.34, 0.36, 0.38, 0.40. The
physical BH candidates. They describe BHs (at least black solid curve marks the equilibrium spin parameter
for a∗ lower than a critical value) and they are an ex- ae∗q. For (cid:15)3 >0, the compact object is more prolate than
tension of the Kerr solution, in the sense that here the Kerr and ae∗q < 1. For (cid:15)3 < 0, the object is more oblate
compact objects are specified by the mass, the spin an- and ae∗q > 1. It is thus clear that observations can po-
gular momentum, and an infinite number of deforma- tentially provide very strong constraints for (cid:15)3 < 0, and
tion parameters (cid:15)i (i = 3, 4, 5, etc.) measuring devia- muchweakerboundsfor(cid:15)3 >0. Inparticular,for(cid:15)3 >0,
tions from the Kerr geometry. When all the deformation themaximumvalueofa∗ maybeevensignificantlylower
parameters vanish, one recovers exactly the Kerr back- than 1, but still η can be very high. Interestingly, for
ground. Theexplicitexpressionofthemetricisreported (cid:15)3 (cid:54)= 0, η does not increases abruptly as a∗ approaches
in Appendix A. The JP metric is not a solution of any ae∗q and therefore the effect of the radiation emitted by
known gravity theory, but it seems to be a very conve- the disk and captured by the BH may be neglected. In-
nient framework to test the Kerr BH hypothesis. deed, we can assume that the radiation captured by the
The NT model for thin accretion disks can be easily BH decreases ae∗q by ∼ 0.002, as in Kerr (actually the
extended to non-Kerr backgrounds. The first important correction to ae∗q should be smaller than 0.002, as the
differenceisthedeterminationoftheISCOradius. Inthe ISCOradiusislarger,seetherightpanelinFig.3). This
Kerr background, circular orbits on the equatorial plane is equivalent to replace the curve of ae∗q in Fig. 2 with
arealwaysverticallystableandtheISCOissetbytheor- the curve ae∗q−0.002. At this point, we have to decrease
bital stability along the radial direction. If the compact η by ∼0.01. Except for the Kerr space-time and for the
objectismoreoblatethanaKerrBH,thatremainstrue. space-times with small deviations from the Kerr geome-
On the contrary, if the compact object is more prolate try, the two effects should not cause significant changes.
(includingthecaseinwhichtheobjectisstilloblate,but Inparticular,westillfindalargeregionwithobjectswith
simply less oblate than a Kerr BH with the same spin) η >0.32, which would be impossible for a Kerr BH in a
both kinds of instabilities are possible. When the ISCO realistic astrophysical context.
radius is marginally stable along the radial direction, as Fig. 3 shows the contour plot of the angular fre-
in Kerr, it is at the minimum of the energy of equatorial quency at the ISCO, Ω , (left panel) and the one of
ISCO
6
5 0.4 0.40
0.30
4 0.3
3 0.2 0.30
3 3
(cid:161) (cid:161)
2 0.1
1 0.40 0
0 -0.1
0.7 0.8 0.9 1 0.92 0.94 0.96 0.98 1 1.02 1.04
a a
* *
FIG. 2. JP space-time with deformation parameter (cid:15) and (cid:15) = 0 for i (cid:54)= 3. Contour plots of the radiative efficiency
3 i
η =1−E : η =0.30 (red solid curve), 0.32 (blue dotted curve), 0.34 (red solid curve), 0.36 (blue dotted curve), 0.38 (red
ISCO
solid curve), 0.40 (blue dotted curve). The black solid curve is the equilibrium spin parameter aeq as inferred from Eq. (2.2)
∗
and represents the maximum value for the spin parameter of the super-massive BH candidates. This means that the region
with a >aeq is not allowed. The right panel is simply the enlargement of the area inside the orange box in the left panel.
∗ ∗
0.4 0.4
0.3 0.3
0.2 0.2
3 3
(cid:161) (cid:161) 1.25
0.1 0.1
0.42
0 0
2.00
0.32
-0.1 -0.1
0.92 0.94 0.96 0.98 1 1.02 1.04 0.92 0.94 0.96 0.98 1 1.02 1.04
a a
* *
FIG.3. JPspace-timewithdeformationparameter(cid:15) and(cid:15) =0fori(cid:54)=3. Leftpanel: contourplotoftheangularfrequencyat
3 i
the ISCO radius: Ω /M =0.32 (red solid curve), 0.34 (blue dotted curve), 0.36 (red solid curve), 0.38 (blue dotted curve),
ISCO
0.40 (red solid curve), 0.42 (blue dotted curve). Right panel: contour plot of the ISCO radius in Boyer-Lindquist coordinates:
r /M = 2.00 (red solid curve), 1.75 (blue dotted curve), 1.50 (red solid curve), 1.25 (blue dotted curve). The black solid
ISCO
curve is the equilibrium spin parameter aeq as inferred from Eq. (2.2).
∗
the ISCO radius in Boyer-Lindquist coordinates, r , parameter (cid:15) and(cid:15) =0for i(cid:54)=4(leftpanel)andfor the
ISCO 4 i
(rightpanel). WhileaccretioninJPspace-timemayhave one with deformation parameter (cid:15) and (cid:15) = 0 for i (cid:54)= 5
5 i
high efficiency, thus mimicking a Kerr BH with a ex- (rightpanel). Forthesetwocases,eventhecontourplots
∗
tremely close to 1 (or even exceeding the efficiency of a ofΩ andr (notshownhere)arefairlysimilarto
ISCO ISCO
KerrBHwitha =1),thesameobjectwouldhaveΩ the ones in Fig. 3.
∗ ISCO
andr moresimilartotheoneexpectedforKerrBHs
ISCO
with lower spin parameter. Ω can be potentially in-
ISCO
ferred in several ways from observations (e.g. from the
V. MANKO-NOVIKOV SPACE-TIMES
variabilityofthesource)andthatmaybeusefultobreak
the degeneracy between a and (cid:15) in η.
∗ 3 TheMNspace-timesarestationary,axisymmetric,and
Lastly, we can consider JP backgrounds with a dif- asymptotically flat exact solutions of the Einstein’s vac-
ferent deformation parameter. Interestingly, one finds uum equations [25]. These metrics can describe the ex-
essentially the same picture. In Fig. 4, I report the con- terior gravitational field of a compact object with mass
tour plots of η for the JP space-time with deformation M,spinparametera ,andarbitrarymass-multipolemo-
∗
7
20 80
0.30
15 60
0.30
4 10 5 40
(cid:161) (cid:161)
5 20
0.40
0 0
0.5 0.6 0.7 0.8 0.9 1 1.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
a a
* *
FIG. 4. As in Fig. 2, for the JP space-time with deformation parameter (cid:15) and (cid:15) = 0 for i (cid:54)= 4 (left panel) and for the one
4 i
with deformation parameter (cid:15) and (cid:15) =0 for i(cid:54)=5 (right panel).
5 i
0.006
0
0.004
0.30
0.40
0.002
0.40
~q -0.1 ~q 0
-0.002
-0.004
-0.2 0.30
-0.006
0.88 0.9 0.92 0.94 0.96 0.98 1 0.97 0.98 0.99 1 1.01
a a
* *
FIG.5. AsinFig.2fortheMNspace-timewithdeformationparameterq˜. Thecompactobjectisoblate(prolate)whenq˜>−1
(q˜<−1), but it is more oblate (prolate) than a Kerr BH with the same value of the spin parameter for q˜>0 (q˜<0).
0 0
1.25
0.32
~q -0.1 ~q -0.1
-0.2 -0.2 2.00
0.88 0.9 0.92 0.94 0.96 0.98 1 0.88 0.9 0.92 0.94 0.96 0.98 1
a a
* *
FIG. 6. As in Fig. 3 for the MN space-time with deformation parameter q˜.
8
ments, whilethecurrent-multipolemomentsarefixedby it is not easy to get an estimate of η, because a mea-
the former. The simplest non-Kerr object has three free surement of the mass accretion rate can be problematic.
parameters: mass M, spin parameter a , and anomalous The Soltan’s argument provides an elegant way to de-
∗
quadrupole moment q˜(see Appendix B). q˜is defined by termine the mean radiative efficiency of AGN, η¯, from
themeanBHmassdensityinthecontemporaryUniverse
Q=(1+q˜)Q , (5.1)
Kerr andtheAGNluminosityperunitvolumeintegratedover
time [20]. There are several sources of uncertainty in
where Q is the mass-quadrupole moment of the compact
the final result, but a conservative bound seems to be
object, while Q = −a2M3 is the one of a Kerr BH
Kerr ∗ η¯ > 0.15 [21]. If the compact objects in AGN are Kerr
with the same spin parameter a . The case q˜ = 0 cor-
∗ BHs, this constraint requires a¯ >0.89. In Ref. [22], the
responds to the Kerr metric, while for q˜ > 0 (q˜ < 0) ∗
authors used a revised version of the Soltan’s argument,
the object is more oblate (prolate) than a Kerr BH. The
in which some important assumptions are not necessary.
propertiesofthissubclassofMNspace-timeswithdefor-
They found a mean radiative efficiency η¯ ≈ 0.30−0.35
mation parameter q˜were studied in [43] and [18]. When
in the redshift interval 0.4<z <2.1.
q˜<0, for a quite restricted set of values of a and q˜, the
∗
plunging region does not connect the ISCO to the com- Recently, it has been proposed a way to estimate the
pactobject[18]. ThatoccurswheninsidetheISCOthere radiativeefficiencyofindividualAGN[23]. Themassac-
is another region with stable circular orbits on the equa- cretion rate M˙ can indeed be determined from the low
torialplaneandenergylowerthanE . Inthesespace- frequency region of the thermal spectrum of the accre-
ISCO
times, the gas’s particles plunging from the ISCO get tion disk of these objects, if the mass M is known. The
trapped between the ISCO and the compact object. As standard accretion disk model cannot reproduce the ob-
thegasneedtoradiateadditionalenergyandangularmo- served spectrum of AGN. That is due to reprocessing,
mentum in order to fall to the compact object, the value Comptonization in a corona, radiative transfer effects in
of the radiative efficiency turns out to be higher than the inner accretion disk, an so on. Here the key-point
the one inferred from Eq. (2.1) or Eq. (2.3). The actual is that this method relies on the validity of the simple
value of η depends on the astrophysical processes. For thin-disk model at relatively large radii, where these ef-
instance, Ref. [18] discussed a simple model in which the fectsarethoughttobeirrelevant. Onecanthenestimate
gas forms a thick disk inside the ISCO and η =1−E , the total luminosity of the source, irrespective of its ex-
in
where E is the specific energy at the inner edge of the act production mechanism, and infer the radiative effi-
in
thick disk, is up to a few percent higher than the one ciency. So, the redistribution of the radiation emitted in
predicted by Eq. (2.1). the accretion process does not affect the measurement of
Fig. 5 shows the radiative efficiency η as computed radiativeefficiency;formoredetails,seeSections2.3and
from Eq. (2.1). The MN metric is written in prolate 4.3 of Ref. [23]. The authors found a strong correlation
spheroidal coordinates, which require a∗ < 1; here the of η with M, raising from η ∼ 0.03 when M ∼ 107 M(cid:12)
region a∗ ≥ 1 is covered by the Malko-Mielke-Sanabria and Lbol/LEdd ∼ 1 to η ∼ 0.4 when M ∼ 109 M(cid:12) and
G´omez solution [44] which can be parametrized by the Lbol/LEdd ∼0.3.
same three parameters of our subclass of the MN metric The most massive objects in AGN with M ∼109 M
(cid:12)
(i.e. M, a∗, and q˜) and was shown to be very similar to maybeexcellentcandidatestotesttheKerrBHhypoth-
theMNspace-timeinthecommonregionofvalidity[42]. esis. The standard criterion to consider thin disks is in-
It is clear that there are strong analogies with the JP deed that L /L does not exceed 0.3, corresponding
bol Edd
space-times. For objects more oblate than Kerr, ae∗q >1 to an opening angle h not larger than 0.05. In this case,
and a high radiative efficiency is possible only for small theestimateofη mayprovideameasurementofthespe-
deviations from the Kerr geometry. For objects more cific energy at the ISCO (or at least a lower bound, as
prolate than Kerr, ae∗q < 1 and when the ISCO radius weareneglectingηk andζE). Theestimateofη canthen
is set by the orbital stability along the vertical direction be used to constrain possible deviations from the Kerr
η can be high even for objects rotating slower. Fig. 6 geometryandmayhavesomeadvantageswithrespectto
shows the angular frequency at the ISCO, ΩISCO, and themostpopulartechniquestomeasuretheBHspinpa-
the ISCO radius in Boyer-Lindquist coordinates, rISCO. rameter, i.e. the continuum fitting method and the Kα
As in the JP background, the same object with high ra- iron line analysis. The continuum fitting method can be
diative efficiency has lower ΩISCO and larger rISCO. The usedonlyforstellar-massBHcandidates,forwhichthere
case of the MN space-time with q˜=0 and an anomalous is no argument to constrain a : the value of the spin pa-
∗
mass-hexadecapolemomenth˜ showssimilarpropertiesin rameter of the BH candidates in X-ray binary systems
η, ΩISCO, and rISCO. should reflect the one at the time of the formation of the
objectanditisdefinitivelyimpossibletopredictitsvalue
relaxing the assumption that GR is the correct theory of
VI. DISCUSSION gravity. The method of the Kα iron line can be used for
the super-massive BH candidates in galactic nuclei (so
The radiative efficiency of a source can be deduced we can impose a < aeq), but it necessarily relies on an
∗ ∗
from the formula η = L /M˙ . In general, however, ad hoc intrinsic surface brightness profile and there is no
bol
9
way to determine it a priori. Although simulations have VII. CONCLUSIONS
the potential to compute the surface brightness profile,
the standard required to test the Kerr background is a
ThereissomeevidencethatthemostmassiveBHcan-
muchhigherlevelofreliabilitythanislikelytobemetat
didates in AGN have a high radiative efficiency and a
any time in the foreseeable future.
thin accretion disk. In this case, they could be excellent
candidates to test GR in the strong field regime and, in
Let us now assume that future GRMHD simulations
particular, the Kerr BH paradigm. A robust observation
confirm the validity of the NT model for thin accretion
of a high radiative efficiency together with the confirma-
disks and that future observations provide robust evi-
tion of the validity of the standard accretion disk model
dencethatthemostmassiveobjectsinAGNhaveavery
for the accretion process onto these objects would con-
high radiative efficiency and a moderate mass accretion
strain possible deviations from the Kerr geometry of the
rate. If we exclude the possibility of objects more pro-
space-time around astrophysical BH candidates. If we
latethanKerr,whichmaybedifficulttoexplaintheoreti-
restrict our attention only to objects more oblate than a
cally3,fromtherightpanelofFig.5wecanseethatq˜can
Kerr BH with the same spin parameter, a radiative effi-
be constrained at the level of ∼10−3. That roughly cor-
ciency η >0.30 requires that the quadrupole moment of
responds to test the mass-quadrupole moment of these the compact object deviates not more than ∼10−3 with
objects with a precision ∼ 10−3, see Eq. (5.1). For a
respect to the one of a Kerr BH. If we allow for the ex-
self-gravitating fluid with reasonable equations of state,
istence of compact objects more prolate than a Kerr BH
q˜ is much larger, i.e. the compact object becomes sig-
(whose existence, however, may be questionable, see the
nificantly more oblate than a Kerr BH when the spin
footnote 3), the constraints is much weaker, at the level
parameter increases. For instance, a neutron star should
of20%. Ontheotherhand,theobservationofanaccret-
have q˜ > 1 [45]. A similar bound may be obtained by
ing BH candidate with η >0.32 might indicate that the
observing the gravitational waves emitted by an EMRI
super-massive BH candidates in galactic nuclei are not
withfuturespace-basedgravitational-wavedetectors(see
the Kerr BH of GR, as there is no realistic astrophysical
the third reference in [11]), even if a direct comparison
mechanism capable to spin up them to a >0.998.
∗
is not possible, as here we are probing the geometry of
For the time being, there are at least two main issues
the space-time very close to the compact objects, where
toaddressbeforeusingthemeasurementoftheradiative
higherordersmultipolemomentsarenotreallynegligible,
efficiency of individual AGN to test GR:
while gravitational wave detectors will study the space-
time at larger distances, and therefore they will be sen- 1. We do not really know if the NT model can de-
sitivetothemass-quadrupolemomentonly. Ifwedonot scribethinaccretiondisksaroundastrophysicalBH
exclude a priori the existence of compact objects more candidates. If it can or it is easy to estimate the
prolate than a Kerr BH, the constraint is much weaker: necessary corrections, the measurement of η could
for η > 0.30, deviations from the Kerr mass-quadrupole provideaninterestingwaytoprobethegeometryof
moments can still be up to 20%. Let us now consider a thespace-timearoundBHcandidates. Ifitcannot,
moreexoticsituation: theNTmodelworksandobserva- it seems to be impossible to test GR (with this or
tions find unambiguously η >0.32. In absence of torque other approaches) from the properties of the radi-
at the inner edge of the disk, it would be difficult to ation emitted by the gas in the accretion disk and
explain this outcome in the Kerr background, and new weshouldwaitfortheadventofgravitationalwave
physics may be invoked. Let us notice, however, that astronomy.
a non-Kerr background can explain high η, say up to
η ∼0.4,buthighervaluesmaynotbenatural,aswestill 2. We need robust and more precise measurements of
have a∗ <ae∗q. η. At present, there are several sources with η >
0.32 at high masses, but we cannot really exclude
they actually have η < 0.32 [23]. There are some
sources of uncertainty in the final estimate of the
radiative efficiency and the most important one is
thedeterminationofthemassoftheBHcandidate.
Moreover,theresultspresentedin[23]arebasedon
a preliminary study and it is necessary to further
3 The true problem is not the existence of compact objects less investigate and verify the validity of the method
oblate than a Kerr BH with the same spin parameter, but that proposed by these authors.
theobjectswearetalkingabouthavetheISCOmarginallysta-
ble along the vertical direction. In the case of the JP metric,
theseobjectsareBHswithtwodisconnectedeventhorizons,one
above and one below the equatorial plane [19]. These objects ACKNOWLEDGMENTS
moreprolatethanKerrandhighradiativeefficiencyhavethere-
fore “two centers of attraction”, one above and one below the
equatorialplane. Itisnotclearatallifsuchasystemcanexist This work was supported by the Humboldt Founda-
andbestable. tion.
10
Appendix A: JP metric correct Newtonian limit we have to impose (cid:15) = (cid:15) = 0,
0 1
while (cid:15) is constrained at the level of 10−4 from current
2
tests in the Solar System [24].
The JP metric is not a solution in any known grav-
ity theory. It is a simple parametrization to describe
thespace-timearoundnon-KerrBHsandwasspecifically
Appendix B: MN metric
proposedtotesttheKerrBHhypothesis[24]. Themetric
was obtained by starting from a deformed Schwarzschild
solution and then by applying a Newman-Janis trans- The MN metric is a stationary, axisymmetric, and
formation. The non-zero metric coefficients in Boyer- asymptotically flat exact solution of the Einstein’s
Lindquist coordinates are: vacuum equations with arbitrary mass-multipole mo-
ments [25]. It does not describe the space-time around
(cid:18) 2Mr(cid:19) a BH and it has naked singularities and closed time-like
g =− 1− (1+h),
tt ρ2 curves at small radii. These pathological features should
beeitherinsidesomesortofexoticcompactobject,whose
2aMrsin2θ
g =− (1+h), exteriorgravitationalfieldwouldbedescribedbytheMN
tφ ρ2 metric,orGRshouldbreakdownclosetothem. Thenon-
(cid:20) 2a2Mrsin2θ(cid:21) zero metric coefficients in prolate spheroidal coordinates
g =sin2θ r2+a2+ +
φφ ρ2 are:
a2(ρ2+2Mr)sin4θ g =−f,
+ h, tt
ρ2 g =fω,
tφ
grr = ∆ρ+2(a12h+shin)2θ , gφφ = kf2 (cid:0)x2−1(cid:1)(cid:0)1−y2(cid:1)−fω2,
gθθ =ρ2, (A1) k2e2γ x2−y2
g = ,
xx f x2−1
where
k2e2γ x2−y2
g = , (B1)
ρ2 =r2+a2cos2θ, yy f 1−y2
∆=r2−2Mr+a2,
where
(cid:88)∞ (cid:18) Mr (cid:19)(cid:18)M2(cid:19)k
h= (cid:15) + (cid:15) . (A2) f =e2ψA/B,
2k ρ2 2k+1 ρ2
k=0 ω =2ke−2ψCA−1−4kα(cid:0)1−α2(cid:1)−1 ,
The metric has an infinite number of free parameters (cid:15)i e2γ =e2γ(cid:48)A(cid:0)x2−1(cid:1)−1(cid:0)1−α2(cid:1)−2 , (B2)
andtheKerrsolutionisrecoveredwhenalltheseparam-
eters are set to zero. However, in order to recover the and
+∞
ψ = (cid:88) αnPn ,
Rn+1
n=1
γ(cid:48) = 1ln x2−1 + (cid:88)+∞ (m+1)(n+1)αmαn(P P −P P )+
2 x2−y2 (m+n+2)Rm+n+2 m+1 n+1 m n
m,n=1
(cid:34)(cid:88)+∞ (cid:32) (cid:88)n x−y+(−1)n−k(x+y) (cid:33)(cid:35)
+ α (−1)n+1−1+ P ,
n Rk+1 k
n=1 k=0
A=(x2−1)(1+ab)2−(1−y2)(b−a)2,
B =[x+1+(x−1)ab]2+[(1+y)a+(1−y)b]2,
C =(x2−1)(1+ab)[b−a−y(a+b)]+(1−y2)(b−a)[1+ab+x(1−ab)],
(cid:34)+∞ (cid:32) n (cid:33)(cid:35)
(cid:88) (cid:88)(x−y)
a=−αexp 2α 1− P ,
n Rk+1 k
n=1 k=0
(cid:34)(cid:88)+∞ (cid:32) (cid:88)n (−1)n−k+1(x+y) (cid:33)(cid:35)
b=αexp 2α (−1)n+ P . (B3)
n Rk+1 k
n=1 k=0