Measuring the Kerr spin parameter of regular black holes from their shadow
MMeasuring the Kerr spin parameter ofregular black holes from their shadow
Zilong Li and Cosimo Bambi Center for Field Theory and Particle Physics & Department of Physics,Fudan University,220 Handan Road, 200433 Shanghai, ChinaE-mail: [email protected], [email protected]
Abstract.
In a previous paper, one of us has showed that, at least in some cases, theKerr-nature of astrophysical black hole candidates is extremely difficult to test and currenttechniques, even in presence of excellent data not available today, cannot distinguish a Kerrblack hole from a Bardeen one, despite the substantial difference of the two backgrounds.In this paper, we investigate if the detection of the “shadow” of nearby super-massive blackhole candidates by near future mm/sub-mm very long baseline interferometry experimentscan do the job. More specifically, we consider the measurement of the Kerr spin parameterof the Bardeen and Hayward regular black holes from their shadow, and we then comparethe result with the estimate inferred from the K α iron line and from the frequency of theinnermost stable circular orbit. For non-rotating black holes, the shadow approach providesdifferent values, and therefore the Kerr black hole hypothesis can potentially be tested. Fornear extremal objects, all the approaches give quite similar results, and therefore it is notpossible to constrain deviations from the Kerr solution. The present work confirms that itis definitively challenging to test this kind of metrics, even with future facilities. However,the detection of a source that looks like a fast-rotating Kerr black hole can put meaningfulconstraints on the nature of the compact object. Keywords: gravity, modified gravity, astrophysical black holes. Corresponding author a r X i v : . [ g r- q c ] J a n ontents In 4-dimensional general relativity, uncharged black holes (BHs) are described by the Kerrsolution and are completely specified by two parameters, the mass M and the spin angularmomentum J [1–3]. The condition for the existence of the event horizon is | a ∗ | ≤
1, where a ∗ = a/M = J/M is the spin parameter . Astrophysical BHs, if they exist, are expectedto be well described by the Kerr metric: initial deviations from the Kerr geometry shouldbe quickly radiated away through the emission of gravitational waves [4, 5], an initially non-vanishing electric charge would be shortly neutralized in their highly ionized environment [6],while the presence of the accretion disk is completely negligible in most cases. AstrophysicalBH candidates are dark compact objects in X-ray binary systems with a mass M ≈ − M (cid:12) and super-massive bodies in galactic nuclei with a mass M ∼ − M (cid:12) [7]. Theyare thought to be the Kerr BHs of general relativity, but their actual nature is still to beverified. Stellar-mass BH candidates are simply too heavy to be neutron or quark starsfor any plausible matter equation of state [8, 9]. At least some of the super-massive BHcandidates at the centers of galaxies are too massive, compact, and old to be clusters ofnon-luminous bodies [10]. The non-observation of electromagnetic radiation emitted by thepossible surface of these objects may also be interpreted as an indication for the existence ofan event horizon [11, 12] (but see [13, 14]). However, there is no evidence that the spacetimegeometry around them is really described by the Kerr solution.The possibility of testing the nature of astrophysical BH candidates with current andnear future observations has recently become a quite active research field (for a review,see [15, 16]). Today, there are two relatively robust techniques to estimate the spin parameterof BH candidates under the assumption that the geometry around them is described by theKerr metric: the so-called continuum-fitting method [17–19] and the analysis of the K α iron line [20–22]. Both the approaches can be used to probe the geometry of the spacetimearound BH candidates and measure the spin parameter and possible deviations from theKerr solution [23–30]. However, it turns out that there is a strong correlation between thespin and possible deformations and that one can only constrain a certain combination ofthese quantities. In other words, the thermal spectrum of a thin accretion disk and theprofile of the K α iron line of a Kerr BH with spin parameter a ∗ can be extremely similar Throughout the paper, we use units in which G N = c = 1, unless stated otherwise. – 1 – practically indistinguishable – from the ones of non-Kerr compact objects with differentspin parameters. In Ref. [31], one of us has showed that, at least for some non-Kerr metrics,the combination of the continuum-fitting method and of the iron line analysis cannot fix thisproblem. Other approaches to test the nature of BH candidates are either not yet mature, likethe case of quasi-periodic oscillations [32, 33], or it is not clear when astrophysical data willbe available, like the case of gravitational waves or observations of a BH binary with a pulsarcompanion [34–40]. The estimate of the power of steady and transient jets can potentiallybreak the degeneracy between spin parameter and deviations from the Kerr solution [41, 42],but at present we do not know the exact mechanism responsible for these phenomena. Arough estimate of possible deviations from the Kerr geometry in the spacetime around super-massive BH candidates can be obtained from considerations on their radiative efficiency andon the possible mechanisms capable of spinning them up and down [43–46].A quite promising technique to test the nature of super-massive BH candidates withnear future very long baseline interferometry (VLBI) facilities is through the observation ofthe “shadow” of these objects [47, 48]. The shadow is a dark area over a bright backgroundappearing in the image of an optically thin emitting region around a BH [49–51]. Whilethe intensity map of the image depends on the details of the accretion process and of theemission mechanisms, the boundary of the shadow is only determined by the metric of thespacetime, since it corresponds to the apparent image of the photon capture sphere as seenby a distant observer. The possibility of testing the nature of supermassive BH candidatesby observing the shape of their shadow has been already discussed in the literature, startingfrom Ref. [52, 53]. In general, very accurate observations are necessary, because the effect ofpossible deviations from the Kerr solution are tiny [54–60].At first approximation, the shape of the shadow of a BH is a circle. The radius of thecircle corresponds to the apparent photon capture radius, which, for a given metric, is set bythe mass of the compact object and its distance from us. These two quantities are usuallyknown with a large uncertainty, and therefore the observation of the size of the shadow canunlikely be used to test the nature of the BH candidate (but see Ref. [60]). The shapeof the shadow is instead the key-point. The first order correction to the circle is due tothe BH spin, as the photon capture radius is different for co-rotating and counter-rotatingparticles. The boundary of the shadow has thus a dent on one side: the deformation is morepronounced for an observer on the equatorial plane (viewing angle i = 90 ◦ ) and decreasesas the observer moves towards the spin axis, to completely disappear when i = 0 ◦ or 180 ◦ .Possible deviations from the Kerr solutions usually introduces smaller corrections.In the present paper, we consider the measurement of the Kerr spin parameter of KerrBHs and non-Kerr regular BHs; that is, we measure the spin parameter a ∗ from the shapeof the shadow of a BH assuming it is of the Kerr kind. We use the procedure proposedin Ref. [61], which is based on the determination of the distortion parameter δ s = D cs /R s ,where D cs and R s are, respectively, the dent and the radius of the shadow. In the caseof non-Kerr BHs, this technique provides the correct value of a ∗ for non-rotating objects,but a quite different spin for near extremal states. We then compare these measurementswith the ones we could infer from the analysis of the K α iron line and the observation ofa hot spot orbiting around the BH candidate. The K α iron line approach is currently theonly relatively robust technique to probe the spacetime geometry around these objects. Theobservations of hot spots orbiting around nearby super-massive BH candidates with mm/sub-mm VLBI facilities will hopefully allow to determine the frequency of test-particles at theinnermost stable circular orbit (ISCO) radius. For non-rotating and slow-rotating objects, the– 2 –hadow approach provides different results with respect to the other two techniques, so thata possible combination of these methods may break the degeneracy between spin parameterand possible deviations from the Kerr solution. All the approaches seem instead to providequite similar measurements for near extremal BHs, which means that their combinationcannot be used to test the spacetime geometry around these objects. Our work confirmsthe difficulty to observationally test the Kerr nature of astrophysical BH candidates. Onlyvery good observations of the shadow, which are capable of measuring simultaneously thespin and possible deformations from the Kerr solution, might be required to test the KerrBH hypothesis. However, in the case of objects that look like very fast-rotating Kerr BHs,interesting constraints on their nature seem to be possible.The content of the paper is as follows. In Section 2, we present our approach to testthe nature of astrophysical BH candidates and we introduce the metrics that will be usedin the rest of the paper. In Section 3, we briefly review the concept and the calculation ofthe BH’s shadow. Section 4 is devoted to the measurement of the Kerr spin parameter: weapply the procedure proposed in Ref. [61] to measure the spin of a Kerr BH to the Bardeenand Hayward BHs. Such a prescription provides the correct value of the spin parameterfor non-rotating objects, but a wrong estimate for fast-rotating BHs. The results are thencompared with the measurements we would obtain from the analysis of the K α iron line andthe hot spot model in Section 5. Summary and conclusions are in Section 6. In Boyer-Lindquist coordinates, the non-vanishing metric coefficients of the Kerr metric are g tt = − (cid:18) − M r Σ (cid:19) , g tφ = − aM r sin θ Σ ,g φφ = (cid:18) r + a + 2 a M r sin θ Σ (cid:19) sin θ , g rr = Σ∆ , g θθ = Σ , (2.1)where Σ = r + a cos θ , ∆ = r − M r + a . (2.2) M is the BH mass and a = J/M is its spin parameter. If we want to test the Kerr natureof an astrophysical BH candidate, it is convenient to consider a more general spacetime, inwhich the central object is described by a mass M , spin parameter a , and one (or more)“deformation paramater(s)”. The latter measure possible deviations from the Kerr solution,which must be recovered when all the deformation parameters vanish. The strategy is thusto calculate some observables in this more general background and then fit the data of thesource to find the allowed values of the spin and of the deformation parameters. If theobservations require vanishing deformation parameters, the compact object is a Kerr BH.If they demand non-vanishing deformation parameters, astrophysical BH candidates are notthe Kerr BH of general relativity and new physics is necessary. In general, however, the resultis that observations allows both the possibility of a Kerr BH with a certain spin parameterand non-Kerr objects with different spin parameters.As non-Kerr metrics, in the present work we will focus on the Bardeen and HaywardBHs [62, 63], which cannot be observationally tested by current techniques (continuum-fittingmethod and iron line analysis), even in presence of excellent data not available today [31].– 3 – g / M a * Bardeen metricBlack HolesHorizonless States g / M a * Hayward metricBlack HolesHorizonless States
Figure 1 . Rotating Bardeen (left panel) and Hayward (right panel) metrics. The red solid linesseparate the regions with BHs from the ones with horizonless objects.
The rotating solutions have the same form of the Kerr metric, with the mass M replaced by m as follows [64, 65]: M → m B = M (cid:18) r r + g (cid:19) / , (2.3) M → m H = M r r + g . (2.4) g can be interpreted as the magnetic charge of a non-linear electromagnetic field or just asa quantity introducing a deviation from the Kerr metric and solving the central singularity.The position of the even horizon is given by the larger root of ∆ = 0 and therefore there isa bound on the maximum value of the spin parameter, above which there are no BHs. Themaximum value of a ∗ is 1 for g/M = 0 (Kerr case), and decreases as g/M increases. Theregions of BHs and horizonless states on the plane ( a ∗ , g/M ) are shown in Fig. 1. In whatfollows, we will restrict the attention to the BH region: even if they can be created [65], thehorizonless states are likely very unstable objects with a short lifetime due to the ergoregioninstability. The shadow of a BH is a dark area over a bright background appearing in the image ofan optically thin emitting region around the compact object. The boundary of the shadowdepends only on the geometry of the background and turns out to correspond to the apparentimage of the photon capture sphere as seen by a distant observer: if one fires a photon insidethe boundary of the shadow, the photon is swallowed by the BH; if outside, the photonreaches a minimum distance from the compact object and then comes back to infinity. Inthis section, we will briefly review the study of the shadow of a BH (for more details, see e.g.Sec. 63 of [49] or Ref. [66]). – 4 – ! M ! a ! M ! a ! M ! " " " " x ! M y ! M Kerr Black Holes i ! a ! M ! a ! M ! a ! M ! " " " " x ! M y ! M Kerr Black Holes i ! a ! M ! a ! M ! a ! M ! " " " " x ! M y ! M Bardeen Black Holes g ! M ! i ! a ! M ! a ! M ! a ! M ! " " " " x ! M y ! M Bardeen Black Holes g ! M ! i ! a ! M ! a ! M ! a ! M ! " " " " x ! M y ! M Hayward Black Holes g ! M ! i ! a ! M ! a ! M ! a ! M ! " " " " x ! M y ! M Hayward Black Holes g ! M ! i ! Figure 2 . Some examples of boundary of shadow of Kerr BHs (top panels), Bardeen BHs (centralpanels), and Hayward BHs (bottom panels) for different values of the spin parameter a ∗ . The viewingangle is i = 60 ◦ (left panels) and 90 ◦ (right panels). – 5 –or a photon, the equation of motion for the radial coordinate r in Boyer-Lindquistcoordinates is Σ (cid:18) drdλ (cid:19) = R , (3.1)where λ is an affine parameter, and R = E r + ( a E − L z − Q ) r + 2 m [( aE − L z ) + Q ] r − a Q , (3.2) Q = p θ + cos θ (cid:18) L z sin θ − a E (cid:19) . (3.3)The parameter m in Eq. (3.2) is equal to: M , for a Kerr BH; m B in Eq. (2.3) for a BardeenBH; m H in Eq. (2.4), for a Hayward BH. E , L z , and Q are constants of motion and are,respectively, the energy, the component of the angular momentum parallel to the BH spin,and the so-called Carter constant. p θ is the canonical momentum coniugate to θ .It is convenient to minimize the number of parameters by introducing the variables ξ = L z /E and η = Q /E . ξ and η are very simply related to the so-called “celestialcoordinates” x and y of the image, as seen by an observer at infinity who receives the lightray, by x = ξ sin i , y = ± ( η + a cos i − ξ cot i ) / , (3.4)where i is the angular coordinate of the observer at infinity. Precisely, x is the apparentperpendicular distance of the image from the axis of symmetry and y is the apparent per-pendicular distance of the image from its projection on the equatorial plane.The radial equation of motion (3.1) depends on θ only in the factor Σ , and is decoupledfrom φ and t . Thus the behavior of R ( r ) determines the type of orbit and the question ofescape versus plunge for given ξ and η . Since motion is only possible when R ( r ) ≥
0, theanalysis of the position of its roots (especially roots in r ≥ r + , where r + is the horizon) isa powerful method of investigation of photon orbits. Qualitatively, there are three kinds ofphoton orbits:(i) R ( r ) may have no roots in r ≥ r + (capture orbits), in which case the photon arrivesfrom infinity and then crosses the horizon;(ii) R ( r ) has real roots in r ≥ r + (scattering orbits), in which case the motion of photonis described by null geodesics which have a turning point ˙ r = 0;(iii) unstable orbits of constant radius, which separate the capture and the scatteringorbits, determined by R ( r ∗ ) = ∂ R ∂r ( r ∗ ) = 0 , and ∂ R ∂r ( r ∗ ) ≥ , (3.5)with r ∗ being the greatest real root of R .The apparent shape of the BH can be found by looking for the unstable orbits. Everyorbit can be characterized by the constants of motion ξ and η , and the set of unstable circularorbits ( ξ c , η c ) can be used to plot a closed curve in the xy plane which represents the boundaryof the BH shadow. The apparent image of the BH is larger than its geometrical size, becausethe BH bends light rays and thus the actual cross section is larger than the geometrical one.– 6 –rom Eqs. (3.2) and (3.5), the equations determining the unstable orbits of constant radiusare R = r + ( a − ξ c − η c ) r + 2 m [ η c + ( ξ c − a ) ] r − a η c = 0 ,∂ R ∂r = 4 r + 2( a − ξ c − η c ) r + 2 m [ η c + ( ξ c − a ) ] = 0 . (3.6)In the case of a Schwarzschild BH ( a = 0 and m = M ), the solution is [49] η c ( ξ c ) = 27 M − ξ c , (3.7)so the apparent image of the BH is a circle of radius √ M (black solid circles in the leftpanels of Fig. 2). For a Kerr BH, one finds ξ c = 1 a ( r − M ) [ M ( r − a ) − r ( r − M r + a )] ,η c = r a ( r − M ) [4 a M − r ( r − M ) ] , (3.8)where r is the radius of the unstable orbit. These two equations determine, parametrically,the critical locus ( ξ c , η c ), which is the set of unstable circular orbits. The boundary of theshadows of Kerr BHs with a/M = 0 .
0, 0 .
7, and 1 . i = 60 ◦ (top left panel) and for one on the equatorial plane (top rightpanel).In the case of the Bardeen and Hayward BHs, the solutions are more complicated. Yet,we can solve Eq. (3.6) with m = m B and m = m H , and obtain the formula of the criticallocus ( ξ c , η c ) for these metrics. In Fig. 2, we show some examples of boundary of shadow ofBardeen BHs (central panels, with deformation parameter g/M = 0 . g/M = 1 .
0) for a ∗ = 0 .
0, 0.2, and 0.3. Sucha low values of the spin parameter with respect to the Kerr case is motivated by the factthat the maximum value of a ∗ is lower than 1 for g/M (cid:54) = 0 and reduces to 0 for g/M ≈ . In the observation of the shadow of a BH, it is helpful to introduce a parameter that approx-imately characterizes its shape [61]. At first approximation, the shape of the shadow of aBH is a circle, so we approximate the shadow by a circle passing through three points, whichare located at the top position, the bottom position, and the most left end of its boundary(the three red points in the left panel of Fig. 3). The radius R s of the shadow is herebydefined by the radius of this circle. On the other hand, when a BH rotates, the differenceof the photon capture radius between co-rotating and counter-rotating particles introduces adent on one side of the shadow. Unlike in the electromagnetic case, in gravity the spin-orbitinteraction term is repulsive when the orbital angular momentum of the photon is parallelto the BH spin (the capture radius thus decreases), and attractive in the opposite case (thecapture radius increases). The dent is more pronounced for fast-rotating objects and it isvery clear for the case of an extremal Kerr BH and a large viewing angle i (the blue-dotted– 7 – s D cs ! ! x ! M y ! M i ! i ! i ! ∆ s a ! M Kerr Black Holes
Figure 3 . Left panel: BH’s shadow with the two parameters that approximately characterized itsshape: the radius R s (defined as the radius of the circle passing through the three red points locatedat the top, bottom, and most left end of the shadow) and the dent D cs (the difference between theright endpoints of the circle and of the shadow). Right panel: Spin parameter a ∗ = a/M as a functionof the distortion parameter δ s = D cs /R s for Kerr BHs and a viewing angle i = 90 ◦ (red dashed curve),60 ◦ (blue dotted curve), and 30 ◦ (black solid curve). curves in the left panels of Fig. 2). The size of the dent is evaluated by D cs , which is thedifference between the right endpoints of the circle and of the shadow (see the left panel ofFig. 3). Thus the distortion parameter δ s of the shadow is defined by δ s = D cs /R s , whichcan be adopted as an observable in astronomical observations [61].In the case of Kerr backgrounds, the exact shape of the shadow depends only on theBH spin parameter, a ∗ , and the line of sight of the distant observer with respect to the BHspin, i . For a given inclination angle i , there is a one-to-one correspondence between a ∗ andthe distortion parameter δ s . If we have an independent estimate of the viewing angle and wemeasure the distortion parameter of the shadow of a Kerr BH, we can infer its spin parameter a ∗ [61]. The right panel of Fig. 3 shows the curves describing the spin parameter a ∗ = a/M as a function of the distortion parameter δ s for Kerr BHs and an inclination angle i = 90 ◦ (red dashed curve), 60 ◦ (blue dotted curve), and 30 ◦ (black solid curve).The same idea can be applied to non-Kerr BHs. If we consider the Bardeen and HaywardBHs with a specific value of the deformation parameter g/M , for a given inclination angle i there is a one-to-one correspondence between the spin a ∗ and the distortion parameter δ s .The counterpart of the right panel in Fig. 3 for the Bardeen and Hayward BHs are shown,respectively, in Fig. 4 and Fig. 5 for several values of g/M . The relation between a ∗ and δ s depends on g/M and it reduces to the Kerr one for g = 0.As in Ref. [31], we can now address the question of what happens if we measure theKerr spin parameter of a non-Kerr BH; that is, we estimate the spin parameter a ∗ from themeasurement of the distortion parameter δ s of the shadow of a Bardeen or Hayward BHassuming it is of Kerr type. In the Kerr metric, there is a one-to-one correspondence betweenthe spin parameter and δ s and therefore, from the measurement of the latter, we can infer– 8 – ! i ! i ! ∆ s a ! M Bardeen Black Holes: g ! i ! i ! i ! ∆ s a ! M Bardeen Black Holes: g ! i ! i ! i ! ∆ s a ! M Bardeen Black Holes: g ! i ! i ! i ! ∆ s a ! M Bardeen Black Holes: g ! Figure 4 . As in the right panel of Fig. 3 for the case of Bardeen BHs with g/M = 0 . the Kerr spin from a Kerr ∗ = a Kerr ∗ ( δ s ) . (4.1)However, in the Bardeen (and Hayward) background the distortion parameter is given by δ s ( a B ∗ , g/M ) and therefore the Kerr spin parameter of a Bardeen BH with spin a B ∗ and charge g/M is a Kerr ∗ = a Kerr ∗ [ δ s ( a B ∗ , g/M )] . (4.2)The result of a similar measurement is reported in Figs. 6 and 7 for the case of Bardeen BHs,and in Fig. 8 for Hayward BHs. The result is independent of the inclination angle i . For– 9 – ! i ! i ! ∆ s a ! M Hayward Black Holes: g ! i ! i ! i ! ∆ s a ! M Hayward Black Holes: g ! i ! i ! i ! ∆ s a ! M Hayward Black Holes: g ! i ! i ! i ! ∆ s a ! M Hayward Black Holes: g ! Figure 5 . As in the right panel of Fig. 3 for the case of Hayward BHs with g/M = 0 .
25 (top leftpanel), 0.50 (top right panel), 0.75 (bottom left panel), and 1.00 (bottom right panel). non-rotating objects, this approach trivially provides the correct spin: for any sphericallysymmetric BH, the shadow is always a circle, δ s = 0, independently of possible deviationsfrom the Kerr background. For slow-rotating BHs, we find a discrepancy between the actualspin parameter of the compact object and the one inferred assuming a Kerr metric. Such adifference increases for higher values of a ∗ and it can be quite significant for large values of g/M . The distortion parameter δ s is just a number and therefore cannot determine both the exactspin a ∗ and the deformation g/M . However, it is remarkable that the shadow of a regular– 10 – ! i ! i ! a " ! Bardeen " a " ! K e rr " g M ! i ! i ! i ! a " ! Bardeen " a " ! K e rr " g M ! i ! i ! i ! a " ! Bardeen " a " ! K e rr " g M ! i ! i ! i ! a " ! Bardeen " a " ! K e rr " g M ! Figure 6 . Spin parameter of a Bardeen BH, a ∗ (Bardeen), against the spin parameter that onewould infer for this object assuming the Kerr background, a ∗ (Kerr), through the determination ofthe distortion parameter of the shadow δ s (red dashed curve, dotted green curve, and yellow solidcurve respectively for a viewing angle i = 90 ◦ , 60 ◦ , and 30 ◦ ), the analysis of the K α iron line (bluesolid curve), and the frequency of a test-particle at the ISCO radius (hot spot model, black solidcurve). g/M = 0 . BH with a large g/M cannot mimic the one of a fast-rotating Kerr BH. In other words, ifwe observe a shadow that looks like the one of a Kerr BH with high spin, we can constrainthe deformation parameter g/M . In general, this is not possible, and we should combine thismeasurement with an independent one, in order to break the degeneracy between a ∗ and g/M .In Ref. [31], one of us has shown that the simultaneous measurement of the Kerr spin via– 11 – ! i ! i ! a " ! Bardeen " a " ! K e rr " g M ! i ! i ! i ! a " ! Bardeen " a " ! K e rr " g M ! Figure 7 . As in Fig. 6 for ( g/M ) = − (0 . (left panel) and − (0 . (right panel). the continuum-fitting and the iron line methods cannot fix this problem for Bardeen BHs. Thetwo techniques provide the same information on the geometry of the spacetime around thecompact object. We can now check if the combination of the Kerr spin parameter measuredwith the shadow approach can be combined with another estimate and if it is possible todistinguish a true Kerr BH from a Bardeen or Hayward one. Near future VLBI facilitieswill be able to image only the shadow of super-massive BH candidates. The analysis of theK α iron line is currently the only technique that can provide a relatively robust estimateof the Kerr spin parameter of these objects (the continuum-fitting method can be usedonly for stellar-mass BH candidates). It is supposed that VLBI experiments will be able toobserve also hot blobs of plasma orbiting around nearby super-massive BH candidates andinfer their angular frequency at the ISCO radius. Such a time measurement can providean independent measurement of the Kerr spin parameter, as in the Kerr metric there is aone-to-one correspondence between BH spin and angular frequency of the ISCO.The profile of the K α iron line depends on the background metric, the geometry ofthe emitting region, the disk emissivity, and the disk’s inclination angle with respect to theline of sight of the distant observer. In the Kerr spacetime, the only relevant parameter ofthe background geometry is the spin parameter a ∗ , while M sets the length of the system,without affecting the shape of the line. In those sources for which there is indication that theline is mainly emitted close to the compact object, the emission region may be thought torange from the radius of the ISCO, r in = r ISCO , to some outer radius r out . In principle, thedisk emissivity could be theoretically calculated. In practice, that is not feasible at present.The simplest choice is an intensity profile I e ∝ r α with index α < i . The dependence of the line profile on a ∗ , i , α , and r out in the Kerr background has been analyzed in detail by many authors, starting with Ref. [20].In the case of non-Kerr backgrounds, see e.g. Ref. [27].The profile of the K α iron line can be obtained by computing the photon flux number– 12 – ! i ! i ! a " ! Hayward " a " ! K e rr " g M ! i ! i ! i ! a " ! Hayward " a " ! K e rr " g M ! i ! i ! i ! a " ! Hayward " a " ! K e rr " g M ! i ! i ! i ! a " ! Hayward " a " ! K e rr " g M ! Figure 8 . As in Fig. 6 for Hayward BHs with g/M = 0 .
25 (top left panel), 0.50 (top right panel),0.75 (bottom left panel), and 1.00 (bottom right panel). density measured by a distant observer, that is N E obs = 1 E obs (cid:90) I obs ( E obs ) d Ω obs = 1 E obs (cid:90) w I e ( E e ) d Ω obs , (5.1)where I obs and E obs are, respectively, the specific intensity of the radiation and the photonenergy as measured by the distant observer, while I e and E e are the same quantities in therest-frame of the emitter. d Ω obs is the solid angle seen by the distant observer and w is theredshift factor w = E obs E e = k α u α obs k β u β e . (5.2)– 13 –ere k α is the 4-momentum of the photon, u α obs = ( − , , ,
0) is the 4-velocity of the distantobserver, and u α e = ( u t e , , , Ω u t e ) is the 4-velocity of the emitter. Ω is the angular velocity forequatorial circular orbits. I e ( E e ) /E = I obs ( E obs ) /E follows from the Liouville’s theorem.Using the normalization condition g µν u µ e u ν e = −
1, one finds u t e = − (cid:112) − g tt − g tφ Ω − g φφ Ω . (5.3)The redshift factor is thus given by w = (cid:112) − g tt − g tφ Ω − g φφ Ω λ Ω , (5.4)where λ = k φ /k t is a constant of the motion along the photon path. Doppler boosting,gravitational redshift, and frame dragging are entirely encoded in the redshift factor w . Asthe K α iron line is intrinsically narrow in frequency, we can assume that the disk emissionis monochromatic (the rest frame energy is E K α = 6 . I e ( E e ) ∝ δ ( E e − E K α ) r α . (5.5)Let us now consider the possibility that an astrophysical BH candidate is a BardeenBH and that we want to measure the spin parameter of this object with the K α iron lineanalysis, assuming that the object is a Kerr BH. In this case, we can use an approach similarto the one discussed in Ref. [27] and define the reduced χ : χ ( a Kerr ∗ , i ) = 1 n n (cid:88) i =1 (cid:2) N Kerr i ( a Kerr ∗ , i ) − N B i ( a B ∗ , g/M, i B ) (cid:3) σ i , (5.6)where the summation is performed over n sampling energies E i and N Kerr i and N B i are thenormalized photon fluxes in the energy bin [ E i , E i + ∆ E ], respectively for the Kerr and theBardeen metric. The error σ i is assumed to be 15% the photon flux N B i , which is roughlythe accuracy of current observations in the best situations. In this paper, all the calculationsare done with an intensity profile index α = − r out = r in + 100 M . Forspecific values of a B ∗ and g/M , we can find the minimum of the reduced χ , and we thusobtain what we call the Kerr spin parameter.In the case of the frequency of a hot spot at the ISCO radius, the idea is the same. Inthe Kerr background, there is a one-to-one correspondence between Ω ISCO and a ∗ , so we canwrite Ω KerrISCO ( a Kerr ∗ ) and the inverse function a Kerr ∗ (Ω KerrISCO ). In the Bardeen (and Hayward)background the frequency at the ISCO radius depends on both the spin a B ∗ and the deforma-tion parameter g/M , so Ω BISCO ( a B ∗ , g/M ). If we measure the frequency of a hot spot at theISCO radius and we assume that the object is a Kerr BH, while it is of Bardeen type, onefinds: a Kerr ∗ = a Kerr ∗ [Ω BISCO ( a B ∗ , g/M )] , (5.7)which is the Kerr spin parameter of the Bardeen BH.Figs. 6-8 show also the possible measurements of the Kerr spin parameter with these twotechniques (blue solid line for the iron line, black solid line for the hot spot). First, it seems– 14 –ike the iron line and hot spot approaches provide essentially the same result. This is becausethe two techniques are essentially sensitive to the properties at the ISCO radius. Second,when we compare the results of these approaches with the measurements inferred from theshadow, we see that the measurements disagree in the case of non-rotating BHs, while thediscrepancy decreases as the spin parameter increases and there is almost no difference fornear extremal states. So, the determination of the distortion parameter of the shadow ofa BH candidate can potentially test the nature of the compact object when combined withthe iron line analysis or the hot spot approach for non-rotating or slow-rotating objects.The degeneracy between a ∗ and g/M cannot be solved in the case of near extremal BHs,which confirms the difficulties, found in Ref. [31], to test this kind of non-Kerr metrics.Lastly, let us notice that here we have always considering an “ideal” observation, neglectingpossible uncertainty in the measurements. We thus adopt an optimistic point of view, andthe difficulties to measure deviations from the Kerr solution are even more challenging. Astrophysical BH candidates are thought to be the Kerr BHs of general relativity, but theactual nature of these objects is still to be verified. The analysis of the thermal spectrumof thin accretion disks and of the profile of the K α iron line are today the only availableapproaches to probe the spacetime geometry around BH candidates and test the Kerr BHhypothesis. However, there is a strong correlation between the spin parameter and possibledeviations from the Kerr solution and it is not possible to check the nature of a specificsource with a single measurement. As shown in Ref. [31], at least for some non-Kerr metrics,the disk’s thermal spectrum and the iron line provide essentially the same information. So,the two measurements may be consistent with the ones of a Kerr BH with a certain value ofthe spin parameter a ∗ even if the object is actually something else with a different spin. Thecombination of the two measurements does not break the degeneracy.In this paper, we have investigated the possibility of measuring the Kerr spin parameterfrom the BH shadow. Near future mm/sub-mm VLBI facilities will be able to observe theemitting region around nearby super-massive BH candidates with a resolution comparable totheir gravitational radius. If the gas is geometrically thick and optically thin, we will observea dark area over a brighter background. While the intensity map of this image dependson the kind of accretion process and the emission mechanisms, the boundary of the shadowis completely determined by the geometry of the spacetime. The shape of the shadow isexpected to be a circle for non-rotating BHs or a viewing angle i = 0 ◦ or 180 ◦ , and slightlydeformed otherwise, as a consequence of the coupling between the spin of the compact objectand the photon angular momentum. Such a deformation can be measured in terms of thedistortion parameter δ s , which, in turn, may provide an estimate of the spin parameter a ∗ .If the compact object is not a Kerr BH, but we assume it is, this technique still providesthe correct value of a ∗ for non-rotating objects, but a wrong measurement for near extremalstates. We have compared these measurements with the ones of the spin that one couldinfer from the iron line and from the determination of the frequency at the ISCO radius (akind of measurement that should become possible and reliable with VLBI facilities). Fornon-rotating BHs, the shadow approach would provide a different result, so the possibleinconsistency between two measurements may be an indication of deviations from the Kerrsolution. All the approaches converge instead to the same value for near extremal BHs.This work thus confirm the intrinsic difficulty to test the Kerr-nature of astrophysical BH– 15 –andidates, even with future facilities. It may however be possible that good measurementsof the shadow, in which one can extract more than one parameter characterizing the shape,are able to break the degeneracy between the spin and possible deviations from the Kerrgeometry for any value of a ∗ . Acknowledgments
This work was supported by the NSFC grant No. 11305038, the Shanghai Municipal Educa-tion Commission grant for Innovative Programs No. 14ZZ001, the Thousand Young TalentsProgram, and Fudan University.
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