Black Hole Parameter Estimation from Its Shadow
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Black Hole Parameter Estimation from Its Shadow
Rahul Kumar and Sushant G. Ghosh Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal,Private Bag 54001, Durban 4000, South Africa
Submitted to ApJABSTRACTThe Event Horizon Telescope (EHT), a global submillimeter wavelength very long baselineinterferometry array, unveiled event-horizon-scale images of the supermassive black hole M87* as anasymmetric bright emission ring with a diameter of 42 ± µ as, and it is consistent with the shadow of aKerr black hole of general relativity. A Kerr black hole is also a solution of some alternative theories ofgravity, while several modified theories of gravity admit non-Kerr black holes. While earlier estimatesfor the M87* black hole mass, depending on the method used, fall in the range ≈ × M ⊙ − × M ⊙ ,the EHT data indicated a mass for the M87* black hole of (6 . ± . × M ⊙ . This offers anotherpromising tool to estimate black hole parameters and to probe theories of gravity in its most extremeregion near the event horizon. The important question arises: Is it possible by a simple techniqueto estimate black hole parameters from its shadow, for arbitrary models? In this paper, we presentobservables, expressed in terms of ordinary integrals, characterizing a haphazard shadow shape toestimate the parameters associated with black holes, and then illustrate its relevance to four differentmodels: Kerr, Kerr − Newman, and two rotating regular models. Our method is robust, accurate,and consistent with the results obtained from existing formalism, and it is applicable to more generalshadow shapes that may not be circular due to noisy data.
Keywords:
Astrophysical black holes (98); Galactic center (565); Black hole physics (159); Gravitation(661); Gravitational lensing (670) INTRODUCTIONBlack holes are one of the most remarkablepredictions of Einstein’s theory of general relativity,which also provides a means to probe them viaunstable circular photon orbits (Bardeen 1973). Ablack hole, due to its defining property at the eventhorizon along with the surrounding photon region,casts a dark region over the observer’s celestial sky,which is known as a shadow (Bardeen 1973;Falcke et al. 2000). Astronomical observations suggestthat each galaxy hosts millions of stellar-mass blackholes, and also a supermassive black hole at thenucleus of the galaxy (Melia & Falcke 2001; Shen et al.2005). However, the majority of these black holes havevery low accretion luminosity and thus are very faint.Due to relatively very large size and close proximity,the black hole candidates at the center of the Milky
Corresponding author: Rahul [email protected]
Way and in the nearby galaxy Messier 87, respectively,Sgr A* and M87*, are prime candidates for black holeimaging (Broderick & Narayan 2006; Doeleman et al.2008, 2012). Probing the immediate environment ofblack holes will not only provide images of theseobjects and the dynamics of nearby matter but willalso enable the study of the strong gravity effects nearthe horizon. The Event Horizon Telescope (EHT) , aglobal array of millimeter and submillimeter radioobservatories, is using the technique of very longbaseline interferometry (VLBI) by combining severalsynchronized radio telescopes around the world. ThisEarth-sized virtual telescope has achieved an angularresolution of 20 µ as, sufficient to obtain thehorizon-scale image of supermassive black holes at agalaxy’s center. The EHT has published the first directimage of the M87* black hole (Akiyama et al.2019a,b,c,d). Further, fitting geometric models to theobservational data and extracting feature parameters http://eventhorizontelescope.org/ Kumar & Ghosh in the image domain indicates that we see emissionfrom near the event horizon that is gravitationallylensed into a crescent shape around the photon ring(Akiyama et al. 2019c,d).It turns out that photons may propagate along theunstable circular orbits due to the strong gravitationalfield of the black hole, and these orbits have a veryimportant influence on quasinormal modes(Cardoso et al. 2009; Hod 2009; Konoplya & Stuchlik2017), gravitational lensing (Stefanov et al. 2010), andthe black hole shadow (Bardeen 1973). Synge (1966)and Luminet (1979), in pioneering works, calculatedthe shadow cast by a Schwarzschild black hole, andthereafter Bardeen (1973) studied the shadows of Kerrblack holes over a bright background, which turn outto deviate from a perfect circle. The past decade sawmore attention given to analytical investigations,observational studies, and numerical simulations ofshadows (see Cunha & Herdeiro 2018). The shadows ofmodified theory black holes are smaller and moredistorted when compared with the Kerr black holeshadow (Bambi & Freese 2009; Johannsen & Psaltis2010; Falcke & Markoff 2013; Broderick et al. 2014;Younsi et al. 2016; Giddings & Psaltis 2018;Mizuno et al. 2018; Long et al. 2019;Konoplya & Zhidenko 2019; Held et al. 2019;Wang et al. 2019; Yan 2019; Kumar et al. 2019;Vagnozzi & Visinelli 2019; Breton et al. 2019).The no-hair theorem states that the Kerr black hole isthe unique stationary vacuum solution of Einstein’sfield equations, however, the exact nature ofastrophysical black holes has not been confirmed(Johannsen & Psaltis 2011; Bambi 2018), and thepossible existence of non-Kerr black holes cannot becompletely ruled out (Johannsen 2013a, 2016). Indeed,the Kerr metric remains a solution in some modifiedtheories of gravity (Psaltis et al. 2008). For rotatingblack holes, significant deviations from the Kerrsolution are found in modified theories(Bambi & Modesto 2013; Berti et al. 2015). TheBardeen perspective of a shadow of a black hole infront of a planar-emitting source was applied to severalblack hole models, e.g., Kerr-Newman black hole(Young 1976; De Vries 2000), Chern-Simons modifiedgravity black hole (Amarilla et al. 2010), Kaluza-Kleinrotating dilaton black hole (Amarilla & Eiroa 2013),rotating braneworld black hole (Amarilla & Eiroa2012; Eiroa & Sendra 2018), regular black holes(Abdujabbarov et al. 2016; Amir & Ghosh 2016;Kumar et al. 2019), and black holes in higherdimensions (Papnoi et al. 2014; Abdujabbarov et al.2015a; Amir et al. 2018; Singh & Ghosh 2018). Theblack hole shadow in asymptotically de-Sitterspacetime has also been analyzed (Grenzebach et al.2014; Perlick et al. 2018; Eiroa & Sendra 2018). Blackhole shadows have been investigated for aparameterized axisymmetric rotating black hole, which generalizes all stationary and axisymmetric black holesin any metric theory of gravity (Rezzolla & Zhidenko2014; Younsi et al. 2016; Konoplya et al. 2016).However, developing a methodological way to estimateparameters from astrophysical observations of a blackhole image is a promising avenue to advance ourunderstanding of black holes. The observationscommonly used for the estimation of the mass and sizeof a black hole are based on the motion of nearby starsand spectroscopy of the radiation emitted from thesurrounding matter in the Keplerian orbits, i.e., stellardynamical and gas dynamical methods(Gebhardt et al. 2000; Schodel et al. 2002; Shafee et al.2006). The dynamical mass measurements from X-raybinaries only provide lower limits of the black hole’smass (H¨aring & Rix 2004; Narayan 2005;Casares & Jonker 2014). Unlike for the mass, effects ofthe black hole’s spin and any possible deviation fromstandard Kerr geometry are manifest only at the smallradii. The two most commonly used model-dependenttechniques to estimate the spin are the analysis of theK α iron line (Fabian et al. 1989) and thecontinuum-fitting method (McClintock et al. 2014).Though black hole parameters have been inferred in anumber of contexts through the gravitational impacton the dynamics of surrounding matter (Matt & Perola1992; Narayan et al. 2008; Broderick et al. 2009;Steiner et al. 2009, 2011; McClintock et al. 2011;Narayan & McClintock 2012; Bambi 2013), the EHTobservations can put stringent bounds on theparameters. Furthermore, it is found that the non-Kerrblack hole shadows strongly depend on the deviationparameter apart from the spin (Atamurotov et al.2013; Johannsen 2013b; Wang et al. 2017, 2018). Thus,shadow observations of astrophysical black holes can beregarded as a potential tool to probe their departurefrom an exact Kerr nature, and in turn, to determinethe black hole parameters (Johannsen & Psaltis 2010).Hioki and Maeda (2009) discussed numericalestimations of Kerr black hole spin and inclinationangle from the shadow observables, which wasextended to an analytical estimation by Tsupko (2017).These observables, namely shadow radius anddistortion parameter, were extensively used in thecharacterization of black holes shadows (De Vries 2000;Amarilla et al. 2010; Amarilla & Eiroa 2012, 2013;Papnoi et al. 2014; Abdujabbarov et al. 2015a, 2016;Amir & Ghosh 2016; Amir et al. 2018; Singh & Ghosh2018; Eiroa & Sendra 2018). However, it was foundthat the distortion parameter is degenerate withrespect to the spin and possible deviations from theKerr solution; a method for discriminating the Kerrblack hole from other rotating black holes using theshadow analysis is presented by Tsukamoto et al.(2014). An analytic description of distortionparameters of the shadow has also been discussed in acoordinate-independent manner (Abdujabbarov et al. lack hole parameter estimation from its shadow G = 1, c = 1, unlessunits are specifically defined. BLACK HOLE SHADOWThe metric of a general rotating, stationary, andaxially symmetric black hole, in Boyer − Lindquistcoordinates, reads (Bambi & Modesto 2013) ds = − (cid:18) − m ( r ) r Σ (cid:19) dt − am ( r ) r Σ sin θdt dφ + Σ∆ dr +Σ dθ + (cid:20) r + a + 2 m ( r ) ra Σ sin θ (cid:21) sin θdφ , (1)and Σ = r + a cos θ ; ∆ = r + a − m ( r ) r, (2)where m ( r ) is the mass function such thatlim r →∞ m ( r ) = M and a is the spin parameter definedas a = J/M ; J and M are, respectively, the angularmomentum and ADM mass of a rotating black hole.Obviously metric (1) reverts back to the Kerr (1963)and Kerr − Newman (Newman et al. 1965) spacetimeswhen m ( r ) = M and m ( r ) = M − Q / r , respectively.Photons moving in a general rotating spacetime (1)exhibit two conserved quantities, energy E and angularmomentum L , associated with Killing vectors ∂ t and ∂ φ . To study the geodesics motion in spacetime (1), weadopt the Carter (1968) separability prescription of theHamilton − Jacobi equation. The complete set of equations of motion in the first-order differential formread (Carter 1968; Chandrasekhar 1985)Σ dtdτ = r + a r − m ( r ) r + a (cid:0) E ( r + a ) − a L (cid:1) − a ( a E sin θ − L ) , (3)Σ drdτ = ± p R ( r ) , (4)Σ dθdτ = ± p Θ( θ ) , (5)Σ dφdτ = ar − m ( r ) r + a (cid:0) E ( r + a ) − a L (cid:1) − (cid:18) a E − L sin θ (cid:19) , (6)with the expressions for R ( r ) and Θ( θ ), respectively, aregiven by R ( r ) = (cid:0) ( r + a ) E − a L (cid:1) − ( r − m ( r ) r + a )( K +( a E − L ) ) , (7)Θ( θ ) = K − (cid:18) L sin θ − a E (cid:19) cos θ. (8)The conserved quantity Q associated with the hiddensymmetry of the conformal Killing tensor is related tothe Carter integral of motion K through Q = K + ( a E − L ) (Carter 1968). One can minimizethe number of parameters by defining twodimensionless impact parameters η and ξ as follows(Chandrasekhar 1985) ξ = L / E , η = K / E . (9)Due to spacetime symmetries, geodesics along t and φ coordinates do not reveal nontrivial features of orbits,therefore the only concern will be mainly for Eqs. (4)and (5). Rewriting Eq. (5) in terms of µ = cos θ , weobtainΣ Z dµ p Θ µ = Z dτ ; Θ µ = η − ( ξ + η − a ) µ − a µ . (10)Obviously η ≥ θ motion, i.e.,Θ µ ≥ θ = 0. However, in the Kerrblack hole, the frame-dragging effects may lead tononplanar orbits as well. Indeed, planar and circularorbits around Kerr black hole are possible only in theequatorial plane ( θ = π/
2) that leads to a vanishingCarter constant ( K = 0). Furthermore, generic boundorbits at a plane other than θ = π/ θ = 0) and cross the equatorial plane while oscillatingsymmetrically about it. These orbits are identified by K > η >
0) and are commonly known as sphericalorbits (Chandrasekhar 1985), and θ motion freezes Kumar & Ghosh r p r p - r p + r + Μ max Η-Ξ - - - Μ Q Μ Figure 1.
Left panel: schematic of a photon region arounda rotating black hole. Right panel: variation of Θ µ with µ for η = 1 and ξ = 1 .
2. Horizontal dashed lines correspondto the maximum and minimum values of Θ µ . only in the equatorial plane. Equation (10) revealsthat the maximum latitude of a spherical orbit, θ max = cos − ( µ max ), depends upon the angularmomentum of photons, i.e., the smaller the angularmomentum of photons the larger the latitude of orbits; µ max correspond to the solution of Θ µ ( µ ) = 0. Onlyphotons with zero angular momentum ( ξ = 0) canreach the polar plane of the black hole ( θ = 0 , µ = 1)and cover the entire span of θ coordinates (see Figure1).Depending on the values of the impact parameters η and ξ , photon orbits can be classified into threecategories, namely scattering orbits, unstable circularand spherical orbits, and plunging orbits. Indeed, theunstable orbits separate the plunging and scatteringorbits, and their radii ( r p ) are given by(Chandrasekhar 1985) R| ( r = r p ) = ∂ R ∂r (cid:12)(cid:12)(cid:12)(cid:12) ( r = r p ) = 0 , ∂ R ∂r (cid:12)(cid:12)(cid:12)(cid:12) ( r = r p ) ≤ . (11)Solving Equation (11) yields the critical locus ( η c , ξ c ) associated with the unstable photon orbits, that fornonrotating black holes are at a fixed radius, e.g., r p = 3 M for a Schwarzschild black hole, and constructa spherical photon sphere. In the rotating black holespacetime, photons moving in unstable circular orbitsat the equatorial plane can either corotate with theblack hole or counterrotate, and their radii can beidentified as the real positive solutions of η c = 0 for r , r − p and r + p , respectively. Photon orbit radii are anexplicit function of black hole spin and lie in the range M ≤ r − p ≤ M and 3 M ≤ r + p ≤ M for the Kerr blackhole, and r − p ≤ r + p due to the Lens − Thirring effect.Whereas, spherical photon orbits (orbits at θ = π/ r − p , r + p ], i.e., for η c > r − p < r p < r + p . Although rotating black holesgenerically have two distinct photon regions, viz., inside the Cauchy horizon ( r − ) and outside the eventhorizon ( r + ), for a black hole shadow we will be onlyfocusing on the latter, i.e., for r p > r + (Grenzebach et al. 2014). The critical impactparameter ξ c is a monotonically decreasing function of r p with ξ c ( r − p ) > ξ c ( r + p ) <
0, such that at r p = r p ( r − p < r p < r + p ) ξ c is vanishing. Even though for orbitsat r p the angular momentum of photons is zero, theystill cross the equatorial plane with nonzero azimuthalvelocity ˙ φ = 0 (Wilkins 1972; Chandrasekhar 1985).A black hole in a luminous background of stars orglowing accreting matter leads to the appearance of adark spot on the celestial sky accounting for thephotons which are unable to reach the observer,popularly known as a black hole shadow. Photonsmoving on unstable orbits construct the edges of theshadow. A far distant observer perceives the shadow asa projection of a locus of points η c and ξ c on thecelestial sphere on to a two-dimensional plane. Let usintroduce the celestial coordinates (Bardeen 1973) α = lim r O →∞ (cid:18) − r O sin θ O dφdr (cid:19) , β = lim r O →∞ (cid:18) r O dθdr (cid:19) . (12)Here, we assume the observer is far away from theblack hole ( r O → ∞ ) and θ O is the angle between theline of sight and the spin axes of black hole, namely,the inclination angle. Since the black hole spacetime isasymptotically flat, we can consider a static observerat an arbitrarily large distance, and this yields α = − ξ c sin θ O , β = ± q η c + a cos θ O − ξ c cot θ O . (13)For an observer in the equatorial plane θ O = π/ α = − ξ c , β = ±√ η c . (14)Solving Eq. (11) for rotating metric (1) and usingEq. (14), the celestial coordinates of the black holeshadow boundary take the following form α = − [ a − r p ] m ( r p ) + r p [ a + r p ][1 + m ′ ( r p )] a [ m ( r p ) + r p [ − m ′ ( r p )]] ,β = ± a [ m ( r p ) + r p [ − m ′ ( r p )]] h r / p h − r p (1 + m ′ ( r p ) )+ m ( r p )[4 a + 6 r p − r p m ( r p )] − r p [2 a + r p − r p m ( r p )] m ′ ( r p ) i / i , (15)and whereas for m ( r ) = M , Eq. (15) yields α = r p ( r p − M ) + a ( M + r p ) a ( r p − M ) ,β = ± r / p (4 a M − r p ( r p − M ) ) / a ( r p − M ) , (16) lack hole parameter estimation from its shadow a = 0) can bedelineated by α + β = 2 r p + [ m ( r p ) + r p m ′ ( r p )][ − r p m ( r p ) + 2 r p m ′ ( r p )][ m ( r p ) + r p [ − m ′ ( r p )]] , (17)which implies that the shadow of a nonrotating blackhole is indeed a perfect circle, and further returns to α + β = 27 M for the Schwarzschild black hole m ( r ) = M . Though the shape of the shadow isdetermined by the properties of null geodesics, it isneither the Euclidean image of the black hole horizonnor that of its photon region, rather it is thegravitationally lensed image of the photon region. Forinstance, the horizon of Sgr A*, with M ≈ . × M ⊙ at a distance d ≈ .
35 kpc, spans an angular size of20 µ as, whereas its shadow has an expected angularsize of ≈ µ as. Whereas, EHT measured the angularsize of the M87* gravitational radius as 3 . ± . µ as,and its crescent-shaped emission region has an angulardiameter of 42 ± µ as, with a scaling factor in therange 10 . − . CHARACTERIZATION OF THE SHADOW VIANEW OBSERVABLESA nonrotating black hole casts a perfectly circularshadow. However, for a rotating black hole, anobserver placed at a position other than in the polardirections witnesses an off-center displacement of theshadow along the direction of black hole rotation.Furthermore, for sufficiently large values of the spinparameter, a distortion appears in the shadow becauseof the Lense-Thirring effect (Johannsen & Psaltis2010). Hioki and Maeda (2009) characterized thisdistortion and shadow size by the two observables δ s and R s , respectively. The shadow is approximated to acircle passing through three points located at the top,bottom, and right edges of the shadow, such that R s isthe radius of this circle and δ s is the deviation of theleft edge of the shadow from the circle boundary(Hioki & Maeda 2009). It was found that theapplicability of these observables was limited to aspecific class of shadows, demanding some symmetriesin their shapes, and they may not precisely work forblack holes in some modified theories of gravity(Abdujabbarov et al. 2015b), which leads to theintroduction of new observables (Schee & Stuchlik2009; Johannsen 2013b; Tsukamoto et al. 2014;Cunha et al. 2015; Abdujabbarov et al. 2015b;Younsi et al. 2016; Tsupko 2017; Wang et al. 2018).EHT observations can constrain the key physicalparameters of the black holes, including the black holemass and other parameters. However, EHTobservations do not give any estimation of angularmomentum (Akiyama et al. 2019a,d). Their measurement of the black hole mass in M87* isconsistent with the prior mass measurement usingstellar dynamics, but is inconsistent with the gasdynamics measurement (Gebhardt et al. 2011;Walsh et al. 2013; Akiyama et al. 2019d). Here, wewould propose new observables for the characterizationof the black hole shadow, which unlike previousobservables (Hioki & Maeda 2009), do not require theapparent shadow shape to be approximated as a circle.We consider a shadow of general shape and size topropose new observables, namely the area ( A ) enclosedby a black hole shadow, the circumference of theshadow ( C ), and the oblateness ( D ) of the shadow.The observables A and C , respectively, are defined by A = 2 Z β ( r p ) dα ( r p ) = 2 Z r + p r − p (cid:18) β ( r p ) dα ( r p ) dr p (cid:19) dr p , (18)and C = 2 Z q ( dβ ( r p ) + dα ( r p ) )= 2 Z r + p r − p vuut (cid:18) dβ ( r p ) dr p (cid:19) + (cid:18) dα ( r p ) dr p (cid:19) ! dr p . (19)The prefactor 2 is due to the black hole shadow’ssymmetry along the α − axis. A and C have dimensionsof [ M ] and [ M ], respectively. A shadow silhouette canbe taken as a parametric curve between celestialcoordinates as a function of r p for r − p ≤ r p ≤ r + p , i.e., aplot of β ( r p ) versus α ( r p ). We can also characterize theshadow of rotating black hole through its oblateness(Takahashi 2004; Grenzebach et al. 2015; Tsupko2017), the measure of distortion (circular asymmetry)in a shadow, by defining the dimensionless parameter D as the ratio of horizontal and vertical diameters: D = α r − α l β t − β b . (20)The subscripts r, l, t , and b stand for right, left, top,and bottom, respectively, of the shadow boundary. Fora spherically symmetric black hole shadow, D = 1,while for a Kerr shadow √ / ≤ D < D = 1 indicates that the shadow has distortionand hence corresponds to a rotating black hole. Inparticular, the quasi-Kerr black hole metric may leadto a shadow with D >
D <
1, depending on thesign of the quadrupole deviation parameters(Johannsen & Psaltis 2010). The definitions of theseobservables require neither any nontrivial symmetry inshadow shape nor any primary curve to approximatethe shadow. It can be expected that an observertargeting the black hole shadow through astronomicalobservations can measure the area, the length of theshadow boundary, and also the horizontal and verticaldiameters. In what follows, we show that these
Kumar & Ghosh
Table 1.
Table representing the values of observables, solidangle, and angular diameter with varying spin parameter forthe Sgr A* black hole shadow. a/M A C D Ω ϑ m (10 m ) (10 m ) (10 − µ as ) ( µ as)0.0 34.079 20.6942 1 2.1818 52.73440.10 34.06 20.6884 0.999443 2.18059 52.7050.20 34.0025 20.671 0.997748 2.1769 52.61560.30 33.9046 20.6413 0.994847 2.17064 52.46260.40 33.7629 20.5984 0.990607 2.16157 52.2390.50 33.572 20.5406 0.984808 2.14934 51.93320.60 33.3227 20.4655 0.977083 2.13338 51.52590.70 32.9998 20.3688 0.966783 2.11271 50.98270.80 32.5742 20.2427 0.952608 2.08546 50.23520.90 31.9754 20.0699 0.931145 2.04713 49.10330.998 30.7793 19.776 0.876375 1.97055 46.2151 observables uniquely characterize the shadow and it ispossible to estimate the black hole parameters fromthese observables.The EHT observations indicated that the M87*black hole shadow is consistent with that of Kerr blackhole, however, the exact nature of the Sgr A* blackhole is still elusive. Astronomical observations haveplace constraints on their masses and distances fromEarth as M = 4 . × M ⊙ and d = 8 .
35 kpc for SgrA* (Ghez et al. 2008; Gillessen et al. 2009;Falcke & Markoff 2013; Reid et al. 2014), and M = (6 . ± . × M ⊙ and d = (16 . ± . ϑ M ) and horizontal (or minor ϑ m ) angulardiameters are not the same and can be defined as ϑ M = β t − β b d , ϑ m = α r − α l d , (21)and the solid angle is Ω = A/d . Clearly, ϑ M is notdependent on black hole spin.Obviously, for a = 0, ϑ M = ϑ m = 52 . µ as for SgrA* and ϑ M = ϑ m = 39 . µ as for M87*. The shadowobservables and angular diameters of the Sgr A* andM87* black hole shadows are calculated for variousvalues of spin parameter a (see Table 1 and Table 2).Nevertheless, the shadow observables for aSchwarzschild black hole take the values A/M = 84 . C/M = 32 . D = 1, whereasfor a maximally rotating Kerr black hole A/M = 76 . C/M = 31 . D = 0 . Table 2.
Table representing the values of observables, solidangle, and angular diameter with varying spin parameter forthe M87* black hole shadow. a/M A C D Ω ϑ m (10 m ) (10 m ) (10 − µ as ) ( µ as)0.0 7.78711 3.12819 1 1.23151 39.61920.10 7.78278 3.12732 0.999443 1.23083 39.59710.20 7.76963 3.12468 0.997748 1.22875 39.530.30 7.74726 3.12019 0.994847 1.22521 39.41510.40 7.7149 3.1137 0.990607 1.22009 39.24710.50 7.67126 3.10498 0.984808 1.21319 39.01730.60 7.6143 3.09362 0.977083 1.20418 38.71130.70 7.54052 3.079 0.966783 1.19252 38.30320.80 7.44327 3.05994 0.952608 1.17714 37.74160.90 7.30644 3.03382 0.931145 1.1555 36.89120.998 6.99973 2.65529 0.866025 1.10699 34.31124. APPLICATION TO VARIOUS BLACK HOLESPACETIMESWe examine several rotating black holes such asKerr − Newman, Bardeen, and nonsingular black holes.In general these black holes are given by metric (1)with an appropriate choice of mass function m ( r ). Weassume that the observer is in the equatorial plane, i.e.,the inclination angle θ O = π/ A or C alongwith D to estimate the black hole parameters. For thesake of brevity, we shall use only A and D for ourpurpose, but shall calculate all three.4.1. Kerr − Newman black hole
We start with a Kerr − Newman black hole, whichencompasses Kerr, Reissner − Nordstrom, andSchwarzschild black holes as special cases. One cananalyze null geodesics to the shadow of aKerr − Newman black hole (Young 1976; De Vries2000). In the case of the Kerr − Newman black hole, themass function m ( r ) has a form m ( r ) = M − Q r . (22)In Figure 2, we have shown the allowed range ofparameters a and Q for the existence of a black holehorizon. The Kerr − Newman black hole shadows aredistorted from a perfect circle and possess a dent onthe left side of shadow (De Vries 2000). This distortionreduces as the observer moves from the equatorialplane to the axis of black hole symmetry, andeventually disappears completely for θ O = 0 , π lack hole parameter estimation from its shadow No Black HoleBlack Hole a (cid:144) M Q (cid:144) M Figure 2.
Allowed parametric space ( a, Q ) for the existenceof a Kerr − Newman black hole. The solid line correspondsfor the extremal black hole with degenerate horizons anddemarcates the black hole case from the no black hole case. (De Vries 2000). It is straightforward to calculate thecelestial coordinates α and β using the m ( r ) inEq. (15). Though for these α and β the observables A , C , and D could not be obtained in exact analytic form,we have calculated them approximately in theAppendix A.In Figures 3 and 4, respectively, charge Q and spinparameter a are plotted with varying observables A , C ,and D . Interestingly, estimated values of black holeparameters decrease with independently increasingobservables. For a far extremal black hole, parametersdecrease rapidly with observables, whereas for a nearextremal black hole, parameters decrease relativelyslowly with increasing D . Therefore, one can concludethat the size of the shadow decreases with an increasein the electric charge, which is consistent with theearlier results (De Vries 2000). On the other hand,Figure 3 suggests that shadows of Kerr − Newman blackholes get more distorted as the charge increase.Shadow observables for Kerr − Newman black holes arenumerically compared with those for Kerr black holesin Figure 4, and it can be inferred that observables forKerr − Newman black holes are smaller than those forKerr black holes for fixed values of a .The apparent shape and size of the Kerr − Newmanblack hole shadow depend on the a and Q (De Vries2000). Next, we see the possibility of estimation of a and Q for the Kerr − Newman black hole by using thetwo observables A and D , expecting that mass M canbe fixed through other astrophysical observations. Weplot the contour map of the observables A and D inthe ( a, Q ) plane (see Figure 5). Each point of thecontour plot in Figure 5 has coordinates ( a, Q ) thatcan be described as a unique intersection of the lines ofconstant A and D . Hence, from Figure 5, it is clear that intersection points give an exact estimation ofparameters a and Q when one has the values of A and D for a Kerr − Newman black hole. In Table 3, we havepresented the estimated values of a and Q for givenshadow observables A and D for the Kerr − Newmanblack hole. 4.1.1.
Kerr Black Hole
When the electric charge is switched off ( Q = 0), theKerr − Newman spacetime becomes Kerr with m ( r ) = M . We plot the spin parameter a (0 ≤ a ≤ A , C , and D in Figure 4. It isevident that with increasing observables A , C , and D the estimated Kerr spin parameter decreases. Figure 4indicates that the black hole shadow gets smaller andmore distorted for a rapidly rotating black hole, asshown in earlier studies as well (Bardeen 1973).Kerr black holes have only two parameters associatedwith them, namely, mass M and spin a , however,presuming the knowledge of only mass through thestellar motion around the black hole, one has only oneunknown parameter i.e., spin. The spin parameter forthe Kerr black hole can be uniquely determined byknowing any one of the shadow observables definedabove (see Figure 4).4.2. Rotating Bardeen Black Hole
The first regular black hole was proposed by Bardeen(1968) with horizons and no curvature singularity − amodification of the Reissner − Nordstrom black hole.The rotating Bardeen black hole (Bambi & Modesto2013) belongs to the prototype non-Kerr family withthe mass function m ( r ) given by m ( r ) = M (cid:18) r r + g (cid:19) / . (23)The Bardeen black hole is an exact solution of theEinstein field equations coupled with nonlinearelectrodynamics associated with the magneticmonopole charge g (Ayon-Beato & Garcia 1999). TheKerr black hole can be recovered in the absence of thenonlinear electrodynamics ( g = 0). For the existence ofa black hole, the allowed values of a and g areconstrained and shown in Figure 6, and the extremalvalues of parameters correspond to those lying on theboundary line. The shadows of rotating Bardeen blackholes get more distorted and their sizes decrease due tothe magnetic charge g (Abdujabbarov et al. 2016).The Bardeen black hole parameters g and a versusthe observables A , C , and D are depicted in Figures 7and 8, respectively. Within the allowed parameterspace, they have a similar behavior to that of theKerr − Newman black hole. The parameters decreasewith increasing observables, however, for a near
Kumar & Ghosh
50 55 60 65 70 75 80 850.00.20.40.60.81.0 A (cid:144) M Q (cid:144) M
26 28 30 320.00.20.40.60.81.0 C (cid:144) M Q (cid:144) M D Q (cid:144) M Figure 3.
Charge parameter Q vs. observables A , C , and D for the Kerr − Newman black hole, for a/M = 0 (solid black curve),for a/M = 0 . a/M = 0 . a/M = 0 .
60 65 70 75 80 850.00.20.40.60.81.0 A (cid:144) M a (cid:144) M
27 28 29 30 31 32 330.00.20.40.60.81.0 C (cid:144) M a (cid:144) M D a (cid:144) M Figure 4.
Spin parameter a vs. observables A , C , and D for the Kerr − Newman black hole, for Kerr black hole
Q/M = 0 . Q/M = 0 . Q/M = 0 . Q/M = 0 . Table 3.
Estimated values of parameters for different black hole models from known shadow observables A and D .Shadow Observable Black Hole ParametersKerr − Newman Bardeen Nonsingular
A/M D a/M Q/M a/M g/M a/M k/M lack hole parameter estimation from its shadow a (cid:144) M Q (cid:144) M Figure 5.
Contour plot of the observables A and D in theplane ( a, Q ) for a Kerr − Newman black hole. Each curve islabeled with the corresponding values of A and D . The solidred lines correspond to the area observable A , and the dashedblue lines correspond to the oblateness observable D . No Black HoleBlack Hole a (cid:144) M g (cid:144) M Figure 6.
The allowed parametric space of a and g forthe existence of a rotating Bardeen black hole. The solidline corresponds to the extremal black hole with degeneratehorizons. extremal black hole, parameters decreasecomparatively slowly with increasing D . Further, theobservables of a rotating Bardeen black hole aresmaller when compared with the Kerr black hole for agiven a , i.e., A ( g = 0) < A ( g = 0) and D ( g = 0) < D ( g = 0) (see Figure 8). An interestingcomparison between shadows of Bardeen and Kerrblack holes shows that for some values of parameters, a Bardeen black hole ( M = 1 , a/M = 0 . , g/M = 0 . M = 0 . , a/M = 0 . A = 69 . , C = 29 . D = 0 . A = 68 . C = 29 . D = 0 . A and C for the two black holes differ by 0 . . . . A and D for the rotating Bardeen blackhole as a function of ( a, g ). In Table 3, we have shownthe estimated values of Bardeen parameters a and g forgiven shadow observables A and D , and compare themwith the estimated values of other black holeparameters. Thus, from Figure 9 and Table 3 it is clearthat if A and D are known from the observations, thisuniquely determines the a and g .4.3. Rotating Nonsingular Black Hole
The Bardeen regular black holes have a de-Sitterregion at the core. Next, we consider a class of rotatingregular black holes with asymptotically Minkowskicores (Simpson & Visser 2020), which have anadditional parameter k = q / M > r >> k ) goes over to a Kerr − Newmanblack hole (Ghosh 2015). Whilst this rotating regularblack hole shares many properties with Bardeenrotating regular black holes, there is also a significantcontrast, and for definiteness, we name it a rotatingnonsingular black hole. It also belongs to the non-Kerrfamily with mass function m ( r ) = M e − k/r . (24)Figure 10 shows the allowed values of parameters a and k for the black hole’s existence. The effect ofvarying observables A , C , and D on the inferredrotating nonsingular black hole parameters k and a aredepicted in Figures 11 and 12, respectively. Thecharacteristic behavior is again similar to that for theKerr − Newman, but the effect on k is visible for bothnonrotating and rotating nonsingular black holes (seeFigures 11 and 12). The estimated values of k show asimilar sharp decreasing behavior with increasing A and C , whereas it slowly decreases with D for nearextremal black holes. The observables for rotatingnonsingular black holes are examined in contrast withthose for Kerr black holes in Figure 12, and for a fixedvalue of a they turn out to be smaller. This indicatesthat shadows of rotating nonsingular black holes aresmaller and more distorted than those of Kerr blackholes (Amir & Ghosh 2016). Contour maps of A and D as a function of ( a, k ) are shown in Figure 13. We0 Kumar & Ghosh
65 70 75 80 850.00.20.40.60.8 A (cid:144) M g (cid:144) M
28 29 30 31 32 330.00.20.40.60.8 C (cid:144) M g (cid:144) M D g (cid:144) M Figure 7.
The magnetic charge parameter g vs. observables A , C , and D for the Bardeen black hole, for a nonrotating Bardeenblack hole a/M = 0 . a/M = 0 . a/M = 0 . a/M = 0 .
65 70 75 80 850.00.20.40.60.81.0 A (cid:144) M a (cid:144) M
28 29 30 31 32 330.00.20.40.60.81.0 C (cid:144) M a (cid:144) M D a (cid:144) M Figure 8.
The spin parameter a vs. observables A , C , and D for the Bardeen black hole, for g/M = 0 . g/M = 0 . g/M = 0 . g/M = 0 . a (cid:144) M g (cid:144) M Figure 9.
Contour plot of the observables A and D in theplane ( a, g ) for a Bardeen black hole. Each curve is labeledwith the corresponding values of A (solid red curve) and D (dashed blue curve). can easily determine the specific points where curves ofconstant A and D intersect each other in the black holeparameter space, yielding the unique values of a and k . No Black HoleBlack Hole a (cid:144) M k (cid:144) M Figure 10.
The allowed parametric space of a and k forthe existence of rotating nonsingular black hole. The solidline corresponds to the extremal black hole with degeneratehorizons. Comparison of Estimated Black Hole Parameters
Applying the method described in section 3, thenumerical values of the three considered rotating blackholes parameters, for a given shadow area A andoblateness D , are summarized in the Table 3. Here, wecompare the estimated black hole parameters for thethree black holes. For a given shadow area A , we findthat the spin parameter decreases with increasing lack hole parameter estimation from its shadow
40 50 60 70 800.00.10.20.30.40.50.60.7 A (cid:144) M k (cid:144) M
22 24 26 28 30 320.00.10.20.30.40.50.60.7 C (cid:144) M k (cid:144) M D k (cid:144) M Figure 11.
Charge parameter k vs. observables A , C , and D for the nonsingular black hole, for a/M = 0 . a/M = 0 . a/M = 0 . a/M = 0 .
50 60 70 800.00.20.40.60.81.0 A (cid:144) M a (cid:144) M
24 26 28 30 320.00.20.40.60.81.0 C (cid:144) M a (cid:144) M D a (cid:144) M Figure 12.
Spin parameter a vs. observables A , C , and D for the nonsingular black hole, for k/M = 0 . k/M = 0 . k/M = 0 . k/M = 0 . oblateness D , and that for a fixed area A andoblateness D we obtain that the spin parameters are a NS > a KN > a Bardeen and the charge parameters are
Q > g > k . For a fixed oblateness D , the chargeparameters Q , g , and k increase and the spinparameter a decreases with a decrease in the area A .For small area A and oblateness D , e.g., A = 55 M and D = 0 .
92, one could estimate parametersassociated with only the rotating nonsingular blackhole (see Table 3). CONCLUSIONThe EHT has obtained the first image of the M87*black hole, and thus its shadow becomes an importantprobe of spacetime structure, parameter estimation,and testing gravity in the extreme region near theevent horizon. Even though most of the available testsare consistent with general relativity, deviations fromthe Kerr black hole (or non-Kerr black hole) arisingfrom modified theories of gravity are not ruled out(Johannsen & Psaltis 2011; Berti et al. 2015). Thesenon-Kerr black holes, in Boyer − Lindquist coordinates,are defined by the metric (1) with mass function m ( r ),and Kerr black holes are included as special case when m ( r ) = M . In this paper, we have proposedobservables, namely, shadow area ( A ), itscircumference ( C ), and oblateness ( D ). Theobservables A and C characterize the size of theshadow, and D defines its shape asymmetry. Theseobservables are calculated for Sgr A* and M87*, a (cid:144) M k (cid:144) M Figure 13.
Contours of constant A and D as a function of( a, k ) for a rotating nonsingular black hole. Each curve islabeled with the corresponding value of A (solid red curve)and D (dashed blue curve). assuming their Kerr nature, and we find that theirangular diameters are approximately 52 µ as and 39 µ as,respectively, and decrease for a rapidly rotating black2 Kumar & Ghosh hole. This is consistent with other predicted results(Falcke & Markoff 2013; Fish et al. 2014;Brinkerink et al. 2019; Akiyama et al. 2019a,b,d).We highlight several other results that are obtained byour analysis.1. The method can estimate, at most, twoparameters by using either A or C along with D ;for example, the Kerr black hole parameters a and θ O can be estimated. In order to estimate asingle parameter, we require any one of theseobservables.2. For given shadow observables, we have estimatedparameters associated with Kerr − Newman ( a, Q ),rotating Bardeen ( a, g ), and rotating nonsingular( a, k ) black holes. Here, our analysis assumes thatthe observer is in the equatorial plane, i.e., at afixed inclination angle θ O = π/ ACKNOWLEDGMENTSS.G.G. would like to thank the DST INDO-SAbilateral project DST/INT/South Africa/P-06/2016and also IUCAA, Pune for the hospitality while thiswork was being done. R.K. would like to thank UGCfor providing SRF, and also Md Sabir Ali andBalendra Pratap Singh for fruitful discussions. Theauthors would like to thank the anonymous reviewerfor providing insightful comments which immenselyhelped to improve the paper.APPENDIX A. ANALYTIC FORM OF OBSERVABLESThe celestial coordinates α and β can be calculated via Equation (15) for a given mass function, and in turn, theyhelp us to calculate observables A , C , and D numerically. Here, we present an approximate and analytic form of A , C , and D obtained from the best fit of the numerical data for the three discussed rotating black holes. For aKerr − Newman black hole it yields lack hole parameter estimation from its shadow A ( a, Q ) M = 84 . − . aM − . a M − . a M + 31 . a M − . a M + 73 . a M − . a M − . QM − . aQM + 52 . a QM − . a QM + 415 . a QM − . a QM + 356 . a QM + 11 . Q M + 43 . aQ M − . a Q M + 1144 . a Q M − . a Q M + 988 . a Q M − . Q M − . aQ M + 786 . a Q M − . a Q M + 1916 . a Q M + 1064 . Q M + 116 . aQ M − . a Q M + 2203 . a Q M − . Q M − . aQ M + 253 . a Q M + 2735 . Q M − . Q M ,C ( a, Q ) M = 32 . − . aM − . a M − . a M + 4 . a M − . a M + 9 . a M − . a M − . QM − . aQM + 7 . a QM − . a QM + 55 . a QM − . a QM + 44 . a QM − . Q M + 7 . aQ M − . a Q M + 187 . a Q M − . a Q M + 137 . a Q M − . Q M − . aQ M + 141 . a Q M − . a Q M + 299 . a Q M + 108 . Q M + 24 . aQ M − . a Q M + 450 . a Q M − . Q M − . aQ M + 54 . a Q M + 250 . Q M − . Q M ,D ( a, Q ) = 1 . − . aM − . a M − . a M + 0 . a M − . a M + 1 . a M − . a M − . aQM + 1 . a QM − . a QM + 10 . a QM − . a QM + 8 . a QM + 1 . aQ M − . a Q M + 30 . a Q M − . a Q M + 24 . a Q M − . aQ M + 22 . a Q M − . a Q M + 49 . a Q M + 3 . aQ M − . a Q M + 62 . a Q M − . aQ M + 8 . a Q M . (A1)Clearly, A , C , and D are functions of spin a and charge Q . For the Bardeen black hole, they depend upon themagnetic charge g in addition to a , and are given by4 Kumar & Ghosh A ( a, g ) M = 84 . − . aM − . a M − . a M + 31 . a M − . a M + 73 . a M − . a M + 0 . gM + 10 . agM − . a gM + 310 . a gM − . a gM + 323 . a gM + 9 . a gM − . g M − . ag M + 1350 . a g M − . a g M + 7243 . a g M − . a g M + 77 . g M + 837 . ag M − . a g M + 21471 . a g M − . a g M − . g M − . ag M + 16337 . a g M − . a g M + 1519 . g M + 1913 . ag M − . a g M − . g M + 289 . ag M − . g M ,C ( a, g ) M = 32 . − . aM − . a M − . a M + 4 . a M − . a M + 9 . a M − . a M + 0 . gM + 1 . agM − . a gM + 41 . a gM − . a gM + 38 . a gM + 4 . a gM − . g M − . ag M + 198 . a g M − . a g M + 1000 . a g M − . a g M + 30 . g M + 128 . ag M − . a g M + 3228 . a g M − . a g M − . g M − . ag M + 2625 . a g M − . a g M + 521 . g M + 302 . ag M − . g M − . ag M + 782 . g M ,D ( a, g ) = 1 . − . aM − . a M − . a M + 0 . a M − . a M + 1 . a M − . a M + 0 . agM − . a gM + 8 . a gM − . a gM + 10 . a gM − . a gM − . ag M + 36 . a g M − . a g M + 197 . a g M − . a g M + 22 . ag M − . a g M + 573 . a g M − . a g M − . ag M + 433 . a g M − . a g M + 55 . ag M − . a g M − . ag M . (A2) lack hole parameter estimation from its shadow a and k and read as A ( a, k ) M = 84 . − . aM − . a M − . a M + 31 . a M − . a M + 73 . a M − . a M − . kM + 0 . akM − . a kM + 139 . a kM − . a kM + 467 . a kM − . a kM − . k M − . ak M + 40 . a k M − . a k M + 1494 . a k M − . a k M + 5 . k M − . ak M + 37 . a k M + 901 . a k M − . a k M − . k M + 3 . ak M − . a k M ,C ( a, k ) M = 32 . − . aM − . a M − . a M + 4 . a M − . a M + 9 . a M − . a M − . kM + 0 . akM − . a kM + 22 . a kM − . a kM + 70 . a kM − . a kM − . k M − . ak M + 38 . a k M − . a k M + 268 . a k M − . a k M + 1 . k M + 27 . ak M − . a k M + 169 . a k M − . a k M − . k M − . ak M + 82 . a k M ,D ( a, k ) = 1 . − . aM − . a M − . a M + 0 . a M − . a M + 1 . a M − . a M + 0 . akM − . a kM + 3 . a kM − . a kM + 11 . a kM − . a kM − . ak M + 1 . a k M − . a k M + 43 . a k M − . a k M + 0 . ak M + 1 . a k M + 28 . a k M − . a k M + 0 . ak M − . a k M . (A3)Here, we have presented the series up to O ( M − ). The nonrotating black hole ( a = 0) casts a perfect circular shadow(Synge 1966; Chandrasekhar 1985), which is also fully consistent with Equations (A1)-(A3), i.e., D (0 , Q ) = D (0 , g ) = D (0 , k ) = 1. B. OBSERVABLES IN ASSOCIATION WITH NOISY DATAThe observables A , C , and D are described in terms of the celestial coordinates ( α, β ), which are easy to calculatefor a given black hole. Astronomical observations may not give a sharp shadow boundary demarcating the bright anddark regions; rather there will be intrinsic uncertainty in determining the shadow boundary because of noise in theobservational data. In such observational data, we consider the set of visibility data points ( α i , β i ) along the hazyshadow boundary. The geometric center ( α G , β G ) of the apparent shadow reads α G = 1 N N X i =1 α i ; β G = 1 N N X i =1 β i , (B4)where N is the total number of data points. In the coordinate system centered at ( α G , β G ), the shadow boundary canbe parameterized by ( α ′ i , β ′ i ) α ′ i = α i − α G ; β ′ i = β i − β G . (B5)Thus, we can calculate the shadow observables, A and C , respectively, as A = N X i =1 | β ′ i − + β ′ i | | α ′ i − α ′ i − | , (B6)6 Kumar & Ghosh
80 82 84 86 880.00.10.20.30.4 A (cid:144) M P H A (cid:144) M L C (cid:144) M P H C (cid:144) M L D P H D L Figure 14.
Probability density distribution of shadow observables for perturbed Kerr black hole shadows. and C = N X i =1 (cid:0) ( α ′ i − α ′ i − ) + ( β ′ i − β ′ i − ) (cid:1) / , (B7)where α ′ = 0 or α = α G and data points are arranged such that | α ′ i | ≥ | α ′ i − | . In this case, the oblateness D becomes D = α ′ r − α ′ l β ′ t − β ′ b , (B8)where ( α ′ l ,
0) and ( α ′ r ,