A coordinate-independent characterization of a black hole shadow
MMon. Not. R. Astron. Soc. , 1–12 (0000) Printed 13 September 2018 (MN L A TEX style file v2.2)
A coordinate-independent characterization of a black hole shadow
A. A. Abdujabbarov, , , L. Rezzolla , and B. J. Ahmedov , , Institute of Nuclear Physics, Ulughbek, Tashkent 100214, Uzbekistan Ulugh Beg Astronomical Institute, Astronomicheskaya 33, Tashkent 100052, Uzbekistan National University of Uzbekistan, Tashkent 100174, Uzbekistan Institut f¨ur Theoretische Physik, Max-von-Laue-Str. 1, D-60438 Frankfurt, Germany Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt, Germany
13 September 2018
ABSTRACT
A large international effort is under way to assess the presence of a shadow in the radio emis-sion from the compact source at the centre of our Galaxy, Sagittarius A ∗ (Sgr A ∗ ). If detected,this shadow would provide the first direct evidence of the existence of black holes and thatSgr A ∗ is a supermassive black hole. In addition, the shape of the shadow could be used tolearn about extreme gravity near the event horizon and to determine which theory of gravitybetter describes the observations. The mathematical description of the shadow has so far useda number of simplifying assumptions that are unlikely to be met by the real observationaldata. We here provide a general formalism to describe the shadow as an arbitrary polar curveexpressed in terms of a Legendre expansion. Our formalism does not presume any knowledgeof the properties of the shadow, e.g., the location of its centre, and offers a number of routes tocharacterize the distortions of the curve with respect to reference circles. These distortions canbe implemented in a coordinate independent manner by different teams analysing the samedata. We show that the new formalism provides an accurate and robust description of noisyobservational data, with smaller error variances when compared to previous approaches forthe measurement of the distortion. Key words: black hole physics – Galaxy: centre – submillimetre: galaxies
There is a widespread belief that the most convincing evidence forthe existence of black holes will come from the direct observa-tions of properties related to the horizon. These could be throughthe detection of gravitational waves from the collapse to a rotatingstar (Baiotti & Rezzolla 2006), from the ringdown in a binary blackhole merger (Berti et al. 2009), or through the direct observationof its ‘shadow’. In a pioneering study, Bardeen (1973) calculatedthe shape of a dark area of a Kerr black hole, that is, its ‘shadow’over a bright background appearing, for instance, in the image ofa bright star behind the black hole. The shadow is a gravitationallylensed image of the event horizon and depends on the closed or-bits of photons around the black hole . Its outer boundary, whichwe will hereafter simply refer to as the shadow, corresponds to theapparent image of the photon capture sphere as seen by a distantobserver. General relativity predicts, in fact, that photons circlingthe black hole slightly inside the boundary of the photon sphere Strictly speaking, also an horizon-less object such as a gravastar (Mazur& Mottola 2004) would lead to a shadow. However, rather exotic assump-tions on the heat capacity of the gravastar’s surface are needed to jus-tify a lack of emission from such a surface (Broderick & Narayan 2007);gravitational-wave emission would unambiguously signal the presence ofan event horizon (Chirenti & Rezzolla 2007). will fall down into the event horizon, while photons circling justoutside will escape to infinity. The shadow appears therefore as arather sharp boundary between bright and dark regions and arisesfrom a deficit of those photons that are captured by the event hori-zon. Because of this, the diameter of the shadow does not depend onthe photons’ energy, but uniquely on the angular momentum of theblack hole. In general relativity and in an idealized setting in whicheverything is known about the emission properties of the plasmanear the black hole, the shadow’s diameter ranges from . r S foran extreme Kerr black hole, to √ r S for a Schwarzschild blackhole, where r S := 2 GM/c is the Schwarzschild radius. In prac-tice, however, the size and shape of the shadow will be influencedby the astrophysical properties of the matter near the horizon and,of course, by the theory of gravity governing the black hole.Besides providing evidence on the existence of black holes,the observation of the black hole shadow and of the deformationsresulting in the case of nonzero spin, is also expected to help de-termine many of the black hole properties (see e.g. Chandrasekhar1998; Falcke & Markoff 2013; Takahashi 2004; Falcke et al. 2000;Doeleman et al. 2009). More specifically, imaging the shadow of ablack hole via radio observations will allow one to test the predic-tions of general relativity for the radius of the shadow and studyastrophysical phenomena in the vicinity of black holes [see Jo-hannsen & Psaltis (2010) and, more recently, Zakharov (2014) for c (cid:13) a r X i v : . [ g r- q c ] S e p Abdujabbarov, Rezzolla & Ahmedov a review]. In addition, it will allow one to set constraints on thevalidity of alternative theories of gravity which also predict blackholes and corresponding shadows (see e.g. Eiroa & Sendra 2014;Tsukamoto et al. 2014; Falcke & Markoff 2013).The possible observation of a black hole shadow has recentlyreceived a renewed attention as the spatial resolution attainableby very long baseline interferometry (VLBI) radio observations issoon going to be below the typical angular size of the event horizonof candidate supermassive black holes (SMBHs), such as the oneat the centre of the Galaxy or the one in the M87 galaxy (Falckeet al. 2000). These observations are the focus of international sci-entific collaborations, such as the Event Horizon Telescope (EHT) or the Black Hole Camera (BHC) , which aim at VLBI obser-vations at 1.3 mm and 0.87 mm of Sagittarius A ∗ (Sgr A ∗ ) andM87. We recall that Sgr A ∗ is a compact radio source at the cen-tre of the Galaxy and the SMBH candidate in our galaxy. In fact,the orbital motion of stars near Sgr A ∗ indicates that its mass is (cid:39) . × M (cid:12) (Ghez et al. 2008; Genzel et al. 2010).Given a distance of 8 kpc from us, the angular size of theSchwarzschild radius of the SMBH candidate in Sgr A ∗ is ∼ µ as , so that the corresponding angular diameter of the shadowis of the order of ∼ µ as . Similarly, with an estimated mass of (cid:39) . × M (cid:12) (Gebhardt et al. 2011) and a distance of 16 Mpc,the M87 galaxy represents an equally interesting SMBH candidate,with an angular size that is of the same order i.e., ∼ µ as . Al-though the resolution achievable at present is not sufficient to ob-serve an image of the shadow of either black hole, it is sufficientlyclose that it is realistic to expect that near-future observations willreach the required resolution. Indeed, future EHT and BHC obser-vations of Sgr A ∗ are expected to go below the horizon scale and tostart to provide precise information on the black hole orientation, aswell as on the astrophysical properties of the accretion flow takingplace on to the black hole (Chan et al. 2015; Psaltis et al. 2015).An extensive literature has been developed to calculate theshadow of the black hole in known space-times, either within gen-eral relativity (Young 1976; Perlick 2004; Abdujabbarov et al.2013), or within alternative theories of gravity (Bambi & Modesto2013; Tsukamoto et al. 2014; Amarilla et al. 2010; Amarilla &Eiroa 2012, 2013; Atamurotov et al. 2013a,b; Schee & Stuchl´ık2009). In most cases, the expression of the shadow as a closed polarcurve is not known analytically, but for the Pleba´nski-Demia´nskiclass of space-times, the shadow has been cast in an analyticform (Grenzebach et al. 2014, 2015).Because the shadow is in general a complex polar curve onthe celestial sky, an obvious problem that emerges is that of thecharacterization of its deformation. For example, in the case of aKerr black hole, the difference in the photon capture radius be-tween corotating and counter-rotating photons creates a “dent” onone side of the shadow, whose magnitude depends on the rotationrate of the black hole. A way to measure this deformation was firstsuggested by Hioki & Maeda (2009) and then further developedby other authors (Bambi & Freese 2009; Bambi & Yoshida 2010;Bambi & Modesto 2013). In essence, in these approaches the shapeof the shadow is approximated as a circle with radius R s and suchthat it crosses through three points located at the poles and at theequator of the shadow’s boundary. The measure of the dent is thenmade in terms of the so called ‘deflection’, that is, the differencebetween the endpoints of the circle and of the shadow, with a di- http://eventhorizontelescope.org/ http://blackholecam.org/ mensionless distortion parameter being given by the ratio of thesize of the dent to the radius R s [cf. Eq. (48)].While this approach is reasonable and works well for a blackhole such as the Kerr black hole, it is not obvious it will workequally well for black holes in more complex theories of gravityor even in arbitrary metric theories of gravity as those consideredby Rezzolla & Zhidenko (2014). Leaving aside the fact in all theseworks the shadow is assumed to be determined with infinite preci-sion (an assumption which is obviously incompatible with a mea-sured quantity), many but not all approaches in characterizing theblack hole shadow and its deformations suffer from at least threepotential difficulties. They often assume a primary shape, i.e., thatthe shadow can be approximated with a circle; exceptions are theworks of Kamruddin & Dexter (2013) and Psaltis et al. (2014).In the first one, a model has been proposed to describe the “cres-cent” morphology resulting from the combined effects of relativis-tic Doppler beaming and gravitational light bending (Kamruddin& Dexter 2013); in the second one, an edge-detection scheme anda pattern-matching algorithm are introduced to measure the prop-erties of the black hole shadow even if the latter has an arbi-trary shape (Psaltis et al. 2014). (ii) They assume that the observerknows the exact position of the centre of black hole . (iii) Theyare restricted to a very specific measure of the distortion and areunable to model arbitrary distortion; exceptions are the works ofJohannsen & Psaltis (2010) and Johannsen (2013), where insteadpolar-averaged distortions have been proposed.To counter these potential difficulties, we present here a newgeneral formalism that is constructed to avoid the limitations men-tioned above. In particular, we assume that the shadow has an ar-bitrary shape and expand it in terms of Legendre polynomials in acoordinate system with origin in the effective centre of the shadow.This approach gives us the advantage of not requiring the knowl-edge of the centre of the black hole and of allowing us to intro-duce a number of parameters that measure the distortions of theshadow. These distortions are both accurate and robust, and canbe implemented in a coordinate independent manner by differentteams analysing the same noisy data.The paper is organized as follows. In Sect. 2 we develop thenew coordinate-independent formalism, where an arbitrary blackhole shadow is expanded in terms of Legendre polynomials. Usingthis formalism, we introduce in Sect. 3 various distortion parame-ters of the shadow. In Sect. 4 we apply the formalism to a numberof black hole space-times by computing the coefficients of the ex-pansion and by showing that they exhibit an exponentially rapidconvergence. We also compare the properties of the different dis-tortion parameters and assess which definition appears to be moreaccurate and robust in general. Section 5 offers a comparison be-tween the new distortion parameters introduced here with the moretraditional ones simulating the noisy data that are expected fromthe observations. Finally, Sect. 6 summarizes our main results theprospects for the use of the new formalism.We use a system of units in which c = G = 1 , a space-like signature ( − , + , + , +) and a spherical coordinate system ( t, r, θ, φ ) . Greek indices are taken to run from 0 to 3. The reconstruction procedures of the observational data does resolve thisproblem, which is however still present when considering and comparingpurely theoretical representations of the shadowc (cid:13) , 1–12 haracterization of a black hole shadow Figure 1.
Schematic representation of the black hole shadow as a genericpolar curve R ψ in a coordinate system ( α, β ) with origin O in the ‘centre’of the shadow. The latter is translated by a vector (cid:126) R with respect to the ar-bitrary coordinate system ( α (cid:48) , β (cid:48) ) with origin O (cid:48) in which the observationsare made. In what follows we develop a rather general formalism to describethe black hole shadow that radio astronomical observations are ex-pected to construct. For all practical purposes, however, we willconsider the problem not to consist of the determination of theinnermost unstable circular orbits for photons near a black hole.Rather, we will consider the problem of characterizing in a mathe-matically sound and coordinate-independent way a closed curve ina flat space, as the one in which the image will be available to us asdistant observers.Assume therefore that the astronomical observations providethe shadow as an one-dimensional closed curve defined by theequation R (cid:48) = R (cid:48) ( ψ (cid:48) ) , (1)where R (cid:48) and ψ (cid:48) can be thought of as the radial and angular coor-dinates in a polar coordinate system with origin in O (cid:48) . In practice,astronomical observations will not be able to provide such a sharpclosed line and a more detailed analysis would need to take the ob-servational uncertainties (which could well be a function of ψ (cid:48) ) intoaccount. We will discuss some of these uncertainties in Sect. 5, butfor the time being we will consider the shadow as an idealized one-dimensional closed curve. A schematic example of the polar curveis shown in Fig. 1, where α (cid:48) and β (cid:48) are the so-called “celestial co-ordinates” of the observer, and represent an orthogonal coordinatesystem with one of the unit vectors being along the line of sight.Of course, there is no reason to believe that such a coordinatesystem is particularly useful, or that in using it a nonrotating blackhole will have a shadow given by a perfect circle. Hence, in orderto find a better coordinate system, and, in particular, one in whicha Schwarzschild black hole has a circular shadow, we define the effective centre of the curve extending the definition of the position of the centre of mass for a solid body to obtain (cid:126) R := (cid:82) π (cid:126) e R (cid:48) ( ψ (cid:48) ) R (cid:48) (cid:2) g R (cid:48) R (cid:48) ( dR (cid:48) /dψ (cid:48) ) + g ψ (cid:48) ψ (cid:48) (cid:3) / dψ (cid:48) (cid:82) π [ g R (cid:48) R (cid:48) ( dR (cid:48) /dψ (cid:48) ) + g ψ (cid:48) ψ (cid:48) ] / dψ (cid:48) , (2)where (cid:126) e R (cid:48) is the radial-coordinate unit vector and where g R (cid:48) R (cid:48) , g ψ (cid:48) ψ (cid:48) are the metric functions of the polar coordinate system ( R (cid:48) , ψ (cid:48) ) . Two important remarks: first, radio observations may wellyield, especially in the nearest future, only a portion of the shadow,namely, the one with the largest brightness. Yet, it is useful to con-sider here the shadow as a closed polar curve since this is the way itis normally discussed in purely theoretical investigations. Second,at least observationally, the location of the centre of the black holeshadow is a free parameter in the image-reconstruction procedureand so already part of the analysis of the observational data. At thesame time, the definition of a centre is useful also in the absenceof actual observational data since it can help in the comparison ofshadows that are built analytically and hence without observationaldata.From the knowledge of the vector (cid:126) R , the coordinate positionof the effective centre can be expressed explicitly in terms of theradial and angular coordinates as R := (cid:18)(cid:90) π R (cid:48) dψ (cid:48) (cid:19) − (cid:34) (cid:18)(cid:90) π R (cid:48) cos ψ (cid:48) dψ (cid:48) (cid:19) + (cid:18)(cid:90) π R (cid:48) sin ψ (cid:48) dψ (cid:48) (cid:19) (cid:35) / , (3) ψ := tan − (cid:32) (cid:82) π R (cid:48) sin ψ (cid:48) dψ (cid:48) (cid:82) π R (cid:48) cos ψ (cid:48) dψ (cid:48) (cid:33) . (4)We note that if the centre of the primary coordinate system O (cid:48) coincides with the black hole origin, then the parameter R exactlycorresponds to the shift of the centre of the shadow with respect tothe black hole position defined by Tsukamoto et al. (2014).Having determined the effective centre of the shadow, it isconvenient to define a new polar coordinate system centred in itwith coordinates ( R, ψ ) . Clearly, the new coordinate system withorigin O is just translated by (cid:126) R with respect to O (cid:48) and, hence, therelation between the two coordinate systems is given by R := (cid:104) (cid:0) R (cid:48) cos ψ (cid:48) − R cos ψ (cid:1) + (cid:0) R (cid:48) sin ψ (cid:48) − R sin ψ (cid:1) (cid:105) / , (5) ψ := tan − R (cid:48) sin ψ (cid:48) − R sin ψ R (cid:48) cos ψ (cid:48) − R cos ψ . (6)Note that we have kept the new axes α and β parallel to the origi-nal ones α (cid:48) and β (cid:48) . This is not strictly necessary but given the ar-bitrarity of the orientation of both sets of axes, it provides a usefulsimplification.A well-defined centre of coordinates allows us now to obtaina robust definition of the reference areal circle as the circle hav-ing the same area as the one enclosed by the shadow. In particular,given the closed parametric curve R = R ( ψ ) , its area will in gen-eral be given by A := (cid:90) ψ ψ dψ (cid:90) R √ g ¯ R ¯ R g ψψ d ¯ R = 12 (cid:90) ψ ψ R dψ = 12 (cid:90) λ λ R ( λ ) (cid:18) dψdλ (cid:19) dλ , (7)where in the second equality we have set g RR = 1 , g ψψ = R ( ψ ) , c (cid:13)000
Schematic representation of the black hole shadow as a genericpolar curve R ψ in a coordinate system ( α, β ) with origin O in the ‘centre’of the shadow. The latter is translated by a vector (cid:126) R with respect to the ar-bitrary coordinate system ( α (cid:48) , β (cid:48) ) with origin O (cid:48) in which the observationsare made. In what follows we develop a rather general formalism to describethe black hole shadow that radio astronomical observations are ex-pected to construct. For all practical purposes, however, we willconsider the problem not to consist of the determination of theinnermost unstable circular orbits for photons near a black hole.Rather, we will consider the problem of characterizing in a mathe-matically sound and coordinate-independent way a closed curve ina flat space, as the one in which the image will be available to us asdistant observers.Assume therefore that the astronomical observations providethe shadow as an one-dimensional closed curve defined by theequation R (cid:48) = R (cid:48) ( ψ (cid:48) ) , (1)where R (cid:48) and ψ (cid:48) can be thought of as the radial and angular coor-dinates in a polar coordinate system with origin in O (cid:48) . In practice,astronomical observations will not be able to provide such a sharpclosed line and a more detailed analysis would need to take the ob-servational uncertainties (which could well be a function of ψ (cid:48) ) intoaccount. We will discuss some of these uncertainties in Sect. 5, butfor the time being we will consider the shadow as an idealized one-dimensional closed curve. A schematic example of the polar curveis shown in Fig. 1, where α (cid:48) and β (cid:48) are the so-called “celestial co-ordinates” of the observer, and represent an orthogonal coordinatesystem with one of the unit vectors being along the line of sight.Of course, there is no reason to believe that such a coordinatesystem is particularly useful, or that in using it a nonrotating blackhole will have a shadow given by a perfect circle. Hence, in orderto find a better coordinate system, and, in particular, one in whicha Schwarzschild black hole has a circular shadow, we define the effective centre of the curve extending the definition of the position of the centre of mass for a solid body to obtain (cid:126) R := (cid:82) π (cid:126) e R (cid:48) ( ψ (cid:48) ) R (cid:48) (cid:2) g R (cid:48) R (cid:48) ( dR (cid:48) /dψ (cid:48) ) + g ψ (cid:48) ψ (cid:48) (cid:3) / dψ (cid:48) (cid:82) π [ g R (cid:48) R (cid:48) ( dR (cid:48) /dψ (cid:48) ) + g ψ (cid:48) ψ (cid:48) ] / dψ (cid:48) , (2)where (cid:126) e R (cid:48) is the radial-coordinate unit vector and where g R (cid:48) R (cid:48) , g ψ (cid:48) ψ (cid:48) are the metric functions of the polar coordinate system ( R (cid:48) , ψ (cid:48) ) . Two important remarks: first, radio observations may wellyield, especially in the nearest future, only a portion of the shadow,namely, the one with the largest brightness. Yet, it is useful to con-sider here the shadow as a closed polar curve since this is the way itis normally discussed in purely theoretical investigations. Second,at least observationally, the location of the centre of the black holeshadow is a free parameter in the image-reconstruction procedureand so already part of the analysis of the observational data. At thesame time, the definition of a centre is useful also in the absenceof actual observational data since it can help in the comparison ofshadows that are built analytically and hence without observationaldata.From the knowledge of the vector (cid:126) R , the coordinate positionof the effective centre can be expressed explicitly in terms of theradial and angular coordinates as R := (cid:18)(cid:90) π R (cid:48) dψ (cid:48) (cid:19) − (cid:34) (cid:18)(cid:90) π R (cid:48) cos ψ (cid:48) dψ (cid:48) (cid:19) + (cid:18)(cid:90) π R (cid:48) sin ψ (cid:48) dψ (cid:48) (cid:19) (cid:35) / , (3) ψ := tan − (cid:32) (cid:82) π R (cid:48) sin ψ (cid:48) dψ (cid:48) (cid:82) π R (cid:48) cos ψ (cid:48) dψ (cid:48) (cid:33) . (4)We note that if the centre of the primary coordinate system O (cid:48) coincides with the black hole origin, then the parameter R exactlycorresponds to the shift of the centre of the shadow with respect tothe black hole position defined by Tsukamoto et al. (2014).Having determined the effective centre of the shadow, it isconvenient to define a new polar coordinate system centred in itwith coordinates ( R, ψ ) . Clearly, the new coordinate system withorigin O is just translated by (cid:126) R with respect to O (cid:48) and, hence, therelation between the two coordinate systems is given by R := (cid:104) (cid:0) R (cid:48) cos ψ (cid:48) − R cos ψ (cid:1) + (cid:0) R (cid:48) sin ψ (cid:48) − R sin ψ (cid:1) (cid:105) / , (5) ψ := tan − R (cid:48) sin ψ (cid:48) − R sin ψ R (cid:48) cos ψ (cid:48) − R cos ψ . (6)Note that we have kept the new axes α and β parallel to the origi-nal ones α (cid:48) and β (cid:48) . This is not strictly necessary but given the ar-bitrarity of the orientation of both sets of axes, it provides a usefulsimplification.A well-defined centre of coordinates allows us now to obtaina robust definition of the reference areal circle as the circle hav-ing the same area as the one enclosed by the shadow. In particular,given the closed parametric curve R = R ( ψ ) , its area will in gen-eral be given by A := (cid:90) ψ ψ dψ (cid:90) R √ g ¯ R ¯ R g ψψ d ¯ R = 12 (cid:90) ψ ψ R dψ = 12 (cid:90) λ λ R ( λ ) (cid:18) dψdλ (cid:19) dλ , (7)where in the second equality we have set g RR = 1 , g ψψ = R ( ψ ) , c (cid:13)000 , 1–12 Abdujabbarov, Rezzolla & Ahmedov
Figure 2.
Schematic representation of the local distortion d ψ between thepolar curve R ψ representing the black hole shadow and representative cir-cles with circumference and area radii R C and R A , respectively. while in the third equality we consider the representation of thecurve in terms of a more generic parameter λ , i.e., R = R ( ψ ( λ )) .If the shadow is a closed curve, the integration limits λ , can befound from the condition ψ ( λ ) = 0 and ψ ( λ ) = 2 π , while theywill be restricted by the actual observational data when only a por-tion of the shadow is available.We can then define the areal radius R A of the reference circlesimply as R A := (cid:18) Aπ (cid:19) / . (8)Similarly, and if simpler to compute, it is possible to define the circumferential radius R c of the reference circle as R C := C π , (9)where the circumference is calculated as C := (cid:90) (cid:0) dR + g ψψ dψ (cid:1) / = (cid:90) λ λ (cid:34)(cid:18) dRdλ (cid:19) + R (cid:18) dψdλ (cid:19) (cid:35) / dλ . (10)An areal radius is particularly useful as it enables one tomeasure two useful quantities, namely, the local deviation of theshadow R ψ := R ( ψ ) from the areal circle, i.e., d ψ := | R A − R ψ | , (11)and its polar average d (cid:104) ψ (cid:105) := 12 π (cid:90) π d ψ dψ = 12 π (cid:90) π | R A − R ψ | dψ . (12)Note that although similar, the areal and the circumferential radiiare in general different and coincide just for a spherically symmet-ric black hole, in which case R A = R c = R ψ , and of course d ψ = 0 = d s . All of these geometrical quantities are shownschematically in Fig. 2. With a well defined and unambiguous set of coordinates ( R, ψ ) , wecan next move to the characterization of the geometrical propertiesof the shadow. To this scope we simply employ an expansion interms of Legendre polynomial, i.e., we define R ψ := ∞ (cid:88) (cid:96) =0 c (cid:96) P (cid:96) (cos ψ ) , (13)where P (cid:96) (cos ψ ) is the Legendre polynomial of order (cid:96) and thecoefficients c (cid:96) of the expansion (13) can be found as c (cid:96) := 2 (cid:96) + 12 (cid:90) π R ( ψ ) P (cid:96) (cos ψ ) sin ψ dψ = 2 (cid:96) + 12 (cid:90) λ λ R ( ψ ) P (cid:96) (cos ψ ) sin ψ (cid:18) dψdλ (cid:19) dλ . (14)The integration limits λ , can be found from the condition ψ ( λ ) =0 and ψ ( λ ) = π , respectively. Using this decomposition, it isstraightforward to measure the differences between the value of theparametrized shadow at two different angles. For example, the rela-tive difference between the shadow at ψ = 0 and at a generic angle ψ = π/m can be computed simply as δ m := R ψ ( ψ = 0) − R ψ ( ψ = π/m ) R ψ ( ψ = 0)=1 − (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) (cos ψ ) | ψ = π/m (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) (cos ψ ) | ψ =0 . (15)When m = 1 , expression (15) simplifies to δ := 1 − (cid:80) ∞ (cid:96) =0 ( − (cid:96) c (cid:96) (cid:80) ∞ (cid:96) =0 c (cid:96) , (16)while, when m = 2 , the difference can still be computed analyti-cally and is given by δ := 1 − BA , (17)where we have introduced the following and more compact nota-tion that will be used extensively in the remainder A := R ψ ( ψ = 0) = ∞ (cid:88) (cid:96) =0 c (cid:96) , (18) B := R ψ ( ψ = π/
2) = ∞ (cid:88) (cid:96) =0 ( − (cid:96) (2 (cid:96) )!2 (cid:96) ( (cid:96) !) c (cid:96) , (19) C := R ψ ( ψ = 3 π/
2) = ∞ (cid:88) (cid:96) =0 ( − (cid:96) c (cid:96) . (20)Analytic expressions for (15) when m > are much harder toderive, but can be easily computed numerically.We note that the parametrization (15) is quite general and al-lows us to recover in a single definition some of the expressionscharacterizing the distortion of the shadow and that have been in-troduced by other authors. For example, the parameter δ n can beassociated to the distortion parameter δ first introduced by Hioki &Maeda (2009) (cf. Fig. 3 of Hioki & Maeda 2009). Similarly, theparameter δ is directly related to the distortion parameter (cid:15) intro-duced by Tsukamoto et al. (2014) (cf. Fig. 3 of Tsukamoto et al.2014).In what follows we will exploit the general expression for thepolar curve representing the black hole shadow to suggest threedifferent definitions that measure in a coordinate-independent man-ner the amount of distortion of the shadow relative to some simple c (cid:13) , 1–12 haracterization of a black hole shadow background curve, e.g., a circle. These expressions are all mathe-matically equivalent and the use of one over the other will dependon the specific properties of the observed shadow. IWe start by considering three points on the polar curve A , B , and D , which occupy precise angular positions at ψ = 0 , π/ , and π/ , respectively (see diagram in Fig. 3). The corresponding dis-tances OA , OB and OD from the centre of coordinates O can thenbe expressed as R A := R ψ ( ψ = 0) = ∞ (cid:88) (cid:96) =0 c (cid:96) P (cid:96) (cos ψ ) | ψ =0 = A , (21) R B := R ψ ( ψ = π/
2) = ∞ (cid:88) (cid:96) =0 c (cid:96) P (cid:96) (cos ψ ) | ψ = π/ = B , (22) R D := R ψ ( ψ = 3 π/
2) = R B , (23)where in the last equality we have exploited the fact that the expan-sion in Legendre polynomials is symmetric with respect to the α axis. Next, we define a new parametric curve for which R A = R B = R D and thus that satisfies the following condition B = A , (24)or, equivalently, for (cid:96) > c (cid:96) − = c (cid:96) (cid:20) ( − (cid:96) (2 (cid:96) )!2 (cid:96) ( (cid:96) !) − (cid:21) . (25)The corresponding polar expression, formulated in terms ofthe Legendre polynomials expansion (13), is therefore given by R ψ, I ( ψ ) = c + ∞ (cid:88) (cid:96) =1 c (cid:96) − × (cid:40) P (cid:96) − (cos ψ ) + (cid:20) ( − (cid:96) (2 (cid:96) )!2 (cid:96) ( (cid:96) !) − (cid:21) − P (cid:96) (cos ψ ) (cid:41) , . (26)To measure the distortion we need a reference curve, whichwe can choose to be the circle passing through the three points A , B , and D and thus with radius R s, I := R A = B = R B = A . (27)We can now compute the deviation of the parametric curve(26) from the corresponding background circle of radius R s, I atany angular position. However, as customary in this type of consid-erations, we can consider the shadow to be produced by a rotatingblack hole with spin axis along the β axis, so that the largest devia-tions will be on the axis of negative α (this is shown schematicallyin the left-hand panel of Fig. A1, when considering the case of aKerr black hole). More specifically, we can define the differencebetween the curves at ψ = π as d s, I := R s, I − R ψ, I ( ψ = π ) = B − ∞ (cid:88) (cid:96) =0 c (cid:96) + ∞ (cid:88) (cid:96) =1 c (cid:96) − =2 ∞ (cid:88) (cid:96) =1 c (cid:96) − . (28)It follows that our first definition for the dimensionless distortion parameter – δ s, I can then be given by δ s, I := d s, I R s, I = 2 (cid:80) ∞ (cid:96) =1 c (cid:96) − B =: ∞ (cid:88) (cid:96) =1 δ (cid:96), I , (29)which reduces to the compact expression δ s, I (cid:39) c c = δ , I . (30)when only the first two coefficients in the expansion are taken intoaccount, i.e., for c (cid:54) = 0 (cid:54) = c and c (cid:96) = 0 , with (cid:96) (cid:62) (we recallthat δ , I = 0 ). We also note that the assumption (27) does not re-strict the analysis to spherically symmetric black hole space-timesand, as we will show in Fig. 6, the distortion parameter (29) can beapplied also to axisymmetric space-times. IIA second possible definition of the distortion parameter is slightlymore general and assumes that the radial distance of points A and B from the centre of coordinates is not necessarily the same,i.e., R A (cid:54) = R B . In this case, one can think of introducing a newpoint E on the α axis (this is shown schematically in the middlepanel of Fig. A1, when considering the case of a Kerr black hole),such that the distances AE = EB and which could therefore serveas the centre of the reference circle. Since the values of the coor-dinates R A and R B are defined by expressions (21) and (22), wecan use the condition AE = EB to find that one can easily findposition of the point E on the α axis is given by R E = (cid:12)(cid:12)(cid:12)(cid:12) R B − R A R A (cid:12)(cid:12)(cid:12)(cid:12) , (31)with the corresponding angular position ψ E being either or π ,i.e., ψ E = cos − (cid:18) R A − R B | R A − R B | (cid:19) . (32)The radius of the circle passing through the three points A , B , and D is R s, II = R B + R A R A , (33)so that the deviation of the shadow at ψ = π from the circle ofradius R s, II can be found using the relation d s, II = 2 R s, II − ( R A + R C ) . (34)Finally, we can introduce the distortion parameter d s, II de-fined as d s, II := B A − C , (35)so that the second dimensionless distortion parameter is expressedas δ s, II := d s, II R s, II = 2 (cid:18) B − A CB + A (cid:19) . (36)The expression for the dimensionless distortion (36) is in thiscase more complex that the one presented in Eq. (29); however,in the simpler case in which only the lowest order coefficients areretained, i.e., if c (cid:54) = 0 (cid:54) = c and c (cid:96) = 0 for (cid:96) (cid:62) , we have δ s, II (cid:39) c c + 2 c c + c = δ , II . (37) c (cid:13) , 1–12 Abdujabbarov, Rezzolla & Ahmedov
Figure 3.
Schematic representation of the distortion parameter – III. Thequantity d s, III measures the difference between the Legendre expandedpolar curve R ψ, III and the reference circle of radius R s, III and passingthrough the points A , B , D , and having centre in point E . Also shown arethe “zero-slope” points S and S (cid:48) . IIIA third and possibly optimal definition of the distortion parameteris one that is meant to consider the case in which the shadow is stillreflection symmetric relative to the α axis, but does not cross the β axis with a zero slope. Rather, the curve admits a point, say S , atangular position < ψ S < π , where it has zero slope relative tothe ( α, β ) coordinate system (see diagram in Fig. 3 and the right-hand panel of Fig. A1 for the case of a Kerr black hole). This pointwill be referred to as the “slope point” of the parametric curve R ψ representing the shadow.To compute the position of this point in the ( α, β ) coordinateswe simply need to find the solution to the equation dβdα (cid:12)(cid:12)(cid:12)(cid:12) ψ S = 0 , (38)or, equivalently, solve for the differential equation dR ψ dψ sin ψ + R ψ cos ψ = 0 . (39)Using the expansion in terms of Legendre polynomials (13),we can rewrite equation (39) as ∞ (cid:88) (cid:96) =0 c (cid:96) P (cid:96) ( x ) x − ∞ (cid:88) (cid:96) =0 c (cid:96) dP (cid:96) ( x ) dx (1 − x ) = 0 , (40)where we have set x := cos ψ . The solutions of (40) provide thepositions of all the possible slope points in the parametric curve,and the solution is unique in the case in which the shadow R ( ψ ) isconvex. The corresponding coordinates of the point S are then R S = ∞ (cid:88) (cid:96) =0 c (cid:96) P (cid:96) ( x S ) , (41) ψ A = cos − ( x S ) . (42)As for the second distortion parameter in Sect. 3.3, we set E to be the centre of the circle passing through the points A , S , and S (cid:48) , where S (cid:48) the point is symmetric to the point S with respect tothe α axis. Using the condition AE = ED , we obtain the solution R D = (cid:12)(cid:12)(cid:12)(cid:12) A − ( (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) ( x S )) (cid:80) ∞ (cid:96) =0 c (cid:96) (1 − P (cid:96) ( x S ) x S ) (cid:12)(cid:12)(cid:12)(cid:12) , (43) ψ D = cos − (cid:18) R D | R D | (cid:19) . (44)Also for this third case, the radius of the circle R s, III pass-ing through the three points A , S , and S (cid:48) , the distortion parameter d s, III , and its dimensionless version δ s, III , have respectively theform R s, III = A − x S A ( (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) ( x S )) + ( (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) ( x S )) (cid:80) ∞ (cid:96) =0 c (cid:96) (1 − P (cid:96) ( x S ) x S ) , (45) d s, III = 2 R s, III − ( R A + R C ) = (cid:32) ∞ (cid:88) (cid:96) =0 c (cid:96) P (cid:96) ( x S ) (cid:33) × ( (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) ( x S ) − x S (cid:80) ∞ (cid:96) =1 c (cid:96) − − A C ) (cid:80) ∞ (cid:96) =0 c (cid:96) (1 − P (cid:96) ( x S ) x S ) , (46) δ s, III = d s, III R s, III = 2 (cid:32) ∞ (cid:88) (cid:96) =0 c (cid:96) P (cid:96) ( x S ) (cid:33) × ( (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) ( x S ) − x S (cid:80) ∞ (cid:96) =1 c (cid:96) − − A C ) A − x S A (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) ( x S ) + ( (cid:80) ∞ (cid:96) =0 c (cid:96) P (cid:96) ( x S )) . (47)We note that this definition is similar to the one proposed byHioki & Maeda (2009), who measure the dimensionless distortionof the shadow as δ s, HM := d s, HM R s, HM , (48)where d s, HM := R ψ ( ψ = π ) − R s, HM , (49)and with R s, HM being the radius of the circle passing through thepoints A , S , and S (cid:48) . The most important difference with respectto the definition of Hioki & Maeda (2009) is that we here expressthe parametric curve in terms of the general Legendre expansion(13), while Hioki & Maeda (2009) assume the knowledge of R ψ at ψ = π .Also in this case, expressions (45)–(47) are not easy to handleanalytically. However, in the simplest case in which the expansion(13) has only two nonvanishing terms, such that c (cid:54) = 0 (cid:54) = c and c (cid:96) = 0 for (cid:96) (cid:62) , Eq. (40) takes the more compact form c x + c x − c = 0 , (50)with solution x S = − c c ± (cid:115) c c + 12 , (51)and where the + or − signs refer to when c > and c < ,respectively. The corresponding quantities R s, III , d s, III and δ s, III have in this case the following form R s, III = 2 c + c (1 + x S ) + 2 c c (1 + x S )2[ c + c (1 + x S )] , (52) d s, III = c (1 + x S ) c + c (1 + x S ) , (53) δ s, III = 2 c (1 + x S )2 c + c (1 + x S ) + 2 c c (1 + x S ) = δ , III . (54) c (cid:13) , 1–12 haracterization of a black hole shadow If the shadow is perfectly circular with radius c , then c = 0 and expressions (52)–(54) show that R s, III = c , d s, III = 0 = δ s, III , as expected.
Having constructed a general formalism that allows us to describein a coordinate-independent manner the black hole shadow and tomeasure its deformation, we are now ready to apply such a formal-ism to the specific case of some well-known space-time metricsreferring to axisymmetric black holes. In particular, we will obvi-ously start with the application of the formalism to a rotating (Kerr)black hole (in Sect. 4.1), to move over to a Bardeen black hole andto a Kerr-Taub-NUT black hole in Sect. 4.2. We note that we donot consider these last two examples of black holes because theyare particularly realistic, but simply because they offer analytic lineelements on which our formalism can be applied.
We start with the Kerr space-time, whose line element in Boyer-Lindquist coordinates reads ds = − (cid:18) − Mr Σ (cid:19) dt − aMr sin θ Σ dtdφ + (cid:18) r + a + 2 a Mr sin θ Σ (cid:19) sin θdφ + Σ∆ dr + Σ dθ , (55)where Σ := r + a cos θ , ∆ := r − Mr + a , (56)with M being the mass of the black hole and a := J/M its specificangular momentum.Since the shape of the shadow is ultimately determined bythe innermost unstable orbits of photons, hereafter we will con-centrate on their equations for photons. In such a space-time, thecorresponding geodesic equations take the form Σ (cid:18) dtdλ (cid:19) = AE − aMrL z ∆ , (57) Σ (cid:18) drdλ (cid:19) = R , (58) Σ (cid:18) dθdλ (cid:19) =Θ , (59) Σ (cid:18) dφdλ (cid:19) = 2 aMrE ∆ + (Σ − Mr ) L z ∆ sin θ , (60)where λ is an affine parameter, R := E r + (cid:0) a E − L z − Q (cid:1) r + 2 M (cid:2) ( aE − L z ) + Q (cid:3) r − a Q , (61) Θ := Q (cid:0) a E − L z csc θ (cid:1) cos θ , (62) A := (cid:0) r + a (cid:1) − a ∆ sin θ , (63)and E and L z are the photon’s energy and the angular momentum,respectively. The quantity QQ = p θ + cos θ (cid:18) L z sin θ − a E (cid:19) (64) is the so-called Carter constant and p θ := Σ dθ/dλ is the canonicalmomentum conjugate to θ .Using these definitions, it is possible to determine the unstableorbits as those satisfying the conditions R (¯ r ) = ∂ R (¯ r ) ∂r = 0 , and ∂ R (¯ r ) ∂r (cid:62) , (65)where ¯ r is the radial coordinate of the unstable orbit. Introducingnow the new parameters ξ := L z /E and η := Q /E , the celestialcoordinates α and β of the image plane of the distant observer aregiven by (Bardeen et al. 1972) α = ξ sin i , β = ± ( η + a cos i − ξ cot i ) / , (66)where i is the inclination angle of the observer’s plane, that is, theangle between the normal to the observer’s plane and the blackhole’s axis of rotation. In the case of the Kerr space-time (55) andafter using the conditions (65), one can easily find that the valuesof ξ and η relative to the circular orbit ( c ) are (Young 1976; Chan-drasekhar 1998) ξ c = M (¯ r − a ) − ¯ r (¯ r − M ¯ r + a ) a (¯ r − M ) , (67) η c = ¯ r [4 a M − ¯ r (¯ r − M ) ] a (¯ r − M ) , (68)Next, to investigate the shape of the black hole shadow weintroduce the generic celestial polar coordinates ( R (cid:48) , ψ (cid:48) ) (cf. Sect.2 and Fig. 1) defined as R (cid:48) =( α + β ) / , (69) ψ (cid:48) = tan − (cid:18) βα (cid:19) . (70)Assuming for simplicity that the observer is in the equatorial plane,i.e., that i = π/ , then in terms of the ( R (cid:48) , ψ (cid:48) ) coordinates theshadow of black hole can be described as (hereafter we will set M = 1 ) R (cid:48) = (2 r + 2 a r − r + a + a r ) / r − , (71) ψ (cid:48) = tan − (cid:18) { r [4 a − r ( r − ] } / r − a − r ( r − r + a ) (cid:19) . (72)Making use of the procedure described in Sect. 2, it is straight-forward to determine the coordinates (3)–(4) of the effective centreof the shadow, and to perform the Legendre expansion (13). To thebest of our knowledge, no analytic expression exists to cast the co-ordinates (71)–(72) as a polar curve R ψ = R ( ψ ) . However, such acurve can be easily constructed numerically and from it the Legen-dre expansion (13) can be computed.Figure 4 summarizes the results of our approach by report-ing in the left-hand panel and in a logarithmic scale the normalizedvalues of the expansion coefficients c (cid:96) as a function of the Legen-dre order (cid:96) . Different curves refer to the different values consid-ered for the dimensionless spin parameter a ∗ := J/M = a/M ,which ranges from a ∗ = 0 . (blue solid line) to a ∗ = 0 . (redsolid line). Interestingly, the series converges very rapidly (essen-tially exponentially) and already with (cid:96) = 4 , the contribution ofhigher-order terms is of the order of − , decreasing further to ∼ − for (cid:96) = 6 . Furthermore, even when considering the moresevere test of a ∗ = 0 . , the expansion coefficient with (cid:96) = 6 is only a factor 2-3 larger than the corresponding coefficient for aslowly rotating black hole. The right-hand panel of Fig. 4 shows adirect measure of the relative differences between the polar curve c (cid:13)000
We start with the Kerr space-time, whose line element in Boyer-Lindquist coordinates reads ds = − (cid:18) − Mr Σ (cid:19) dt − aMr sin θ Σ dtdφ + (cid:18) r + a + 2 a Mr sin θ Σ (cid:19) sin θdφ + Σ∆ dr + Σ dθ , (55)where Σ := r + a cos θ , ∆ := r − Mr + a , (56)with M being the mass of the black hole and a := J/M its specificangular momentum.Since the shape of the shadow is ultimately determined bythe innermost unstable orbits of photons, hereafter we will con-centrate on their equations for photons. In such a space-time, thecorresponding geodesic equations take the form Σ (cid:18) dtdλ (cid:19) = AE − aMrL z ∆ , (57) Σ (cid:18) drdλ (cid:19) = R , (58) Σ (cid:18) dθdλ (cid:19) =Θ , (59) Σ (cid:18) dφdλ (cid:19) = 2 aMrE ∆ + (Σ − Mr ) L z ∆ sin θ , (60)where λ is an affine parameter, R := E r + (cid:0) a E − L z − Q (cid:1) r + 2 M (cid:2) ( aE − L z ) + Q (cid:3) r − a Q , (61) Θ := Q (cid:0) a E − L z csc θ (cid:1) cos θ , (62) A := (cid:0) r + a (cid:1) − a ∆ sin θ , (63)and E and L z are the photon’s energy and the angular momentum,respectively. The quantity QQ = p θ + cos θ (cid:18) L z sin θ − a E (cid:19) (64) is the so-called Carter constant and p θ := Σ dθ/dλ is the canonicalmomentum conjugate to θ .Using these definitions, it is possible to determine the unstableorbits as those satisfying the conditions R (¯ r ) = ∂ R (¯ r ) ∂r = 0 , and ∂ R (¯ r ) ∂r (cid:62) , (65)where ¯ r is the radial coordinate of the unstable orbit. Introducingnow the new parameters ξ := L z /E and η := Q /E , the celestialcoordinates α and β of the image plane of the distant observer aregiven by (Bardeen et al. 1972) α = ξ sin i , β = ± ( η + a cos i − ξ cot i ) / , (66)where i is the inclination angle of the observer’s plane, that is, theangle between the normal to the observer’s plane and the blackhole’s axis of rotation. In the case of the Kerr space-time (55) andafter using the conditions (65), one can easily find that the valuesof ξ and η relative to the circular orbit ( c ) are (Young 1976; Chan-drasekhar 1998) ξ c = M (¯ r − a ) − ¯ r (¯ r − M ¯ r + a ) a (¯ r − M ) , (67) η c = ¯ r [4 a M − ¯ r (¯ r − M ) ] a (¯ r − M ) , (68)Next, to investigate the shape of the black hole shadow weintroduce the generic celestial polar coordinates ( R (cid:48) , ψ (cid:48) ) (cf. Sect.2 and Fig. 1) defined as R (cid:48) =( α + β ) / , (69) ψ (cid:48) = tan − (cid:18) βα (cid:19) . (70)Assuming for simplicity that the observer is in the equatorial plane,i.e., that i = π/ , then in terms of the ( R (cid:48) , ψ (cid:48) ) coordinates theshadow of black hole can be described as (hereafter we will set M = 1 ) R (cid:48) = (2 r + 2 a r − r + a + a r ) / r − , (71) ψ (cid:48) = tan − (cid:18) { r [4 a − r ( r − ] } / r − a − r ( r − r + a ) (cid:19) . (72)Making use of the procedure described in Sect. 2, it is straight-forward to determine the coordinates (3)–(4) of the effective centreof the shadow, and to perform the Legendre expansion (13). To thebest of our knowledge, no analytic expression exists to cast the co-ordinates (71)–(72) as a polar curve R ψ = R ( ψ ) . However, such acurve can be easily constructed numerically and from it the Legen-dre expansion (13) can be computed.Figure 4 summarizes the results of our approach by report-ing in the left-hand panel and in a logarithmic scale the normalizedvalues of the expansion coefficients c (cid:96) as a function of the Legen-dre order (cid:96) . Different curves refer to the different values consid-ered for the dimensionless spin parameter a ∗ := J/M = a/M ,which ranges from a ∗ = 0 . (blue solid line) to a ∗ = 0 . (redsolid line). Interestingly, the series converges very rapidly (essen-tially exponentially) and already with (cid:96) = 4 , the contribution ofhigher-order terms is of the order of − , decreasing further to ∼ − for (cid:96) = 6 . Furthermore, even when considering the moresevere test of a ∗ = 0 . , the expansion coefficient with (cid:96) = 6 is only a factor 2-3 larger than the corresponding coefficient for aslowly rotating black hole. The right-hand panel of Fig. 4 shows adirect measure of the relative differences between the polar curve c (cid:13)000 , 1–12 Abdujabbarov, Rezzolla & Ahmedov
Figure 4.
Left-hand panel:
Magnitude of the expansion coefficients c (cid:96) of the polar curve and shown as a function of the expansion order. Note the veryrapid (exponential) convergence of the expansion. The coefficients are computed for a Kerr black hole and different lines refer to different values of the spinparameter a ∗ . Right-hand panel:
Relative differences between the polar curve for the black hole shadow as constructed from expressions (71)–(72) and thecorresponding curve obtained from the expansion, i.e., − (cid:80) n c (cid:96) P (cid:96) /R ψ . Different lines refer to different truncations of the expansion and show that threecoefficients are sufficient to obtain deviations of a few percent. Figure 5.
Magnitude as a function of the expansion order of the three different distortion parameters δ s, I – δ s, III defined as in Eqs. (29), (36), and (47), thusmeasuring the difference between the Legendre expanded polar curves R ψ, I – R ψ, III and the reference circles of radii R s, I – R s, III . All curves refer to a Kerrblack hole and different colours are used to represent different values of the spin parameter. for the black hole shadow as constructed from expressions (71)–(72) and the corresponding curve obtained from the expansion,i.e., − (cid:80) n c (cid:96) P (cid:96) /R ψ . Remarkably, already when considering thefirst three terms in the expansion, i.e., c , c , and c , the relativedifference is of a few percent (blue line), and this further reduces to − when the expansion is truncated at n = 4 (black line).In summary, Fig. 4 demonstrates that when considering a Kerrblack hole, the approach proposed here provides a coordinate inde-pendent and accurate representation of the black hole shadow andthat a handful of coefficients is sufficient for most practical pur-poses. In the following Sections we will show that this is the casealso for other axisymmetric black holes.Before doing that, we show in Fig. 5 the values of the dimen-sionless distortion parameters as computed for the shadow of a Kerrblack hole and for increasing values of the expansion index (cid:96) . Thethree different panels are relative respectively to the parameters (29), (36), and (47), with the different curves referring to valuesof the spin parameter a ∗ , ranging from a ∗ = 0 . (blue solid line)to a ∗ = 0 . (red solid line). As one would expect, for all values of a ∗ , each of the three distortion parameters decreases as the expan-sion includes higher-order terms. At the same time, because largerrotation rates introduce larger distortions in the shadow, they alsolead to larger values of the distortion parameters for a fixed valueof (cid:96) . Finally, Fig. 6 offers a comparative view of the different dis-tortion parameters for specific values of the spin parameter, withthe left-hand and right-hand panels referring to a ∗ = 0 . and a ∗ = 0 . , respectively. This view is rather instructive as it showsthat the different definitions lead to significantly different valuesof the distortion, despite they all refer to the same parametric polarcurve. Furthermore, it helps appreciate that the distortion parameter δ s, II is systematically smaller than the other two and hence not the c (cid:13) , 1–12 haracterization of a black hole shadow Figure 6.
Comparative view of the different distortion parameters δ s, I (red line), δ s, II (blue line), and δ s, III (light-blue line). The left-hand and right-handpanels show the values of the distortion parameters as a function of the expansion order (cf. Fig. 5), and refer to a Kerr black hole with spin a ∗ = 0 . and a ∗ = 0 . , respectively. optimal one. This is because a larger value of the distortion param-eter will increase the possibility of capturing the complex structureof the shadow. The fact that the curves for δ s, I and δ s, III intersectfor the Kerr black hole considered at (cid:96) = 5 implies that both dis-tortion parameters (29) and (47) are useful indicators, with δ s, III being the recommended choice for expansions having (cid:96) (cid:62) . We continue our application of the formalism developed in Sects. 2and 3 by considering the space-time of a rotating Bardeen blackhole (Bardeen 1968). We recall that in Boyer-Lindquist coordi-nates, the metric of a Kerr and of a Bardeen black hole differuniquely in the mass, which needs to be modified as (Bambi &Modesto 2013; Tsukamoto et al. 2014) M → m = M (cid:18) r r + g (cid:19) / , (73)where the parameter g is the magnetic charge of the nonlinear elec-trodynamic field responsible for the deviation away from the Kerrspace-time.The impact parameters ξ and η relative to the circular orbit arein this case (Tsukamoto et al. 2014) ξ c = m [(2 − f )¯ r − fa ] − ¯ r (¯ r − m ¯ r + a ) a (¯ r − fm ) , (74) η c = ¯ r { − f ) a m − ¯ r [¯ r − (4 − f ) m ] } a (¯ r − fm ) , (75)and can be taken to define the shadow of black hole. Note that m and f are functions of the unstable circular radius ¯ r and are givenby m = m (¯ r ) = M (cid:18) ¯ r ¯ r + g (cid:19) / , (76) f = f (¯ r ) = ¯ r + 4 g ¯ r + g . (77)In complete analogy, we can consider a Kerr-Taub-NUT black hole with nonvanishing gravitomagnetic charge n and specific an-gular momentum a := J/M . The corresponding metric is givenby (Newman et al. 1963) ds = − (cid:0) ∆ − a sin θ (cid:1) dt + Σ (cid:18) dr ∆ + dθ (cid:19) + 1Σ (cid:2) (Σ + aχ ) sin θ − χ ∆ (cid:3) dφ + 2Σ (∆ χ − a (Σ + aχ ) sin θ ) dtdφ, (78)where the functions ∆ , Σ , and χ are now defined as ∆ := r + a − n − Mr ,
Σ := r + ( n + a cos θ ) ,χ := a sin θ − n cos θ . (79)In this case, the impact parameters ξ and η for the circularorbits are given by (Abdujabbarov et al. 2013) ξ c = a (1 + ¯ r ) + ¯ r (¯ r −
3) + n (1 − r ) a (1 − ¯ r ) , (80) η c = 1 a (¯ r − (cid:26) ¯ r [4 a − ¯ r (¯ r − ] − n (cid:2) r a + (1 − r )( n (1 − r ) − r + 4 a ¯ r + 2¯ r ) (cid:3)(cid:27) , (81)and define the shadow of the Kerr-Taub-NUT black hole.Applying the formalism described in Sect. 2, it is possible tocompute the coefficients of the Legendre expansion (13) also for aBardeen and for a Kerr-Taub-NUT black hole. The numerical val-ues of these coefficients are reported as a function of the expansionorder (cid:96) in Fig. 7, where the left-hand panel refers to a Bardeenblack hole, while the right-hand one to a Kerr-Taub-NUT blackhole. More specifically, the different lines in the left-hand panel re-fer to different values of the magnetic charge: g = 0 . (red line), g = 0 . (blue line), and g = 0 . (light-blue line); all lines referto a fixed value of the rotation parameter a ∗ = 0 . . Very similar isalso the content of the right-hand panel of Fig. 7, which is howeverrelative to a Kerr-Taub-NUT black hole. The different curves in this c (cid:13) , 1–12 Abdujabbarov, Rezzolla & Ahmedov
Figure 7.
Left-hand panel:
Magnitude of the expansion coefficients c (cid:96) as a function of the expansion order (cid:96) for the different values of the magnetic chargeof a Bardeen black hole: g = 0 . (red line), g = 0 . (blue line), and g = 0 . (light-blue line). All lines refer to a fixed value of the rotation parameter a ∗ = 0 . (cf. left panel of Fig. 4). Right-hand panel:
The same as in the left-hand panel but for a Kerr-Taub-NUT black hole. Different curves refer to thedifferent values of the NUT parameter: n = 0 . (red line), n = 0 . (blue line), and n = 0 . (light-blue line). All lines refer to a fixed value of the rotationparameter a ∗ = 0 . . case refer to the different values of the NUT parameter: n = 0 . (red line), n = 0 . (blue line), and n = 0 . (light-blue line);once again, all lines refer to a fixed value of the rotation parameter a ∗ = 0 . .In analogy with what shown in the left-hand panel of Fig. 4for a Kerr black hole, also for these black holes the series con-verges very rapidly and already with (cid:96) = 4 , the contribution ofhigher-order terms is of the order of − , decreasing further to ∼ − for (cid:96) = 6 , even when considering higher larger mag-netic charges or NUT parameters. Furthermore, in analogy with theright-hand panel of Fig. 4, we have checked that the relative differ-ences between the polar curves for the shadow constructed fromexpressions (74)–(75) and (80)–(81), and the corresponding curveobtained from the expansion, i.e., − (cid:80) n c (cid:96) P (cid:96) /R ψ , is below − when n = 2 (not shown in Fig. 7); this difference further reducesto − when the expansion is truncated at n = 4 .In conclusion, also Fig. 7 demonstrates that the approach pro-posed here provides a coordinate independent and accurate repre-sentation of black hole shadows in space-times other than the Kerrone. All of the considerations made so far have relied on the assumptionthat the shadow is a well-defined one-dimensional curve (cf. dis-cussion in Sect 2). In practice, however, this is not going to be thecase as the astronomical observations will have intrinsic uncertain-ties that, at least initially, will be rather large. It is therefore naturalto ask how the formalism presented here will cope with such un-certainties. More precisely, it is natural to ask whether it will stillbe possible to determine the effective centre of a noisy polar curveand then determine from there its properties. Although a very ob-vious and realistic problem, this concern is systematically ignoredin the literature, where the shadow is traditionally assumed to haveno uncertainty due to the observational measurements.While awaiting for actual observational data, we can straight- forwardly address this issue in our formalism and mimic the nois-iness in the observational data by considering the polar curve asgiven by the Legendre expansion (13), where however the differ-ent coefficients c (cid:96) are artificially perturbed. More specifically, weexpress the shadow via the new expansion R ψ = ∞ (cid:88) (cid:96) =0 c (cid:96) (1 + ∆ c ) P (cid:96) (cos ψ ) , (82)where ∆ c is a random real number chosen uniformly in the range [ − ∆ max , ∆ max ] . In this way, our putative polar curve represent-ing the shadow will be distorted following a random distribu-tion and we have optimistically assumed a variance of only 5%,i.e., ∆ max = 0 . . Of course, there is no reason to expect that theerror distribution in the actual observational data will be uniform,but assuming a white noise is for us the simplest and less arbitrarychoice.With the setup described above and the formalism discussedin the previous Sections, we have considered a reference shadow ofa Kerr black hole with spin parameter a/M = 0 . and have repro-duced it after truncating the expansion (82) at (cid:96) = 9 , which is morethan sufficient given the accuracy obtained at this order (cf. Fig. 4).We have therefore constructed a very large number of such realiza-tions of the observational shadow after making use of N tot = 10 draws of the random deformation ∆ c . For each putative observedreconstructed shadow we have computed the distortion parameters δ s, I − δ s, III defined in Eqs. (29), (36), and (47), as well as the dis-tortion definition of Hioki & Maeda (2009) and defined in Eq. (48).For each of the shadow realizations we have therefore com-puted the measurement error as (cid:15) ∗ := δ s − δ s, ∗ , (83)where δ s is the exact distortion of the background Kerr solutionand measuring the relative difference of the shadow at ψ = 0 and ψ = π . On the other hand, δ s, ∗ is given by either δ s, I − δ s, III or δ s, HM .Figure 8 shows the distributions of the errors computed in this c (cid:13) , 1–12 haracterization of a black hole shadow Figure 8.
Comparison of probability density distributions of the errors (cid:15) ∗ computed in the measurement of the distortion parameters δ s, I − δ s, III fora Kerr black hole shadow reconstructed using a perturbed expansion [cf. Eq.(82)]. Also shown is the distribution of the error measured when using thedistortion parameter introduced by Hioki & Maeda (2009) and that has alarger variance. way for the four different possible definitions of the distortion pa-rameters, with the black line referring to the distortion parameterin Eq. (48), and the red, blue and light-blue lines referring to thedefinitions (29), (36), and (47), respectively. Note that the valuesof the probability densities distributions are reported in such a waythat N tot (cid:90) ∞−∞ dn = 1 N tot (cid:90) ∞−∞ (cid:18) dnd(cid:15) ∗ (cid:19) d(cid:15) ∗ = 1 . (84)The distributions reported in Fig. 8 are rather self-explanatory.Clearly, all the different definitions are centred on (cid:15) ∗ = 0 , indicat-ing that on average they provide a good measurement of the distor-tion. On the other hand, the variance of the different distribution israther different. Overall, the distortion parameters δ s, I − δ s, III havecomparable variances, with a slightly smaller variations for the def-inition δ s, I . However, the variance of the distortion parameter for δ s, HM is almost twice as large as the others and it essentially spansthe variation that we introduce in the random distortion ∆ c .These results are rather reassuring as they indicate that new defini-tions are not only accurate, but also robust with respect to randomwhite noise. Furthermore, they appear to be superior to other dis-tortion measurements suggested in the past.As a final remark we note that the introduction of the pertur-bations in the expansion (82) also has the effect of changing theposition of the effective centre of the shadow and hence the valuesof (cid:126) R and ψ [cf. Eqs. (2) and (4)]. Fortunately, such variationsrepresent only a high-order error, which is much smaller than thosemeasured by the distortion parameters, with maximum measuredvariance of the order of − . As a result, the distortions reportedin Fig. 8 are genuine measurements of the shadow and not artefactsintroduced by the changes in the effective centres. The radio-astronomical observations of the shadow of a black holewould provide convincing evidence about the existence of blackholes. Further, the study of the shadow could be used to learn aboutextreme gravity near the event horizon and determine the correcttheory of gravity in a regime that has not been explored directly sofar. A number of different mathematical descriptions of the shadowhave already been proposed, but all make use of a number of sim-plifying assumptions that are unlikely to be offered by the real ob-servational data, e.g., the ability of knowing with precision the lo-cation of the centre of the shadow.To remove these assumptions we have developed a new gen-eral and coordinate-independent formalism in which the shadow isdescribed as an arbitrary polar curve expressed in terms of a Leg-endre expansion. Our formalism does not presume any knowledgeof the properties of the shadow and offers a number of routes tocharacterize the properties of the curve. Because our definition ofthe shadow is straightforward and unambiguous, it can be imple-mented by different teams analysing the same noisy data.The Legendre expansion used in our approach converges ex-ponentially rapidly and we have verified that a number of coeffi-cients less than ten is sufficient to reproduce the shadow with aprecision of one part in , both in the case of a Kerr black holewith spin parameter of a/M = 0 . , and in the case of Bardeenand Kerr-Taub-NUT black holes with large magnetic charges andNUT parameters. Furthermore, the use of a simple Legendre ex-pansion has allowed us to introduce three different definitions ofthe distortion of the shadow relative to some reference circles. Thecomparison of the different definitions has allowed us to determinewhich of them is best suited to capture the complex structure of theshadow and its distortions.Finally, again exploiting the advantages of the Legendre ex-pansion, we were able to simulate rather simply the presence ofobservational random errors in the measurements of the shadow.Constructing a large number of synthetically perturbed shadows,we have compared the abilities of the different parameters to mea-sure the distortion in the more realistic case of a noisy shadow.Overall, we have found that our new definitions have error distri-butions with comparable variances, but also that these are about afactor of 2 smaller than the corresponding variance measured whenusing more traditional definitions of the distortion. Given these re-sults, the approach proposed here could be used in all those studiescharacterizing the shadows of black holes as well as in the analysisof the data from experimental efforts such as EHT and BHC. ACKNOWLEDGEMENTS
It is a pleasure to thank A. Grenzebach, Y. Mizuno, Z. Younsi, andA. Zhidenko for useful discussions and comments. We are alsograteful to the referee, D. Psaltis, for comments and suggestionsthat have improved the presentation. This research was partiallysupported by the Volkswagen Stiftung (Grant 86 866) and by theERC Synergy Grant “BlackHoleCam – Imaging the Event Horizonof Black Holes” (Grant 610058). AAA and BJA are also supportedin part by the project F2-FA-F113 of the UzAS and by the ICTPthrough the projects OEA-NET-76, OEA-PRJ-29. AAA and BJAthank the Institut f¨ur Theoretische Physik, Goethe Universit¨at forwarm hospitality during their stay in Frankfurt. c (cid:13) , 1–12 Abdujabbarov, Rezzolla & Ahmedov
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APPENDIX A: GRAPHICAL REPRESENTATION OFDISTORTION PARAMETERS: KERR BLACK HOLES
To help in the visualization of the different distortion parametersintroduced in Section 3, we here present their graphical represen-tation when they are applied to the shadow of a Kerr black holeshadow with spin parameter a ∗ = 0 . . The left-hand panel ofFig. A1 represents the distortion parameter I in Sect. 3, where thereference circle is defined in a such a way that the points B, D and the centre of the reference circle are all on the vertical axis β .Hence, the difference of the radial coordinates of the point C of theshadow boundary and of the left-hand point of the circle intersect-ing the axis α corresponds to the distortion parameter d s, I .Similarly, the middle panel of Fig. A1 refers to the distortionparameter II, where the reference circle passes through the points A and B , which are on the axis β . The centre of the reference cir-cle is on the point E and does not coincide with the centre of thecoordinate system. The position of the reference circle centre E isinstead defined by Eq. (31). The difference of the radial coordinatesof the point C of the shadow and of the left-hand point of the circleintersecting with the axis α corresponds to the distortion parameter d s, II .Finally, the right-hand panel of Fig. A1 corresponds to the dis-tortion parameter III, where the reference circle passes through theslope points S, S (cid:48) , and the right-hand point A of the shadow in-tersecting the axis α . The positions of the slope points are definedby solving Eq. (39). The centre of the reference circle does not co-incide with the origin of the coordinate system and its position isdefined by Eq. (43). The difference of the radial coordinates of thepoint C of the shadow and of the left-hand point of the circle inter-secting the axis α corresponds to the distortion parameter d s, III .Note that the radius of the reference circle radius depends onthe definition used for the distortion parameter; hence the distortionparameters are also different for each definition. c (cid:13) , 1–12 haracterization of a black hole shadow Figure A1.
Schematic representations of the distortion parameters I, II and III when applied to the shadow of a Kerr black hole. The left-hand panel refers tothe definition I, where the centre of the reference circle as well as the points B and D (which are not slope points) are on the coordinate axis β . The middlepanel refers instead to definition II and in this case the centre of the reference circle E is displaced along the α axis. Finally, the right-hand panel shows therepresentation of the definition III, where we consider the reference circle passing through the point A and the slope points B and D are not on the β axis.The centre of the reference circle E is also displaced and not exactly at the centre of the coordinate system.c (cid:13)000