Cuspy and fractured black hole shadows in a toy model with axisymmetry
Wei-Liang Qian, Songbai Chen, Cheng-Gang Shao, Bin Wang, Rui-Hong Yue
aa r X i v : . [ g r- q c ] F e b Cuspy and fractured black hole shadows in a toy model withaxisymmetry
Wei-Liang Qian , , , Songbai Chen , Cheng-Gang Shao , Bin Wang , , and Rui-Hong Yue Center for Gravitation and Cosmology,College of Physical Science and Technology,Yangzhou University, 225009, Yangzhou, China Institute for theoretical physics and cosmology,Zhejiang University of Technology, 310032, Hangzhou, China Escola de Engenharia de Lorena, Universidade de S˜ao Paulo, 12602-810, Lorena, SP, Brazil Institute of Physics and Department of Physics,Hunan Normal University, 410081, Changsha, Hunan, China MOE Key Laboratory of Fundamental Physical Quantities Measurement,Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF, and School of Physics,Huazhong University of Science and Technology, 430074, Wuhan, Hubei, China and School of Aeronautics and Astronautics,Shanghai Jiao Tong University, 200240, Shanghai, China (Dated: Feb. 2nd, 2021)Cuspy shadow was first reported for hairy rotating black holes, whose metricsdeviate significantly from the Kerr one. The non-smooth edge of the shadow isattributed to a transition between different branches of unstable but bounded orbits,known as the fundamental photon orbits, which end up at the light rings. In searchingfor a minimal theoretical setup to reproduce such a salient feature, in this work,we devise a toy model with axisymmetry, a Kerr black hole enveloped by a thinrotating dark matter shell. Despite its simplicity, we show rich structures regardingfundamental photon orbits explicitly in such system. We observe two disconnectedbranches of unstable spherical photon orbits, and the jump between them gives riseto a pair of cusps in the resultant black hole shadow. Besides the cuspy shadow,we explore other intriguing phemomena when the Maxwell construction cannot beestablished. We find that it is possible to have an incomplete arc of Einstein rings anda “fractured” shadow. The potential astrophysical significance of the correspondingfindings is addressed.
I. INTRODUCTION
The bending of light rays owing to the spacetime curvature constitutes one of the mostinfluential predictions of General Relativity. At its extreme form, the shadow casted bya black hole [1–6] is widely considered as an essential observable in the electromagneticchannel. As the boundary of a black hole shadow is determined by the critical gravitationallensing of the radiation from nearby celestial bodies, it bears crucial information on spacetimegeometry around the black hole. With the prospect to directly probe the underlying theoryof gravity in the strong-field region, the topic has aroused much renewed curiosity in thepast decade [7–19] (for a concise review of the topic, see [20]). In particular, it is beingtargeted by the Event Horizon Telescope [21–23], whose recent developments open up a newavenue with promising possibilities.The black hole shadow is defined by the set of directions in the observer’s local skywhere the ingoing null geodesics are originated from the event horizon. In other words,no radiation is received by the observer at a certain solid angle due to the presence of theblack hole. Intuitively, the shape of the black hole shadow can be derived by analyzingthe lower bound of the free-fall orbits that circulating the black hole in a compact spatialregion. Such a bound is closely associated with a specific type of null geodesics, dubbedfundamental photon orbits (FPOs) [5, 11], first proposed by Cunha et al. . One may, by andlarge, argue that the edge of the shadow is furnished by the collection of the light rays thatbarely skim the unstable FPOs. This is because a null geodesic that slightly deviates from anunstable FPO might marginally escape to the spatial infinity after orbiting the black hole fora multitude of times. When tracing back in time, it is either originated from the black holehorizon or emanated by some celestial light source. While the former constitutes part of theshadow by definition, a light ray associated with the latter, on the other hand, contributesto the image of the relevant celestial body. In practical calculations, background radiationsources are placed on a sphere, referred to as the celestial sphere [2], (infinitely) further awayfrom both the observer and the black hole. Owing to the significant gravitational lensing,an infinite number of (chaotic) images of the entire celestial sphere pile up in the vicinity ofthe shadow edge [24–27].In the case of the Schwarzschild black hole, the relevant FPOs are the light rings (LRs),forming a photon sphere. While for the Kerr one, the role of the FPO is carried by thespherical orbits [28]. The LRs are circular planar null geodesics, which by definition, is aparticular type of FPOs associated with the axisymmetry of the relevant spacetimes. Inparticular, it is understood that unstable LRs play a pivotal role in the strong gravitationallensing as well as shadow formation [5, 11, 25, 29, 30]. Stable LRs, being rather contrary tothe nomenclature, might leads to the accumulation of different modes when the spacetimeis perturbed [31]. Such a system is subsequently prone to nonlinear instabilities [32]. In thecase of the Kerr black hole, the two LR solutions, restricted to the equatorial plane, are bothunstable. From the observer’s viewpoint, on the shadow edge, they mark the two endpointsin the longitudinal direction. The analyses of the black hole shadow in Kerr spacetime aresimplified by the fact that the corresponding FPOs are of constant radius. This is becausethe geodesic is Liouville integrable and separates in the Boyer-Lindquist coordinates [33].In a generically stationery and axisymmetric spacetime, however, the separation of vari-ables is often not feasible for the geodesic motion by choosing a specific coordinate chart.As a result, the FPOs become more complicated and have to be evaluated numerically.Nonetheless, it was pointed out [11] the stability of LRs can be studied by employing thePoincar´e maps. Recently, Kerr black hole metrics with Proca hair was investigated, and aquantitatively novel shadow with cuspy edge was spotted [11]. Instead of a smooth shadow,the black hole silhouette is characterized by a pair of cusps at the boundary. The authorsattributed the above feature to the sophisticated FPO structure, and in particular, to aninterplay between stable and unstable FPOs. To be specific, when compared with the caseof Kerr metric, an additional stable branch of FPOs appears, which attaches both of its endsto that of two unstable branches of FPOs. Consequently, a point of the cusp corresponds toa sudden transition between two FPOs from those unstable branches. More lately, a similarcharacteristic was also reported [12] in rotating non-Kerr black holes [34]. For both cases,the metrics involved are rather complicated and possess a stable branch of FPOs. Apartfrom the existing efforts, it still seems not very clear what is a minimal theoretical setup toreproduce a cuspy shadow edge.If instead of vacuum, the black hole is surrounded by an accretion disk, a trespassing pho-ton is likely subjected to inelastic scatterings. As a result, it will deviate from its geodesicor even be entirely absorbed by an opaque disk. However, if the disk is composed purelyof dark matter, it is transparent to the photon. This is because no observational signatureregarding the interaction between the photon and dark matter particles has yet turned up,in any experiment designated to direct dark matter detection. Nonetheless, the gravitationaleffect of the dark matter may still impact the null geodesics, and subsequently, the resultantblack hole shadow. For a spherical galactic black hole surrounded by a thick dark mattercloud, it was argued that observable deviation from its Schwarzschild counterpart mightbe expected [17]. More lately, in studying the rotating dirty black holes [18], the authorsfound that although the existence of the dark matter modifies the size of the shadow, theD-shaped contour almost remains unchanged. On the other hand, although the physical na-ture remains largely elusive, in literature, many interesting substructures in the dark matterhalos and sub-halos have been speculated [35–39]. Among a large variety of alternatives,the venerable ΛCDM model indicates that a discontinuity in the matter distribution mightbe triggered by the presence of the dark matter [37, 38]. In this context, the exploration ofthe rich substructure regarding the dark matter halo is closely related to our understandingof the underlying physics. Indeed, the resultant substructures predicted by the theoreticalmodels may, in turn, serve to discriminate between different interpretations about the natureof dark matter. In this regard, it is not clear whether a discontinuity in the matter distribu-tion may further appreciably distort the black hole shadow, particularly to the extent suchmodification becomes potentially observable.In fact, recently, it was shown analytically [40] that a discontinuity in the effective poten-tial significantly affects the asymptotic properties of quasinormal modes. As the last phaseof a merger process, such a dramatic change in the quasinormal ringing may potentially leadto observable effects. In fact, discontinuity is also present at the surface of compact celestialobjects, and the numerical calculations of the curvature modes have indeed confirmed sucha nontrivial consequence [41]. Considering that the quasinormal frequencies at the eikonallimit are closely connected with the shadow and photon sphere in spherical metrics [15, 42],it is natural to ask whether some discontinuity out of the horizon of the black hole mightlead to meaningful implications in the context of the black hole shadow. This is the primarymotivation of the present study.The present study continues to pursue further discussions on black hole shadows con-cerning the role of dark matter surrounding the black holes. By simplifying the dark matterenvelope to a thin shell wrapped around the black hole, the mass distribution is concen-trated in an infinitesimal layer so that we have a sharp discontinuity in the effective po-tential. We will show that for such stationary axisymmetric configuration, the spacetimepossesses two branches of unstable FPOs but not any stable FPO. Our analysis reveals asudden jump between the different branches, which results in a cusp on the boundary ofthe black hole shadow. Moreover, it is argued that the transition point can be determinedusing the Maxwell construction, which is reminiscent of the Gibbs conditions for the phasetransition. We also investigate other intriguing possibilities regarding different model param-eterizations, which include the cases involving an incomplete arc of the Einstein rings andfractured shadow edge. The astrophysical significance of the present findings is addressed.The remainder of the manuscript is organized as follows. In the next section, we presentour model and the mathematical framework for evaluating the null geodesics as well asthe associated celestial coordinates. We study the properties of the FPOs and discuss therelevant criterion to determine the black hole shadow. In Sec. III, for a specific choice ofthe metric, we show that the Maxwell construction can be utilized to locate the transitionpoint on the shadow edge, where the cusp is present. The discussions also extend to othermeaningful metric parameterizations, where one elaborates on two additional intriguingscenarios. Further discussions and concluding remarks are given in the last section.
II. A DARK MATTER SHELL TOY MODEL
In this section, we first present the proposed toy model and then proceed to discuss theFPOs of the relevant metric as well as their connection with the black hole shadow. Forthe purpose of the present study, we consider a stationary axisymmetric metric with thefollowing form in the Boyer-Lindquist coordinates ( t, r, θ, ϕ ) ds = − (cid:18) − M r Σ (cid:19) dt − M ar sin θ Σ dtdϕ + Σ∆ dr + Σ dθ + (cid:18) r + a + 2 M a r sin θ Σ (cid:19) sin θdϕ , (1)where ∆ = r − M r + a , Σ = r + a cos θ, and M = M BH r ≤ r sh M TOT r sh < r < ∞ ,a = a BH r ≤ r sh a TOT r sh < r < ∞ , (2)where M BH , a BH , M TOT and a TOT are constants, r sh is the location of thin layer of darkmatter. For both the regions r ≤ r sh and r sh < r < ∞ , the metric coincides with that of aKerr one, satisfying the Einstein’s equation in vaccum.Inside the dark matter envelop, the above spacetime metric describes a Kerr black holewith the mass M = M BH and angular momentum J = a BH M BH sitting at the center.For an observer sitting far away ( r ≫ r sh ), they are essentially dealing with a rotatingspacetime with the mass M = M TOT and angular momentum per unit mass a = a TOT .The discontinuity at r = r sh indicates a rotating (infinitesimally) thin shell of mass M sh = M TOT − M BH wrapping around the central black hole. We note that the above metric can bereadily derived as the limit of the case, which possesses a continuous matter distribution [18].The latter can be obtained by employing the Newman-Janis algorithm [43, 44].The null geodesic of a photon in a pure Kerr spacetime satisfying the following systemof equations [28] Σ dtdλ = ( a + r )(( a + r ) E − aL )∆ + aL − a E sin θ, Σ dϕdλ = a (( a + r ) E − aL )∆ + L csc θ − aE, Σ drdλ = ±√ R, Σ dθdλ = ±√ Θ , (3)where R = − ∆( Q + ( L − aE ) ) + (( a + r ) E − aL ) , Θ = Q + cos θ (cid:18) a E − L sin θ (cid:19) . Here E , L , and Q are the energy, angular momentum, and the Carter constant of thephoton. For our present case, the analysis of the null-geodesic motion can be achievedby implementing a simple modification. For light rays propagating inside a given region,namely, r < r sh or r > r sh , its motion is governed by Eq. (3) with the metric parametersgiven by Eq. (2). The difference occurs when the light ray crossing the dark matter layer at r = r sh . For instance, let us consider a free photon that escapes from the inside of the thinshell. Due to the singularity in the derivatives of the metric tensor at r = r sh , the photon’strajectory will suffer a deflection as it traverses the shell. However, the values of E , L ,and Q remain unchanged during the process and thus can be utilized to unambiguouslymatch the geodesics on both sides of the shell. This is because these constants of motion arederived by the corresponding Killing objects implied by the axisymmetry of the spacetimein question. Subsequently, when given one point on the trajectory, a null-geodesic motionis entirely determined by a pair of values, η = LE ,ξ = QE . (4)This can be easily seen by rescaling the affine parameter λ → λ ′ = λE in Eq. (3). Moreover,due to the axisymmetry, the separation of variables is still feasible for the present case,and therefore, all the FPOs are spherical orbits as for the Kerr metric. The spherical orbitsolution can be obtained by analyzing the effective potential associated with the radialmotion, which separates from those of angular degrees of freedom. To be specific, the thirdline of Eq. (3) can be rewritten as Σ ˙ r + V eff = 0 , (5)where the effective potential V eff reads V eff E = (cid:0) a + r − aξ (cid:1) − (cid:0) r + a − r (cid:1) (cid:0) η + ( ξ − a ) (cid:1) . (6)Similar to the analysis of the planetary motion in Newtonian gravity, the spherical orbitsare determined by the extremum of the effective potential, namely, V eff = ∂ r V eff = 0. Onefinds η = r − M r + a r + M a a ( M − r ) ,ξ = r (3 r + a − η ) r − a , (7)where r is the radius of the spherical orbit. These orbits are unstable since the encounteredextremum is a local maximum. Moreover, the fact that all FPO are spherical orbits is relatedto the uniqueness of the above local maximum.On the other hand, for an observer located at (asymptotically flat) infinity with zenithalangle θ , the boundary of the black hole is governed by those null geodesics that marginallyreach them. By assuming that the entire spacetime is flat, the “visual” size of the blackhole can be measured by slightly “tilting their head” (or in other words, by an infinitesimaldisplacement of their location). To be specific, when projected onto the plane perpendicularto the line of sight, the size of the image in the equatorial plane and on the axis of symmetrycan be obtained by the derivatives of the angular coordinates ( ϕ, θ ) [2]. These derivativescan be calculated explicitly using the asymptotical form of the geodesic, namely, Eq. (3)evaluated at the limit r → ∞ . We have α = lim r →∞ − r sin θ dϕdr (cid:12)(cid:12)(cid:12)(cid:12) θ → θ ! = − η csc θ ,β = lim r →∞ r dθdr (cid:12)(cid:12)(cid:12)(cid:12) θ → θ ! = ± q ξ + a cos θ − η cot θ , (8)where the pair ( η, ξ ) are dictated by the geodesic of the photon in question. The coordinatesin terms of α and β are often referred to as the celestial coordinates in the literature [2]. Bycollecting all coordinate pairs ( α, β ), one is capable of depicting the apparent silhouette ofthe black hole.As discussed above, the relevant null geodesics that potentially contribute to the shadowedge are the FPOs. In contrary to the evaluation of the celestial coordinates, which involvesthe asymptotic behavior of the metric, the FPOs are determined by spacetime properties inthe vicinity of the horizon. Although all the FPOs for our metric are spherical orbits, thepresence of a rotating thin shell leads to some interesting implications.In what follows, let us elaborate on the properties of the FPO and their connection withthe black hole shadow. First, consider a FPO solution for the pure Kerr spacetime with M = M BH , a = a BH . It will also be qualified as a FPO for the metric defined in Eq. (2), ifand only if the radius of the corresponding spherical orbit r satisfies r < r sh . Likewise, aFPO solution with r < r sh for the pure Kerr spacetime with M = M TOT , a = a TOT doesnot exist physically for the metric under consideration.Secondly, we note that not every
FPO contributes to the edge of the black hole shadow.Let us consider, for instance, a photon moves along a spherical orbit right outside the shellwith r = r sh + ε . When its trajectory is perturbed and let us assume that the photon spiralsslightly inward. As the photon conserves its values of ( η, ξ ), at the moment it intersects theinfinitesimally thin shell, the trajectory is promptly deflected from the tangential directionperpendicular to the radius. This implies that it no longer stays in the vicinity of anyspherical orbits, namely, the FPO for the region r < r sh . Subsequently, the photon willspiral into the event horizon rather quickly instead of critically orbiting the black hole foran extensive number of times beforehand. This, in turn, indicates the photon is mappedonto a pixel disconnected from those associated with the FPOs of the region r < r sh , whichconstitute the shadow edge. The above heuristic arguments can be restated in terms ofthe fact that the pair of values ( η, ξ ) for an FPO of the outer region r > r sh does not, ingeneral, corresponds to that of an FPO of the inner region r < r sh . Therefore, the photonwhich skims the thin layer of dark matter on the outside, by and large, does not contributeto the edge of the black hole shadow. Now, one may proceed to consider a peculiar case,where the pair of values ( η, ξ ) for an FPO in the outer region matches that of an FPO inthe inner region. Therefore when the trajectory of the former is perturbed and the photoneventually traverses the thin shell, it will still stay in the vicinity of the latter and eventuallycontributes to the edge of the shadow. Since the values of ( η, ξ ) for both FPO are the same,according to Eq. (8), they also contribute to the same pixel in the celestial coordinates.This is precisely the Maxwell condition that we will explore further in the next section. It isworth noting that, even if an FPO does not directly contribute to the shadow edge, it is stillsubjected to strong gravitational lensing and therefore possibly leads to a nontrivial effect.Moreover, we note that the inverse of the above statement is still valid. In other words,the edge of the black hole shadow is entirely furnished by the FPOs in either region ofthe spacetime. If some FPOs in the outer region r > r sh contributes to the shadow edge,the section of the shadow boundary is identical to those of a Kerr black hole with M = M TOT , a = a TOT . However, if some FPOs in the inner region r < r sh contributes to theshadow edge, due to Eq. (8), the corresponding section of the black hole silhouette is differentfrom that of the Kerr black hole that sits inside the thin shell.Before proceeding further, we summarize the key features regarding the FPOs in thepresent model and their connection with the black hole shadow edge as follows • The null-geodesic motion is determined by a pair of conserved quantity ( η, ξ ). • The black hole shadow is a projection of asymptotic light rays onto a plane perpen-dicular to the observer’s line of sight, and any point on its edge is governed by thetwo-dimensional orthogonal (celestial) coordinates consisting of ( α, β ). • The boundary of the black hole shadow is determined entirely by the unstable FPOs,but some FPO may not contribute to the shadow edge. • Due to the presence of the thin shell, some formal FPO solutions for the pure Kerrspacetime are not physically relevant. • When the values of ( η, ξ ) of a particular FPO on one branch match those of anotherFPO on a different branch, both FPOs contribute to the same point in the celestialcoordinates, probably on the shadow edge.
III. THE MAXWELL CONSTRUCTION AND BLACK HOLE SHADOW
In the last section, we discuss the close connection between the unstable FPOs and theblack hole shadow edge. It is pointed out that the null-geodesic motion can be determinedin terms of the pair of values ( η, ξ ). As this is the same number of degrees of freedom tolocate a specific point on the celestial coordinates, the dual ( η, ξ ) of an FPO can be usedto map onto the corresponding point on the shadow edge in the celestial coordinates. Tobe specific, the transition point on the shadow edge can be identified by matching ( η, ξ ) fortwo FPOs from different branches, namely, η ( r cuspBH ) = η ( r cuspTOT ) ,ξ ( r cuspBH ) = ξ ( r cuspTOT ) , (9)where r cuspBH < r cuspTOT are two distinct FPO solutions, belong to the two distinct unstablebranches of FPOs. Since the established condition is between two sets of quantities, it isreminiscent of the Maxwell construction (e.g. in terms of the chemical potentials) in a two-component system [45, 46]. On a rather different ground, such a construction was derivedfrom the Gibbs conditions for the phase transition in a thermodynamic system. For thepresent context, the pair ( η, ξ ) determined by Eq. (9) is mapped to ( α, β ) in the celestialcoordinates, which subsequently gives rise to a cusp on the shadow edge. Such a salientfeature is similar to what has been discovered earlier [11, 12] using more sophisticated blackhole metrics.The present section is devoted to investigating different scenarios emerging from the pro-posed model. We show that, due to an interplay between different branches of unstableFPOs and the location of the discontinuity introduced by the thin shell, the resultant blackhole shadow presents a rich structure. The following discussions will be primarily concen-trated on three sets of model parameters, given in Tab. I. The choice of the parameters aimsat enumerating all relevant features in the present model, in terms of the feasibility of theMaxwell construction, as well as the different roles carried by the FPO. In particular, in thefirst case, the Maxwell construction can be established. Besides, the parameters are chosenso that unstable FPOs contribute both to black hole shadow edge and metastable states,after the unphysical ones are excluded. In the other two cases, on the other hand, one can-not find such a transition between different branches of FPOs via the Maxwell construction.However, two physically interesting scenarios are observed for these cases. The second set ofparameters leads to an incomplete section of Einstein rings, while the third set gives rise toa fractured black hole shadow. For simplicity, we keep most of the parameters unchanged.For all three cases, we set M BH = 1 . a BH = 0 .
95, and M TOT = 1 .
1. The only differencebetween the first and second cases is the value of a TOT , while that between the second andthird cases is the value of r sh . By using these parameters, the four rightmost columns listthe calculated radii of LRs. The latter correspond to the radial bounds for the sphericalorbits if there were no constraints associated with the thin shell.We first consider the first set of model parameters given in Tab. I, and the calculatedcuspy black hole shadow is shown in Fig. 1. To give a more transparent presentation, forthe figures, we adopt the following conventions. The FPOs associated with the edge of theblack hole shadow are shown by solid curves. Meanwhile, the FPOs that are valid nullgeodesics of the metric but irrelevant to the shadow are depicted in dashed curves. The graydotted curves are unphysical FPO solutions. As discussed in the last section, they must beexcluded due to the physical constraint related to the thin shell. It is observed that the set M BH a BH M TOT a TOT r sh r − BH r +BH r − TOT r +TOT . .
95 1 . .
25 3 .
38 1 .
386 3 .
955 2 .
997 3 . . .
95 1 . .
85 4 .
00 1 .
386 3 .
955 2 .
057 4 . . .
95 1 . .
85 3 .
35 1 .
386 3 .
955 2 .
057 4 . r − BH , r +BH and r − TOT , r +TOT , for the regions r < r sh and r > r sh of the metric, respectively. r η ( ξ ) Maxwellconstruction - - α - - - β cuspshadow shadowunphysical metastable Fig. 1. (Color online) The Maxwell construction and the corresponding black hole shadow withcusp. The blue curves denote the FPOs associated with the metric for the region r < r sh , while theorange ones are those for the region r > r sh . The solid blue and orange curves (labeled “shadow”)are the collections of FPOs that contribute to the shadow edge. The dashed blue (barely visible)and orange curves (labeled “metastable”) represent those FPOs that do not directly give rise to theshadow edge. The dotted gray curves (labeled “unphysical”) correspond to the FPO solutions thatare not physically permitted. Left: The Maxwell construction, shown in the dashed red rectangle,establishes the transition point between the two branches of unstable FPOs in terms of η and ξ as functions of orbit radius r . The curves with dark colors (dark blue and dark orange) are for η = η ( r ), while those with light colors (light blue and light orange) are for ξ = ξ ( r ). The unstableFPOs excluded from the shadow edge by the Maxwell construction are denoted as “metastable”due to their resemblance to the metastable states in a thermodynamical system. Right: Thecorresponding shadow edge is shown in solid blue and orange curves, where the transition point islabeled by “cusp”. The eyelash shape extension of the shadow edge, shown in dashed curves, maystill lead to a strong gravitational lensing effect. resultant spacetime is featured by two disconnected branches of unstable FPOs. The FPOsassociated with the inner region ( r < r sh ) are shown in solid and dashed blue curves. Thoseassociated with the outer region ( r > r sh ) are represented by solid and dashed orange curves.As shown in the left plot of Fig. 1, the Maxwell construction, Eq. (9), is indicated by thedashed red rectangle. It corresponds to the transition point (labeled “cusp”) in the rightplot. However, it is worth noting that, different from previous studies [11, 12], the present0 r η ( ξ ) No Maxwell construction - - α - - - β shadow metastableunphysical Fig. 2. (Color online) The black hole shadow surrounded by an incomplete section of Einsteinrings. We adopt the same convention for the line types and colors as Fig. 1. Left: No Maxwellconstruction can be established for this case. Right: The resultant black hole shadow edge is shownby the closed solid curve. The unstable FPOs, associated with the metric in the region r > r sh ,form a section of arc, which might lead to an incomplete ring. metric does not possess any stable FPO. Therefore, the latter is not a necessary conditionfor the presence of the cusp.The Maxwell construction give r cuspBH ≃ . < r cuspTOT ≃ . r sh = 3 . r − TOT < r sh < r cuspTOT and r cuspBH < r sh < r +BH , implies that the “eyelash” is present for both branches after theremoval of unphysical FPOs. For instance, the dashed orange eyelash shown in the left plotof Fig. 1 corresponds to the FPOs with their orbital radii r cuspBH < r < r sh .Now, we move to consider the other two scenarios where the Maxwell construction cannotbe encountered. In both cases, the resulting black hole shadow does not possess any cusp,but still, noticeable features are observed. In Fig. 2, we present the results obtained for thesecond set of metric parameters given in Tab. I. From the left plot, the Maxwell constructioncan not be established, and the resultant boundary of the black hole shadow is subsequentlydetermined by the metric of the Kerr black hole sitting at the center. However, since r sh
IV. FURTHER DISCUSSIONS AND CONCLUDING REMARKS
To summarize, in this work, we showed that rich features concerning the black hole shadowcan be obtained using a simple but analytic toy model. The model we devised consists ofa thin shell of rotating dark mass wrapping around a Kerr black hole while preserving theaxisymmetry of the system. It is found that the resulting metric possesses two disconnectedbranches of unstable FPOs. Moreover, their interplay with the location of the dark matterlayer leads to various features such as the cuspy and fractured black hole shadow edge. Interms of the Maxwell construction, an analogy was made between the transition amongdifferent branches of FPOs and that occurs in a thermodynamic system. In particular, wehave investigated three different spacetime configurations aiming at illustrating exhaustivelyall the features of the present model. The first set of parameters is designated to the casewhere the Maxwell construction can be established. The parameters are particularly chosenso that both unstable branches contribute to form an enclosed shape, which subsequentlydefines the contour of the black hole shadow. The point of transition corresponds to a pairof cusps on the shadow edge. Moreover, the remaining FPOs are mapped onto the eyelashshape extension of shadow edge on the celestial coordinates, reminiscent of the metastablestates in a thermodynamical system. The other two sets of parameters are dedicated tothe cases where the Maxwell construction cannot be established. The second set leads toa scenario where the shadow, solely defined by one branch of FPOs related to the Kerrblack hole sitting at the center, is enclosed by an incomplete arc of Einstein rings. For thethird set of parameters, again, both unstable branches of FPOs contribute to the shadow.However, since there is no Maxwell construction, the two sections of the shadow edge areapparently disconnected, giving rise to a fractured shadow edge. The above choices of metricparameters are representative for different physical outcomes implied by proposed model. Interms of which, we show that interesting physics can be realized in a rather straightforwardframework.In the previous sections, we have considered a simplified scenario where a discontinuity isplanted by including a thin layer of dark matter surrounding the black hole at a given radialcoordinate. To a certain extent, the proposed metric is somewhat exaggerated when com-pared to the cuspy dark matter halos [37, 38]. However, the main goal of the present studyis to illustrate that some intereting features of the black hole shadow can be understood interms of a barebone approach. Moreover, one may argue that most of our results will remainvalid when one generalizes the metric given in Eq. (1)-(1) to that regarding a more realisticmatter distribution. To be specific, one may consider a thin but continuous matter distribu-tion is used to replace the dark matter shell with infinitesimal thickness while maintainingthe axisymmetry. For the case where the Maxwell construction can be encountered, such asthat studied in Fig. 1, the two endpoints of the metastable part of the FPO branches will beconnected (probably by a branch of stable FPOs). On the left plot of Fig. 1, this correspondsto a curve that joins continuously between the endpoint of the dashed yellow curve and thatof the dashed blue curve. It is noted that the Maxwell construction will remain unchangedas long as the section of the metric involving the rectangle stays the same. This is indeed thecase if the matter distribution is confined inside the interval [ r cuspBH , r cuspTOT ]. Similar argumentscan be given to the other two cases where the Maxwell construction cannot be established.In particular, as discussed in the last section, for the scenario investigated in Fig. 3, a finite3thickness is required to properly evaluate the shadow edge between the two rings. The mainadvantage to introduce an infinitesimally thin layer of dark matter is that the discontinuitybrings mathematical simplicity, as well as a more transparent interpretation of the relevantphysics content. In a recent paper [18], the black hole shadow was investigated regardingthe effect of dark matter on a Kerr black hole. While considering a finite layer of continuousmatter distribution, it was observed that the D-shaped black hole shadow mostly remainsunchanged except for an overall scaling of its size. When compared with the present work,the encountered novelty is largely due to two key factors, namely, the thickness of the darkmatter shell and its nonvanishing angular momentum. We also note that the second equal-ity of Eq. (8), and subsequently, the entire equation, is valid for any asymptotically Kerrspacetime. As a result, the Maxwell construction utilized in the present study is valid forany axisymmetric metric which asymptotically approaches a Kerr solution. Based on theabove discussions and the astrophysical significance of the Kerr-type metrics, we argue thatour findings are meaningful and potentially valid on a rather general ground. Moreover, theongoing advance of the observational astrophysics enlightens an optimistic perspective onboth the black hole quasinormal modes and shadow as promising observables. Therefore, itis worthwhile to explore the subject further, inclusively extend the study to more realisticmetrics. ACKNOWLEDGMENTS
WLQ is thankful for the hospitality of Huazhong University of Science and Technology,where a significant portion of this work was carried out. We gratefully acknowledge thefinancial support from Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP),Funda¸c˜ao de Amparo `a Pesquisa do Estado do Rio de Janeiro (FAPERJ), Conselho Nacionalde Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), Coordena¸c˜ao de Aperfei¸coamento dePessoal de N´ıvel Superior (CAPES), and National Natural Science Foundation of China(NNSFC) under contract Nos. 11805166, 11775036, and 11675139. A part of this work wasdeveloped under the project Institutos Nacionais de Ciˆencias e Tecnologia - F´ısica Nucleare Aplica¸c˜oes (INCT/FNA) Proc. No. 464898/2014-5. This research is also supported bythe Center for Scientific Computing (NCC/GridUNESP) of the S˜ao Paulo State University(UNESP). [1] J. L. Synge, Mon. Not. Roy. Astron. Soc. , 463 (1966).[2] C. DeWitt and B. S. DeWitt, editors,
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