Can different black holes cast the same shadow?
Haroldo C. D. Lima Junior, Luís C. B. Crispino, Pedro V. P. Cunha, Carlos A. R. Herdeiro
MMistaken identity:can different black holes cast the same shadow?
Haroldo C. D. Lima Junior ∗ and Luís C. B. Crispino † Faculdade de Física, Universidade Federal do Pará, 66075-110, Belém, PA, Brazil
Pedro V. P. Cunha ‡ and Carlos A. R. Herdeiro § Departamento de Matemática da Universidade de Aveiro and Centre forResearch and Development in Mathematics and Applications (CIDMA),Campus de Santiago, 3810-183 Aveiro, Portugal (Dated: February 2021)We consider the following question: may two different black holes (BHs) cast exactly the sameshadow? In spherical symmetry, we show the necessary and sufficient condition for a static BH tobe shadow-degenerate with Schwarzschild is that the dominant photonsphere of both has the sameimpact parameter, when corrected for the (potentially) different redshift of comparable observers inthe different spacetimes. Such shadow-degenerate geometries are classified into two classes. The firstshadow-equivalent class contains metrics whose constant (areal) radius hypersurfaces are isometricto those of the Schwarzschild geometry, which is illustrated by the Simpson and Visser (SV) metric.The second shadow-degenerate class contains spacetimes with different redshift profiles and anexplicit family of metrics within this class is presented. In the stationary, axi-symmetric case,we determine a sufficient condition for the metric to be shadow degenerate with Kerr for far-awayobservers. Again we provide two classes of examples. The first class contains metrics whose constant(Boyer-Lindquist-like) radius hypersurfaces are isometric to those of the Kerr geometry, which isillustrated by a rotating generalization of the SV metric, obtained by a modified Newman-Janisalgorithm. The second class of examples pertains BHs that fail to have the standard north-south Z symmetry, but nonetheless remain shadow degenerate with Kerr. The latter provides a sharpillustration that the shadow is not a probe of the horizon geometry. These examples illustrate thatnon-isometric BH spacetimes can cast the same shadow, albeit the lensing is generically different. CONTENTS
I. Introduction 1II. Shadow-degeneracy in spherical symmetry 2A. Class I of shadow-degenerate spacetimes 3B. Class II of shadow-degenerate spacetimes 4III. Illustrations of shadow-degeneracy (static) 4A. Class I example: the SV spacetime 4B. Class II example: A spacetime with A (cid:54) = 1 Z symmetry 13VI. Conclusions 14 ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] Acknowledgments 15References 15
I. INTRODUCTION
Strong gravity research is undergoing a golden epoch.After one century of theoretical investigations, the lastfive years have started to deliver long waited data on thestrong field regime of astrophysical black hole (BH) can-didates. The first detection of gravitational waves froma BH binary merger [1] and the first image of an astro-physical BH resolving horizon scale structure [2–4] haveopened a new era in testing the true nature of astrophys-ical BHs and the hypothesis that these are well describedby the Kerr metric [5].In both these types of observations a critical questionto correctly interpret the data is the issue of degener-acy. How degenerate are these observables for differentmodels? This question, moreover, is two-fold. Thereis the practical issue of degeneracy, due to the observa-tional error bars. Different models may predict differentgravitational waves or BH images which, however, are in-distinguishable within current data accuracy. But thereis also the theoretical issue of degeneracy. Can differ-ent models predict the same phenomenology for some(but not all) observables? For the case of gravitationalwaves this is reminiscent of an old question in spectralanalysis: can one hear the shape of a drum [6]? (See a r X i v : . [ g r- q c ] F e b also [7]). Or, in our context, can two different BHs beisospectral? For the case of BH imaging, this is the ques-tion if two different BHs capture light in the same way,producing precisely the same silhouette. In other words,can two different BH geometries have precisely the sameshadow [8]? The purpose of this paper is to investigatethe latter question.The BH shadow is not an imprint of the BH’s eventhorizon [9]. Rather, it is determined by a set of boundnull orbits, exterior to the horizon, which, in the non-spherical cases include not only light rings (LRs) but alsonon-planar orbits; in Ref. [10] these were dubbed funda-mental photon orbits . In the Kerr case these orbits areknown as spherical orbits [11], since they span differentlatitudes at constant radius, in Boyer-Lindquist coordi-nates [12].It is conceivable that different spacetime geometriescan have, in an appropriate sense, equivalent fundamen-tal photon orbits without being isometric to one another.In fact, one could imagine extreme situations in whichone of the spacetimes is not even a BH. It is knownthat equilibrium BHs, in general, must have LRs [14],and, consequently, also non-planar fundamental photonorbits [15]. But the converse is not true: spacetimes withLRs need not to have a horizon - see e.g.
Ref. [16].In this paper we will show such shadow-degenerate ge-ometries indeed exist and consider explicit examples. Forspherical, static, geometries, a general criterion can be es-tablished for shadow-degenerate geometries. The latterare then classified into two distinct equivalence classes.Curiously, an interesting illustration of the simplest classis provided by the ad hoc geometry introduced by Simp-son and Visser (SV) [17], describing a family of space-times that include the Schwarzschild BH, regular BHsand wormholes. In the stationary axi-symmetric case weconsider the question of shadow degeneracy with the Kerrfamily within the class of metrics that admit separabil-ity of the Hamilton-Jacobi (HJ) equation. We providetwo qualitatively distinct classes of examples. The firstone is obtained by using a modified Newman-Janis al-gorithm [18] proposed in [19]; we construct a rotatingversion of the SV spacetime, which, as shown here, turnsout to belong to a shadow-degenerate class including theKerr spacetime. A second class of examples discussesBHs with the unusual feature that they do not possessthe usual north-south Z symmetry present in, say, theKerr family, but nonetheless can have the same shadowas the Kerr spacetime. This provides a nice illustrationof the observation in [9] that the shadows are not a probeof the event horizon geometry.This paper is organized as follows. In Section II we dis-cuss a general criterion for shadow-degeneracy in spheri-cal symmetry and classify the geometries wherein it holds A rough, but not precise, imitation of the BH shadow by a dy-namically robust BH mimicker was recently discussed in [13]. into two equivalence classes. Section III discusses illus-trative examples of shadow-degenerate metrics for bothclasses. For class I, the example is the SV spacetime.We also consider the lensing in shadow-degenerate space-times, which is generically non-degenerate. Section IVdiscusses shadow degeneracy for stationary geometriesadmitting separability of the HJ equation. Section V dis-cusses illustrative examples of shadow-degenerate metricswith Kerr, in particular constructing a rotating general-ization of the SV spacetime and a BH spacetime without Z symmetry, and analysing their lensing/shadows. Wepresent some final remarks in Section VI. II. SHADOW-DEGENERACY IN SPHERICALSYMMETRY
Let us consider a spherically symmetric, asymptoti-cally flat, static BH spacetime. Its line element can bewritten in the generic form: ds = − V ( R ) A ( R ) dt + dR B ( R ) V ( R ) + R d Ω . (1)Here, V ( R ) ≡ − m/R is the standard Schwarzschildfunction, where m is a constant that fixes the BH horizonat R = 2 m . The constant m needs not to coincide withthe ADM mass, denoted M . The two arbitrary radialfunctions A ( R ) , B ( R ) are positive outside the horizon,at least C and tend asymptotically to unity: lim R →∞ A ( R ) , B ( R ) = 1 . (2)In the line element (1), d Ω ≡ (cid:2) dθ + sin θ dϕ (cid:3) is themetric on the unit round 2-sphere.Due to the spherical symmetry of the metric (1), wecan restrict the motion to the equatorial plane θ = π/ .Null geodesics with energy E and angular momentum j have an impact parameter λ ≡ jE , (3)and spherical symmetry allows us to restrict to λ (cid:62) . Following Ref. [16], we consider the Hamiltonian H = g µν p µ p ν , to study the null geodesic flow associated tothe line element (1). One introduces a potential term V ( R ) (cid:54) such that H = V + g RR ˙ R = 0 . (4)The dot denotes differentiation regarding an affine pa-rameter. Clearly, a radial turning point ( ˙ R = 0) is only The class of metrics (1) may describe non-BH spacetimes as well. The case λ = 0 has no radial turning points. possible when V = 0 for null geodesics ( H = 0 ). Onecan further factorize the potential V as V = j g tt (cid:18) λ − H (cid:19) (cid:18) λ + H (cid:19) (cid:54) , (5)which introduces the effective potential H ( R ) : H ( R ) = (cid:112) A ( R ) V ( R ) R (cid:62) . (6)In spherical symmetry, the BH shadow is determined byLRs. In the effective potential (6) description, a LR cor-responds to a critical point of H ( R ) , and its impact pa-rameter is the inverse of H ( R ) at the LR [16]: H (cid:48) ( R LR ) = 0 , λ LR = 1 H ( R LR ) , (7)where prime denotes radial derivative.The BH shadow is an observer dependent concept. Letus therefore discuss the observation setup. We consideran astrophysically relevant scenario. Firstly, the BH isdirectly along the observer’s line of sight. Secondly, theobserver is localized at the same areal radius, R obs , in theSchwarzschild and non-Schwarzschild spacetimes, bothhaving the same ADM mass M . Finally, the observer issufficiently far away so that no LRs exist for R > R obs in both spacetimes, but R obs needs not to be at infinity. The connection between the impact parameter λ of a generic light ray and the observation angle β with respectto the observer-BH line of sight is [20]: λ = sin β H (cid:0) R obs (cid:1) . (8)The degeneracy condition, that is, for the shadow edgeto be the same, when seen by an observer at the sameareal radius R obs in comparable spacetimes ( i.e. with thesame ADM mass, M ), is that the observation angle β coincides in both cases, for both the metric (1) and theSchwarzschild spacetime. This implies that the impactparameter of the shadow edge in the generic spacetimemust satisfy: λ LR = √ M √ A obs , (9)where we have used that, for Schwarzschild, the LR im-pact parameter is λ Schw LR = √ M and A obs ≡ A ( R obs ) . This is a geometrically significant coordinate, and thus can becompared in different spacetimes. If a LR exists for
R > R obs a more complete analysis can bedone, possibly featuring a BH shadow component in the oppositedirection to that of the BH, from the viewpoint of the observer.However, this will not be discussed here in order to focus on amore astrophysical setup.
Hence, only for the cases wherein A = 1 does shadow de-generacy amounts to having the same impact parameterin the Schwarzschild and in the non-Schwarzschild space-times, for an observer which is not at spatial infinity. Ingeneral, the different gravitational redshift at the "same"radial position in the two spacetimes must be accountedfor, leading to (9).In the next two subsections we will distinguish twodifferent classes of shadow degenerate spacetimes (classI and II), using the results just established. It is im-portant to remark that the fact that two non-isometricspacetimes are shadow-degenerate does not imply thatthe gravitational lensing is also degenerate. In fact, it isgenerically not.This will be illustrated in Section III with some con-crete examples for the two classes of shadow-degeneratespacetimes. An interesting example of shadow, but notlensing, degeneracy can be found in [21], albeit in a dif-ferent context. A. Class I of shadow-degenerate spacetimes
Specializing (7) for (1) yields λ LR = R LR (cid:112) A ( R LR ) V ( R LR ) . (10)and R LR = 3 m + R λ A (cid:48) ( R LR ) A ( R LR ) , (11)where A (cid:48) ≡ dA/dR . The notorious feature is that re-gardless of B ( R ) , if A ( R ) = 1 , then it holds for the back-ground (1) that m = M and R LR = 3 M , λ LR = √ M , (12)which are precisely the Schwarzschild results. The condi-tion A = 1 is thus sufficient to have the same shadow asSchwarzschild, since (9) is obeyed. Spacetimes (1) with A = 1 define the equivalence Class I of shadow degener-ate spacetimes.If the spacetime (1) with A ( R ) = 1 but B ( R ) non triv-ial describes a BH, it will have precisely the same shadowas the Schwarzschild spacetime. Such family of space-times is not fully isometric to Schwarzschild. But its con-stant (areal) radius hypersurfaces are isometric to thoseof Schwarzschild and thus have overlapping R = constantgeodesics, which explains the result. This possibility willbe illustrated in section III A.These observations allow us to anticipate some shadow-degenerate geometries also for stationary, axially sym-metric spacetimes. If the fundamental photon orbits are"spherical", not varying in some appropriate radial coor-dinate, as for Kerr in Boyer-Lindquist coordinates, anygeometry with isometric constant radial hypersurfaceswill, as in the static case, possess the same fundamen-tal photon orbits, and will be shadow-degenerate withKerr. We shall confirm this expectation in an illustrativeexample in Section V. B. Class II of shadow-degenerate spacetimes If A ( R ) (cid:54) = 1 the solution(s) of (7) will not coincide, ingeneral, with (12). In particular, there can be multiplecritical points of H ( R ) , i.e. multiple LRs around the BH.This raises the question: if multiple LRs exist, which willbe the one to determine the BH shadow edge?We define the dominant LR as the one that determinesthe BH shadow edge. To determine the dominant LRfirst observe that:1. Since g tt < outside the horizon, the condition /λ (cid:62) H ( R ) must be satisfied along the geodesicmotion, see Eq. (5). In particular, at LRs, thesmaller the impact parameter λ is, the larger thepotential barrier H ( R LR ) is.2. The radial motion can only be inverted ( i.e. have aturning point) when V = 0 ⇐⇒ /λ = H .3. The function H ( R ) vanishes at the horizon, H| R =2 m = 0 .The BH’s shadow edge is determined by critical light raysat the threshold between absorption and scattering bythe BH, when starting from the observer’s position R obs .Considering points 1-3 above, these critical null geodesicsoccur at a local maximum of H , i.e. a LR. The infallingthreshold is provided by the LR that possesses the largest value of /λ , since this corresponds to the largest po-tential barrier in terms of H ( R ) . Any other photon-spheres that might exist with a smaller value of /λ donot provide the critical condition for the shadow edge, al-though they might play a role in the gravitational lensing.Hence, the dominant photonsphere has the smallest valueof the impact parameter λ , and shadow degeneracy withSchwarzschild is established by constraining the smallestLR impact parameter λ , via Eq. (9).Combining Eq. (9) and λ H ( R ) (cid:54) , yields the neces-sary and sufficient conditions on A ( R ) in order to haveshadow-degeneracy with Schwarzschild. Explicitly, theseare: (i): A ( R ) (cid:54) R ( R − m ) (cid:18) A obs M (cid:19) , (13) (ii): the previous inequality must saturate at least onceoutside the horizon for some R domLR < R obs . At suchdominant LR, located at R domLR , (9) is guaranteed tohold. See [22, 23] for a related discussion.
Observe that M < R obs so that the observer is outsidethe LR in the Schwarzschild spacetime. Spacetimes (1)with A (cid:54) = 1 obeying the two conditions above define theequivalence Class II of shadow-degenerate spacetimes.One example will be given in section III B. We remarkthat class I of shadow-degenerate spacetimes is a partic-ular case of class II.
III. ILLUSTRATIONS OFSHADOW-DEGENERACY (STATIC)A. Class I example: the SV spacetime
The SV spacetime is a static, spherically symmetricgeometry that generalizes the Schwarzschild metric withone additional parameter b , besides the ADM mass M ,proposed in [17]. It is an ad hoc geometry. Its associatedenergy-momentum tensor, via the Einstein equations, vi-olates the null energy condition, and thus all classicalenergy conditions. Nonetheless, it is a simple and illus-trative family of spacetimes that includes qualitativelydifferent geometries. It is given by the following line ele-ment: ds = − (cid:18) − M √ r + b (cid:19) dt + (cid:18) − M √ r + b (cid:19) − dr + (cid:0) r + b (cid:1) d Ω , (14)where the coordinates have the following domains: −∞ (cid:54) r (cid:54) ∞ , −∞ (cid:54) t (cid:54) ∞ , < θ < π and − π (cid:54) ϕ < π . Depending on the value of the additionalparameter b , which without loss of generality we assumeto be non-negative, the spacetime geometry describes: ( i ) the Schwarzschild geometry ( b = 0 ); ( ii ) a regular BH( < b < M ) with event horizon located at r h = ± (cid:112) (2 M ) − b ; (15) ( iii ) a one-way traversable wormhole geometry with anull throat at r t = 0 ( b = 2 M ); or ( iv ) a two-waytraversable wormhole geometry with a timelike throatat r t = 0 , belonging to the Morris-Thorne class [24]( b > M ).The coordinate system in (14) is relevant to observethat r = 0 is not a singularity. Thus the geometry canbe extended to negative r . However, it hides some otherfeatures of the geometry, since the radial coordinate in(14) is not the areal radius for b (cid:54) = 0 . Introduce the arealradius R as R ≡ r + b . (16)The SV spacetime reads, in ( t, R, θ, ϕ ) coordinates, ds = − V ( M ) dt + dR V ( M ) B SV ( R ) + R d Ω , (17)where V ( M ) = 1 − MR , (18) B SV ( R ) ≡ (cid:18) − b R (cid:19) . (19)The geometry is now singular at R = 2 M ≡ R h , and R = b ≡ R t . (20)For < b < M , R h > R t , the null hypersurface R = R h is a Killing horizon and the event horizon of the space-time. It can be covered by another coordinate patch andthen another coordinate singularity is found at R = R t .This is again a coordinate singularity, as explicitly shownin the coordinate system (14). It describes a throat orbounce. A free falling observer bounces back to a grow-ing R , through a white hole horizon into another asymp-totically flat region - see Fig. 4 in [17]. Thus, in thecoordinate system ( t, R, θ, ϕ ), b (cid:54) R < ∞ . The othercoordinate ranges are the same as before.As it is clear from Eq. (20), the areal radius of theevent horizon (when it exists) is b -independent. More-over, since the SV spacetime is precisely of the type (1)with A ( R ) = 1 and B ( R ) = B SV ( R ) , it follows from thediscussion of the preceding section that Eq. (12) holdsfor the SV spacetime. Thus, whenever the SV space-time describes a BH ( < b < M ) it is class I shadow-degenerate with a Schwarzschild BH, for an equivalentobserver. This result can be also applied to the worm-hole geometry if the LR is located outside the throat,i.e. the LR must be covered by the R coordinate range( b ≤ M ). For b > M , the LR is located at the throatand Eq. (12) does not hold [25].The LR in this spacetime has the same areal ra-dius as in Schwarzschild, R LR = 3 M . However, theproper distance between the horizon and the LR, alonga t, θ, ϕ = constant curve is b -dependent: ∆ R = (cid:90) R LR R h √ g RR dR = (cid:26) (cid:39) . M , for b = 0 , → ∞ , for b = 2 M . (21)It is a curious feature of the SV spacetime that the spa-tial sections of the horizon and the photonsphere have b -independent proper areas, respectively π (2 M ) and π (3 M ) . But the proper distance between these sur-faces is b -dependent and diverges as b → M .Let us now consider the gravitational lensing in the SVspacetime. We set the observer’s radial coordinate equalto R obs = 15 M . In the top panel of Fig. 1 we plot thescattered angle ∆ ϕ on the equatorial plane, in units of π , as a function of the observation angle β . We choosethree distinct values of b , including the Schwarzschildcase ( b = 0 ). For large observation angles, the scatteredangle is essentially the same for different values of b . Aslight difference arises near the unstable LR, character-ized by the divergent peak, as can be seen in the insetof Fig. 1 (top panel). In this region, for a constant β , the scattered angle increases as we increase b . We notethat the LR, and hence the shadow region, is indepen-dent of b , as expected. In the bottom panel of Fig. 1 weshow the trajectories of light rays for the observation an-gle β = 0 . and different values of b . The event horizon,for the BH cases, is represented by the dashed circle. Wenotice that higher values of b lead to larger scatteringangles. Shadowregion (cid:1)(cid:0) / ( (cid:3) ) (cid:4) b=0b=1.99Mb=2.5M -15-10-5 0 5 10 15 -15 -10 -5 0 5 10 15O y / M x/M b=0b=1.99Mb=2.5M FIG. 1. Top panel: Scattering angle for null geodesics as afunction of the observation angle, for the Schwarzschild BH( b = 0 ), SV BH ( b = 1 . M ) and SV wormhole ( b = 2 . M )cases. The shaded area corresponds to the shadow region inthe observer’s local sky. Bottom panel: The trajectories de-scribed by null geodesics in the Cartesian plane for the samechoices of b and with observation angle β = 0 . . The ob-server O is located at ( x = 15 M, y = 0) , as shown in thebottom panel. The dashed circle represents the event horizonof the BH cases.
We show in Fig. 2 the shadow and gravitational lens-ing of the Schwarzschild BH, SV BH and SV wormholespacetimes, obtained using backwards ray-tracing. Inthe backward ray-tracing procedure, we numerically in- (a) Schwarzschild ( b = 0 )(b) SV BH ( b = 1 . M )(c) SV wormhole ( b = 2 . M ) FIG. 2. The shadow and gravitational lensing of the:Schwarzschild BH (top panel), SV BH spacetime with b =1 . M (middle panel), and SV wormhole spacetime with b = 2 . M (bottom panel). The observer is located on theequatorial plane and at the radial coordinate R obs = 15 M .The angle of view is equal to ◦ . We present, in the topright corner of each panel, a zoom of the region next to theshadow edge. tegrate the light rays from the observer position, back-wards in time, until the light rays are captured by theevent horizon or scattered to infinity. The results wereobtained with two different codes: a C ++ code developedby the authors, and the PYHOLE code [26] which wasused as a cross-check. In this work we only show the ray-tracing results obtained with the C ++ code, since theyare essentially the same as the ones obtained with thePYHOLE code. The light rays captured by the BH areassigned a black color in the observer’s local sky. For thescattered light rays, we adopt a celestial sphere with fourdifferent colored quadrants (red, green, blue and yellow).A grid with constant latitude and longitude lines, anda bright spot behind the BH/wormhole are also presentin the celestial sphere. This setup is similar to the onesconsidered in Refs. [26–28]. On the other hand, the lightrays captured by the wormhole throat are assigned witha different color pattern (purple, cyan, dark green anddark yellow) on the celestial sphere on the other side ofthe throat, but without the grid lines and the bright spot.The white ring surrounding the shadow region in Fig. 2corresponds to the lensing of the bright spot behind theBH, known as Einstein ring [29]. It is manifest in theSchwarzschild and SV BHs, as well as in the SV wormholecase. The Einstein ring has a slightly different angularsize in the three configurations shown in Fig. 2. This canbe confirmed in Fig. 1, since on the equatorial plane, theEinstein ring is formed by the light rays scattered withangle ∆ ϕ = π , which corresponds to close values of β forthe three cases. Due to the spherical symmetry, a similaranalysis holds for light rays outside the equatorial plane,which explains the formation of the Einstein ring withsimilar angular size in Fig. 2.Inside the Einstein ring, the whole celestial sphere ap-pears inverted. Next to the shadow edge, there is a sec-ond Einstein ring (corresponding to a circular grid ringbest seen in the inset of the figures between the yel-low and green patches), in this case corresponding tothe point behind the observer, ∆ ϕ = 2 π in Fig. 1. Aslight difference between the three configurations is alsoobserved in this region, as can be seen in the inset ofFig. 2. In between the second Einstein ring and theshadow edge there is an infinite number of copies of thecelestial sphere. B. Class II example: A spacetime with A (cid:54) = 1 As a concrete example of the class II of shadow-degenerate BH spacetimes with respect to theSchwarzschild BH, as seen by an observer at radialcoordinate R obs , we consider (using m = M = 1 units)the following A ( R ) function: A = 1 + (cid:18) R LR − b + b (cid:19)(cid:32) a (cid:20) R LR R (cid:21) + a (cid:20) R LR R (cid:21) (cid:33) , (22) a = − a − R (cid:0) − u + u (cid:1) ,a = −
36 + 24 R LR − R + R (cid:0) u − (cid:1) ,b = 27( R LR − ,b = R u (cid:2) (7 − R LR ) + (2 R LR − u (cid:3) . We have here introduced a free parameter R LR whichsets the radial position of the dominant photonsphere(after suitably restricting the range of R LR ). This space-time is also modified by the quantity u ≡ R LR /R obs ,which depends on the observer location. In particular,the choice u = 0 corresponds to setting the observerat spatial infinity. This spacetime reduces to theSchwarzschild case for R LR = 3 (provided than B = 1 ),since it implies A ( R ) = 1 . However, for R LR (cid:54) = 3 thespacetime is not Schwarzschild.Not every parameter combination { R LR , R obs } yieldsan acceptable spacetime for our analysis. Consideringthe discussion in Section II B, the spacetime outside thehorizon must satisfy both A > and Eq. (13), togetherwith < R LR < R obs and < R obs . These conditionsfix the allowed range of parameters.For concreteness, we can set R obs = 15 , which leads tothe following allowed range for the parameter R LR : R LR ∈ [2 . , . . (23)In contrast, the function B ( R ) can be left fairly un-constrained. Curiously, for some values of R LR in therange (23) there can be three photonspheres (two unsta-ble, and one stable) outside the horizon. However, theLR at R = R LR is always the dominant one by construc-tion, as can be seen in Fig. 3, where the horizontal dashedlines correspond to /λ LR - see Eq. (9). Each line inter-sects the maximum point of the associated potential H ,that determines the dominant LR location. H ( R ) R R LR =2.227R LR =2.37R LR =3 FIG. 3. The effective potential H (R) for the class II ofshadow-degenerate BHs, with A ( R ) given by Eq. (22), fordifferent dominant LR radius R LR . The horizontal lines cor-respond to the critical impact parameters for each R LR , seeEq. (9). Let us now consider the gravitational lensing in thisspacetime, assuming for simplicity B ( R ) = 1 . In thetop panel of Fig. 4 we plot the scattered angle, in unitsof π , as a function of the observation angle β , forthe two parameter values R LR = { . , . } . TheSchwarzschild case R LR = 3 is also included in Fig. 4,for reference. First taking R LR = 2 . , there are twodiverging peaks on the scattering angle, related to theexistence of two unstable LRs (there exists also a stableLR which leaves no clear signature in the scatteringplot). In contrast to the first case, R LR = 2 . containsonly a single diverging peak. However, it presents alocal maximum of the scattering angle next to β = 0 . ,as can be seen in the inset of Fig. 4 (top panel). Thislocal maximum leads to a duplication of lensed images.Importantly, the shadow region, corresponding to theshaded area in Fig. 4, is the same for the different valuesof R LR , as expected.On the bottom panel of Fig. 4 we show the trajectoriesdescribed by null geodesics, for different values of the pa-rameter R LR , given the same observation angle β = 0 . .Curiously, when R LR = 2 . , the trajectory followed bythe light ray presents two inflection points when repre-sented in the ( x = R cos ϕ, y = R sin ϕ ) plane, cf. Fig. 4(bottom panel). Such inflection points are determined by d y/dx = 0 , which yields R − R d Rdϕ + 2 (cid:18) dRdϕ (cid:19) = 0 ⇔ ¨ R = j R , (24)where the last equivalence holds for j (cid:54) = 0 . This is also thecondition for the curve R = R ( ϕ ) to have vanishing cur-vature. On the other hand, the equations of motion (4)and ˙ ϕ = j/R , give, for B ( R ) = 1 , ¨ R = j R + E ddR (cid:20) A ( R ) + 2 λ R (cid:21) . (25)Equating the last two results yields R λ A (cid:48) ( R ) A ( R ) = 0 , (26)as the condition at an inflection point. Observe the simi-larity with (11) (recall m = 1 here); the latter has ¨ R = 0 ,unlike (24). For the Schwarzschild case, A ( R ) = 1 , andthere are no inflection points. But for A ( R ) given by (22)and R LR = 2 . it can be checked that the function insquare brackets in (25) has a local maximum, which ex-plains the existence of inflection points.To further illustrate the effect of different choices ofthe parameter R LR , we display the shadows and gravita-tional lensing in Fig. 5, obtained numerically via back-wards ray-tracing. Despite identical shadow sizes, thegravitational lensing can be quite different for each valueof the parameter R LR . For instance, although Einsteinrings are present in all cases depicted, they have different Shadowregion (cid:1)(cid:0) / ( (cid:3) ) (cid:4) R LR =2.227R LR =2.37R LR =3 -15-10-5 0 5 10 15 -15 -10 -5 0 5 10 15O y / M x/M R LR =2.227R LR =2.37R LR =3 FIG. 4. Top: The scattered angle of null geodesics, as a func-tion of the observation angle β , for R LR = { . , . , } .The shaded area corresponds to the shadow region in the ob-server’s local sky. The horizontal dotted line represents thelight rays scattered at ∆ ϕ = 2 π (located behind the observer).Since there is a black dot in the celestial sphere right behindthe observer, the gravitational lensing displays black Einsteinrings (see Fig. 5). Bottom: The corresponding trajectories de-scribed by null geodesics in the equatorial plane in Cartesian-like coordinates with an observation angle β = 0 . . Theobserver O is located at ( x = 15 , y = 0) . The dashed circlerepresents the event horizon of these class II shadow degen-erate examples. angular diameters. This is best illustrated by looking atthe white circular rings, which are mapping the point inthe colored sphere directly behind the BH.There are also some curious features of the lensingthat can be anticipated from the scattering angle plotin Fig. 4 (top panel). For example, for a parameter R LR = 2 . there are multiple copies of the celestialsphere very close to the shadow edge that are not easilyidentifiable in Fig. 5(a). This is due to light rays scat- tered with angles greater than π having an observationangle β very close to the shadow edge. The divergingpeak in the scattering angle also has a clean signature inthe image, in the form of a very sharp colored ring whichis just a little smaller in diameter than the white circle.Additionally, taking the parameter R LR = 2 . , we canfurther expand our previous remark on the effect of thelocal maximum of the scattering angle, which introducesan image duplication of a portion of the colored spheredirectly behind the observer. This feature is best seenin Fig. 5(b) as an additional colored ring structure thatdoes not exist in Fig. 5(c). IV. SHADOW-DEGENERACY IN STATIONARYBHS
Let us now investigate possible rotating BHs with Kerrdegenerate shadows. We assume that the spacetime isstationary, axi-symmetric and asymptotically flat. Incontrast to the spherically symmetric case, the generalmotion of null geodesics in rotating BH spacetimes is notconstrained to planes with constant θ . This introducesan additional complication in the analysis of the geodesicmotion, and in many cases it is not possible to computethe shadows analytically. For Kerr spacetime, there isan additional symmetry that allows the separability ofthe HJ equation, thus the shadow problem can be solvedanalytically. This symmetry is encoded in the so-calledCarter constant, that arises due to a non-trivial Killingtensor present in the Kerr metric [30]. Here, we shall in-vestigate shadow degeneracy specializing for rotating BHspacetimes that admit separability of the null geodesicequation. A. Spacetimes admitting HJ separability
Following the strategy just discussed, we need the gen-eral form of the line element for a rotating BH space-time that admits separability of the HJ equation. Itis known that rotating spacetimes obtained through aNewman-Janis and modified Newman-Janis algorithmsallow separability of the HJ equation [19, 31, 32]. Sinceit is not guaranteed, however, that every spacetime al-lowing separability of the HJ equation can be obtainedby such algorithms, we pursue a different strategy. InRef. [33], Benenti and Francaviglia found the form of themetric tensor for a spacetime admitting separability ofnull geodesics in n-dimensions (see also [34]). This resultis based on the following general assumptions: Although this strategy implies a loss of generality it seems un-likely (albeit a proof is needed) that a spacetime without sep-arability can yield precisely the same shadow as the Kerr one,which is determined by separable fundamental photon orbits. (a) Class II example ( R LR = 2 . )(b) Class II example ( R LR = 2 . )(c) Schwarzschild ( R LR = 3 ) FIG. 5. The shadow and gravitational lensing for the classII example of shadow degenerate BHs, with A ( r ) given inEq. (22), for R LR = 2 . (top panel), R LR = 2 . (middlepanel), and R LR = 3 (bottom panel). The observer positionand angle of view are the same as in Fig. 2. • There exist (locally) z independent commutingKilling vectors X α .• There exist (locally) n − z independent commutingKilling tensors K a , satisfying [ K a , X α ] = 0 . (27)• The Killing tensors K a have in common n − z com-muting eigenvectors X a such that g ( X a , X α ) = 0 , [ X a , X α ] = 0 . (28)We are interested in four-dimensional spacetimes ( n = 4 )admitting two Killing vectors ( z = 2 ), associated to theaxi-symmetry and stationarity. Hence the metric ten-sor that admits separability of the HJ equation is givenby [33]: ∂ s = g ab ∂ a ∂ b = 1˜ A ( R ) + ˜ B ( θ ) (cid:104)(cid:16) ˜ A ( R ) + ˜ B ( θ ) (cid:17) ∂ t + 2 (cid:16) ˜ A ( R ) + ˜ B ( θ ) (cid:17) ∂ t ∂ ϕ + ˜ A ( R ) ∂ R + ˜ B ( θ ) ∂ θ + (cid:16) ˜ A ( R ) + ˜ B ( θ ) (cid:17) ∂ ϕ (cid:105) . (29)For our purpose, it is convenient to rewrite the functions ˜ A i ( R ) and ˜ B i ( θ ) as ˜ A = R A , ˜ B = a cos θB , ˜ A = − ( R + a ) ∆ A , ˜ B = a sin θB , ˜ A = − amR ∆ A , ˜ B = B − , ˜ A = ∆ A , ˜ B = B , ˜ A = − a ∆ A , ˜ B = 1sin θ B , since we can recover the Kerr spacetime by simply taking A i = 1 and B i = 1 . The function ∆ is given by ∆ = R − mR + a , (30)where m and a are constants that fix the BH event hori-zon at R h = m + (cid:112) m − a . (31)Similarly to the spherically symmetric case, m and a neednot to coincide with the ADM mass and total angular mo-mentum per unit mass, respectively. The metric tensorin terms of A i and B i assumes the following form: ∂ s = 1 (cid:101) Σ (cid:20) − (cid:18) ( R + a ) ∆ A − a sin θB (cid:19) ∂ t − (cid:18) amR ∆ A + 1 − B (cid:19) ∂ t ∂ ϕ + ∆ A ∂ R + B ∂ θ + (cid:18) − a ∆ A + 1sin θ B (cid:19) ∂ ϕ (cid:21) , (32)In Eq. (32), we have (cid:101) Σ = R A + a cos θB . (33)0We assume that A i ( R ) are positive outside the eventhorizon, and at least C . Nevertheless, the general resultfound by Benenti and Francaviglia may also describe non-asymptotically flat spacetimes. Hence, we need to imposeconstraints on the 10 functions present in the metric ten-sor (32), in order to describe asymptotically flat BHs. Forthis purpose it is sufficient that they tend asymptoticallyto unity, namely: lim R →∞ A i ( R ) = 1 . (34)The metric far away from the BH is given by [35]: d ˜ s = − (cid:18) − MR (cid:19) dt − J sin θR dtdϕ + dR + R d Ω , (35)where J denotes the ADM angular momentum. Ex-panding the metric tensor (32) for R (cid:29) m and R (cid:29) a ,and comparing with the BH metric far away from theBH, we find that B ( θ ) = B ( θ ) = B ( θ ) = 1 , (36)while B ( θ ) and B ( θ ) are left unconstrained. Hence weconclude that the metric tensor for a spacetime admittingseparability and asymptotically flatness is given by ∂ s = 1 (cid:101) Σ (cid:20) − (cid:18) ( R + a ) ∆ A − a sin θB (cid:19) ∂ t − amRA ∆ ∂ t ∂ ϕ + ∆ A ∂ R + ∂ θ + (cid:18) − a ∆ A + 1sin θ (cid:19) ∂ ϕ (cid:21) . (37) B. Fundamental Photon Orbits
We are now able to study the problem of shadow de-generacy in stationary and axi-symmetric BHs, using theconcrete metric form (37) for spacetimes with a separa-ble HJ equation. It is straightforward to compute thefollowing geodesic equations for the coordinates { R, θ } : (cid:101) Σ A ˙ RE = R ( R ) , (38) (cid:101) Σ ˙ θE = Θ( θ ) , (39)where R ( R ) = (cid:0) R + a (cid:1) A − λamRA + a λ A − ∆ (cid:0) η + a + λ (cid:1) , (40) Θ( θ ) = η + a (cid:0) − sin θB (cid:1) − λ tan θ . (41) The constant parameters λ and η are given by λ = jE , η = QE . (42)As before, { E, j } are the energy and angular momentumof the photon respectively. The quantity Q is a Carter-like constant of motion, introduced via the separabilityof the HJ equation.The analysis of the Spherical Photon Orbits (SPOs) isparamount to determine the BH’s shadow analytically.These orbits are a generalization of the LR orbit (photonsphere) that was discussed in the introduction. SPOshave a constant radial coordinate R and are characterizedby the following set of equations: R ( R ) = 0 , (43) d R ( R ) dR = 0 . (44)In general, the set of SPOs in the spacetime (37) willnot coincide with the Kerr one. However if we set A = A = A = B = 1 , (45)Eq. (40) is identical to the Kerr case in Boyer-Lindquistcoordinates, the same will hold for the set of equa-tions (43)-(44), regardless of A , B and A . One mightthen have the expectation that (45) is a sufficient condi-tion to be shadow-degenerate with Kerr. Although thiswill turn out to be indeed true, caution is needed in deriv-ing such a conclusion, since the influence of the observerframe also needs to be taken into account.The solutions to Eqs. (43)-(44) with (45) are the fol-lowing two independent sets: η ∗ = − R a ; λ ∗ = R + a a , (46)and η = − R (cid:0) R − M R + 9 M R − a M (cid:1) a ( R − M ) , (47) λ = − (cid:0) R − M R + a R + M a (cid:1) a ( R − M ) . (48)The first set { λ ∗ , η ∗ } is unphysical because (41) is notsatisfied for real values. In contrast, the second set ofEqs. (47)-(48) is physically relevant: it defines the con-stants of motion for the different SPOs as a function ofthe parametric radius R ∈ [ R , R ] . In the latter, R and R are defined as the roots of Eq. (47), given by R k M = 2 + 2 cos (cid:26)
23 arccos (cid:20) (2 k − | a | M (cid:21)(cid:27) , (49)where k ∈ { , } . Importantly, the physical range R ∈ [ R , R ] follows from the requirement that Θ (cid:62) (from Eq. (41)). The set of equations (47)-(48) havebeen extensively analysed in the literature for the Kerrmetric [11, 36]. Remarkably, this set of orbits does notdepend on the functions A , B and A .1 C. Shadows
The BH shadow edge in the spacetime (37) is deter-mined by the set of SPOs discussed above. To analysethe former, it is important to obtain the components ofthe light ray’s 4-momentum p µ , as seen by an observerin a local ZAMO (Zero Angular Momentum Observer)frame (see discussion in Ref. [20]). In the following ex-pressions, all quantities are computed at the observer’slocation: p ( t ) = (cid:114) g ϕϕ g tϕ − g tt g ϕϕ (cid:18) E + g tϕ g ϕϕ j (cid:19) , (50) p ( R ) = p R √ g RR = √ g RR ˙ R, (51) p ( θ ) = p θ √ g θθ = √ g θθ ˙ θ, (52) p ( ϕ ) = j √ g ϕϕ . (53)One generically requires two observation angles { α, β } ,measured in the observer’s frame, to fully specify thedetection direction of a light ray in the local sky. Theseangles are defined by the ratio of the different componentsof the light ray’s four momentum in the observer frame.Following [20], we can define the angles α, β to satisfy: sin α = p ( θ ) p ( t ) , tan β = p ( ϕ ) p ( R ) . (54)In addition, one can expect the angular size of an ob-ject to decrease like α ∼ /R circ in the limit of far awayobservers, where the circumferential radius R circ is ameasure of the observer’s distance to the horizon [20, 37].Given this asymptotic behaviour, it is useful to introducethe impact parameters: X = − R circ β, Y = R circ α . (55)By construction, these quantities { X, Y } are expectedto have a well defined limit when taking R circ → ∞ .The relation between { X, Y } and the constants of nullgeodesic motion { λ, η } can be obtained by consideringEqs. (40)-(41) for ˙ R and ˙ θ , and then combining themwith Eqs. (50)-(54).We can compute the shadow edge expression in thelimit of far-away observers ( R circ → ∞ ): X = − λ sin θ , (56) Y = ± (cid:115) η + a cos θ − λ tan θ . (57) The quantity R circ is computed by displacing the observer to theequatorial plane ( θ = π/ ), while keeping its coordinate R fixed; R circ = √ g ϕϕ at that new location. The shadow degeneracy occurs when the quantities { X, Y } coincide in the non-Kerr and Kerr spacetimes,for an observer located at the same circumferentialradius R circ and polar angle θ . This will certainlybe the case if the set of SPOs coincides in both Kerrand the non-Kerr geometry for observers that are veryfar-away. Thus recalling Eq. (45), we note that the latteris a sufficient condition for shadow degeneracy at infinity.We conclude with the following line element obtainedfrom (37) together with (45): ds = − (cid:0) ∆ − a sin θ (cid:1) (cid:101) ΣΣ dt + (cid:101) Σ A ∆ dR + (cid:101) Σ dθ − amR sin θ (cid:101) ΣΣ dtdφ + (cid:104)(cid:0) R + a (cid:1) − a ∆ sin θ (cid:105) (cid:101) ΣΣ sin θdφ , (58)where Σ = R + a cos θ, (59) (cid:101) Σ = R A + a cos θB . (60)This geometry will be shadow degenerate with respectto the Kerr spacetime for very far-away observers, withvery weak constrains on A , B and A . V. ILLUSTRATIONS OFSHADOW-DEGENERACY (STATIONARY)A. Rotating SV spacetime
An example of a rotating, stationary and axi-symmetric BH with a Kerr degenerate shadow, can beobtained by applying the method proposed in [19] to thestatic SV geometry. This method consists on a varia-tion of the Newman-Janis algorithm (NJA) [18]. Startingfrom a static seed metric, we can generate via this mod-ified NJA a rotating circular spacetime that can be al-ways expressed in Boyer-Lindquist-like coordinates. Thiscomes in contrast to the standard NJA, for which thelatter is not always possible. However, the method intro-duces a new unknown function ( Ψ below) that may befixed by additional (physical) arguments, for instance, in-terpreting the component of the stress-energy tensor T µν to be those of a fluid [19].Applying the modified NJA discussed in Ref. [19] to theseed metric (14) (class I spacetime - SV), we introduce As this paper was being completed a rotating version of the SVspacetime was independently constructed, using the NJA, in [38,39]. F ( r ) = (cid:18) − M √ r + b (cid:19) , (61) K ( r ) = r + b . (62)The latter contains the same functional structure of someof the metric elements of the seed metric (14). Combin-ing these functions F, K with a complex change of coor-dinates (which introduces a parameter a ), we obtain apossible rotating version of the SV metric (14), namely: ds = − (cid:0) F K + a cos θ (cid:1) ( K + a cos θ ) Ψ dt + Ψ F K + a dr − a sin θ (cid:34) K − F K ( K + a cos θ ) (cid:35) Ψ dtdϕ + Ψ dθ (63) +Ψ sin θ (cid:34) a sin θ K − F K + a cos θ ( K + a cos θ ) (cid:35) dϕ , where Ψ is an undefined function that can be fixed byan additional requirement. Assuming a matter-sourcecontent of a rotating fluid, we can check that setting Ψ = r + a cos θ + b leads to an Einstein tensor thatsatisfies the suitable fluid equations detailed in Ref. [19].We may then rewrite the line element (63) in terms ofthe radial coordinate R = √ r + b , which leads to themore compact form: ds = − (cid:20) − M R Σ (cid:21) dt + Σ∆ dR B SV ( R ) − M a R sin θ Σ dt dϕ + Σ dθ ++Σ (cid:20) a sin θ Σ (Σ + 2 M R ) (cid:21) sin θ dϕ , (64)where Σ = R + a cos θ, (65) ∆ = R − M R + a . (66)For simplicity, we shall designate the geometry (64) asthe rotating SV spacetime. Curiously, the line element(64) is precisely the Kerr one, except for the extra factor B SV ( R ) in the g RR component. For a = 0 , the SV lineelement (17) is recovered; for b = 0 we recover the Kerrmetric in Boyer-Lindquist coordinates.An important remark is in order. Depending on the co-ordinate choice of the seed metric (14), the latter mightbe mapped to a different final geometry. For instance,had we applied the modified NJA to the seed SV met-ric in the coordinates of Eq. (17) rather than those ofEq. (14), then we would have obtained a rotating space-time different than that of (64).Let us now examine some of the properties of thespinning SV geometry (64).
1. Singularities and horizons
The line element (64) presents singularities at: R ± = M ± (cid:112) M − a , (67) R t = b. (68)These are coordinate singularities and the spacetime isregular everywhere. In particular (67) are Killing hori-zons that only exist if R ± > R t , or M ± (cid:112) M − a (cid:62) b. (69)Adopting the positive sign in Eq. (69), a BH only existsif M + √ M − a > b and a < M ; for M + √ M − a = b the geometry describes a wormhole with a throat, whichcan be nulllike, spacelike or timelike, depending on thevalue of a and b [38]. These singularities R ± canbe removed by writing the line element in Eddington-Finkelstein-like coordinates. On the other hand, the sin-gularity R t can be removed by writing the line elementin the coordinates given by Eq. (63), in which R t corre-sponds to the radial coordinate r = 0 .In order to inquire if the geometry (64) is regu-lar everywhere, let us consider some curvature invari-ants: the Kretschmann scalar ( K = R µ ν β ρ R µ ν β ρ ),the Ricci scalar ( R µ ν g µ ν ), and the squared Ricci tensor( R µ ν R µ ν ). The full expressions of the curvature invari-ants are too large to write down. However, we can writethem in a compact form as: K = P ( R, θ )2 R Σ , (70) R µν g µν = b Q ( R, θ ) R Σ , (71) R µν R µν = b S ( R, θ )2 R Σ , (72)where P ( R, θ ) , Q ( R, θ ) and S ( R, θ ) are polynomials ofpowers of R , sin θ and cos θ . We observe that the curva-ture invariants are all finite in the range of the radial co-ordinate R ( b < R < ∞ ), for b (cid:54) = 0 . The Carter-Penrosediagram, as well as further properties of this rotating SVgeometry can be found in Ref. [38].
2. Shadow and lensing
The rotating SV geometry is a particular case ofEq. (58), since A = 1 , B = 1 , A = B SV ( R ) . (73)Hence, this BH geometry is shadow degenerate with Kerrspacetime, regardless of b . A plot of the shadow edge ispresented in Fig. 6 (top panel) for the rotating SV space-time with different values of b . As was previously men-tioned, the shadow does not depend on the parameter b .Notwithstanding, the dependence on b through B SV ( R ) does not generically result in a Kerr degenerateshadow geometry. In particular, if one applies the modi-fied NJA to the class II shadow degenerate example (22),the resulting rotating BH geometry is not shadow degen-erate. B. Black holes without Z symmetry It has been pointed out that the BH shadow is not aprobe of the event horizon geometry [9]. As a second ex-ample of shadow-degeneracy in rotating, stationary andaxi-symmetric BHs, allowing separability of the HJ equa-tion, we shall provide a sharp example of the previousstatement. We shall show that a rotating, stationaryand axi-symmetric BH without Z symmetry (i.e. with-out a north-south hemispheres discrete invariance) can,nonetheless, be shadow degenerate with the Kerr BH.Geometries within the family (58) are Z symmetric ifinvariant under the transformation θ → π − θ, (74)which maps the north and south hemispheres into oneanother. The Kerr [5], Kerr-Newman [40], Kerr-Sen [41]and the rotating SV spacetimes are examples of BHs with Z symmetry. But BHs without Z symmetry are alsoknown in the literature. One example was constructedin Ref. [42]; this property was explicitly discussed inin Ref. [43], where the corresponding shadows were alsostudied.The general line element displaying shadow degen-eracy, under the assumptions discussed above is givenin Eq. (58). It has a dependence on the θ coordinatethrough the function B (see Eq. (33)). B ( θ ) needs notbe invariant under Z reflection: B ( θ ) (cid:54) = B ( π − θ ) ; (75)then, the BH geometry is shadow degenerate and displaysno Z symmetry.In order to provide a concrete example, B ( θ ) can bechosen to be cos θB ( θ ) = α + α cos θ + α cos θ + α cos θ, (76)while A and A are left unconstrained. In Eq. (76), α , α and α are constant parameters for which (cid:101) Σ > . If α = α = α = 0 , and α = 1 , (77)we recover the Kerr metric (provided that A = A =1 ). In Fig. 8, we show the Euclidean embedding of the -8-6-4-2 0 2 4 6 8-8 -6 -4 -2 0 2 4 6 8 Y XObserved black hole shadow for (cid:1) = (cid:2) /2Kerrb=0.3Mb=0.5M (a) Rotating SV -8-6-4-2 0 2 4 6 8-8 -6 -4 -2 0 2 4 6 8 Y XObserved black hole shadow for (cid:1) = (cid:2) /2Kerr( (cid:3) , (cid:3) , (cid:3) , (cid:3) )=(0, -2.6, 1, 0)( (cid:3) , (cid:3) , (cid:3) , (cid:3) )=(0, -4.2, 1, 2) (b) Rotating BH without Z symmetry FIG. 6. Top: Shadow of the rotating SV spacetime, for dif-ferent values of the parameter b and a = 0 . M . Bottom:Shadow of a BH without the Z symmetry, for different valuesof α i and the same value of the BH spin ( a = 0 . M ). Theobserver is located at spatial infinity. event horizon geometry of this BH spacetime for a/m =0 . . We also show the corresponding result for the KerrBH [44] (top panel). In the middle panel, we have chosen For higher values of the spin parameter a/m , it may be impos-sible to globally embed the spatial sections of the event horizon FIG. 7. The shadow and gravitational lensing for the rotatingSV spacetime, for a = 0 . M and different values of b . Inthe top panel, we have the Kerr spacetime ( b = 0 ). In themiddle and bottom panels, we have b = 0 . M and b = M ,respectively. The observer position and angle of view are thesame as in Figs. 2 and 5. Although the images look the same,there is a subtle difference in the gravitational lensing ob-served mainly next to the shadow edge. -2-1 0 1 2 -2 -1 0 1 2-2-1 0 1 -2 -1 0 1 2-2-1 0 1 -2 -1 0 1 2 FIG. 8. Euclidean embedding of the (spatial sections of the)event horizon geometry for the shadow degenerate BHs with-out Z symmetry. In the top panel, we show the Kerr re-sult, while in the middle and bottom panels we have cho-sen ( α , α , α , α ) = (0 , − . , , and ( α , α , α , α ) =(0 , − . , , , respectively. a/m = 0 . for all cases. the constants ( α , α , α , α ) = (0 , − . , , , whilein the bottom panel we have chosen ( α , α , α , α ) =(0 , − . , , . We note that the Z symmetry is clearlyviolated. Nonetheless, the shadow is always degeneratewith the Kerr BH one, as can be seen in the bottom panelof Fig. 6. VI. CONCLUSIONS
The imaging of the M87 supermassive BH by theEvent Horizon Telescope collaboration [2–4] has estab-lished the effectiveness of Very Large Baseline Interfer-ometry to probe the strong gravity region around astro-physical BHs. In due time, one may expect more BHs, geometry in Euclidean 3-space. This is also the case for the Kerrspacetime [44] - see also [45].
ACKNOWLEDGMENTS
The authors thank Fundação Amazônia de Amparoa Estudos e Pesquisas (FAPESPA), Conselho Nacionalde Desenvolvimento Científico e Tecnológico (CNPq) andCoordenação de Aperfeiçoamento de Pessoal de Nível Su-perior (Capes) - Finance Code 001, in Brazil, for partialfinancial support. This work is supported by the Cen-ter for Research and Development in Mathematics andApplications (CIDMA) through the Portuguese Founda-tion for Science and Technology (FCT - Fundação paraa Ciência e a Tecnologia), references UIDB/04106/2020and UIDP/04106/2020. We acknowledge support fromthe projects PTDC/FIS-OUT/28407/2017, CERN/FIS-PAR/0027/2019 and PTDC/FIS-AST/3041/2020. Thiswork has further been supported by the Euro-pean Union’s Horizon 2020 research and innova-tion (RISE) programme H2020-MSCA-RISE-2017 GrantNo. FunFiCO-777740. The authors would like to ac-knowledge networking support by the COST ActionCA16104. [1] B. P. Abbott et al. (Virgo, LIGO Scientific), Observationof Gravitational Waves from a Binary Black Hole Merger,Phys. Rev. Lett. , 061102 (2016), arXiv:1602.03837[gr-qc].[2] K. Akiyama et al. (Event Horizon Telescope), First M87Event Horizon Telescope Results. I. The Shadow of theSupermassive Black Hole, Astrophys. J. , L1 (2019).[3] K. Akiyama et al. (Event Horizon Telescope), First M87Event Horizon Telescope Results. V. Physical Origin ofthe Asymmetric Ring, Astrophys. J. , L5 (2019).[4] K. Akiyama et al. (Event Horizon Telescope), First M87Event Horizon Telescope Results. VI. The Shadow andMass of the Central Black Hole, Astrophys. J. , L6(2019).[5] R. P. Kerr, Gravitational field of a spinning mass as anexample of algebraically special metrics, Phys. Rev. Lett. , 237 (1963).[6] M. Kac, Can one hear the shape of a drum?, Am. Math.Mon. , 1 (1966).[7] S. H. Völkel and K. D. Kokkotas, Scalar Fields andParametrized Spherically Symmetric Black Holes: Canone hear the shape of space-time?, Phys. Rev. D ,044026 (2019), arXiv:1908.00252 [gr-qc].[8] H. Falcke, F. Melia, and E. Agol, Viewing the shadow ofthe black hole at the galactic center, Astrophys. J. ,L13 (2000), arXiv:astro-ph/9912263 [astro-ph].[9] P. V. Cunha, C. A. Herdeiro, and M. J. Rodriguez, Doesthe black hole shadow probe the event horizon geometry?, Phys. Rev. D , 084020 (2018), arXiv:1802.02675 [gr-qc].[10] P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu,Fundamental photon orbits: black hole shadows andspacetime instabilities, Phys. Rev. D , 024039 (2017),arXiv:1705.05461 [gr-qc].[11] E. Teo, Spherical photon orbits around a kerr black hole,General Relativity and Gravitation , 1909 (2003).[12] R. H. Boyer and R. W. Lindquist, Maximal analytic ex-tension of the Kerr metric, J. Math. Phys. , 265 (1967).[13] C. A. R. Herdeiro, A. M. Pombo, E. Radu, P. V. P.Cunha, and N. Sanchis-Gual, The imitation game: Procastars that can mimic the Schwarzschild shadow, (2021),arXiv:2102.01703 [gr-qc].[14] P. V. Cunha and C. A. Herdeiro, Stationary black holesand light rings, Phys. Rev. Lett. , 181101 (2020),arXiv:2003.06445 [gr-qc].[15] J. Grover and A. Wittig, Black Hole Shadows and In-variant Phase Space Structures, Phys. Rev. D , 024045(2017), arXiv:1705.07061 [gr-qc].[16] P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Lightring stability in ultra-compact objects, Phys. Rev. Lett. , 251102 (2017), arXiv:1708.04211 [gr-qc].[17] A. Simpson and M. Visser, Black-bounce to traversablewormhole, JCAP , 042, arXiv:1812.07114 [gr-qc].[18] E. T. Newman and A. I. Janis, Note on the Kerr spinningparticle metric, J. Math. Phys. , 915 (1965). [19] M. Azreg-Aïnou, Generating rotating regular black holesolutions without complexification, Phys. Rev. D ,064041 (2014), arXiv:1405.2569 [gr-qc].[20] P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, andH. F. Runarsson, Shadows of Kerr black holes with andwithout scalar hair, Proceedings, 3rd Amazonian Sympo-sium on Physics: Belem, Brazil, September 28-October2, 2015 , Int. J. Mod. Phys. D , 1641021 (2016),arXiv:1605.08293 [gr-qc].[21] S. Chen, M. Wang, and J. Jing, Polarization effects inKerr black hole shadow due to the coupling between pho-ton and bumblebee field, (2020), arXiv:2004.08857 [gr-qc].[22] R. Konoplya and A. Zhidenko, General parametrizationof black holes: the only parameters that matter, Phys.Rev. D , 124004 (2020), arXiv:2001.06100 [gr-qc].[23] L. C. S. Leite, C. F. B. Macedo, and L. C. B. Crispino,Black holes with surrounding matter and rainbow scat-tering, Phys. Rev. D , 064020 (2019), arXiv:1901.07074[gr-qc].[24] M. Morris and K. Thorne, Wormholes in space-time andtheir use for interstellar travel: A tool for teaching gen-eral relativity, Am. J. Phys. , 395 (1988).[25] H. C. D. Lima Junior, C. L. Benone, and L. C. B.Crispino, Scalar absorption: Black holes versuswormholes, Phys. Rev. D , 124009 (2020),arXiv:2006.03967 [gr-qc].[26] P. V. P. Cunha, J. Grover, C. Herdeiro, E. Radu,H. Runarsson, and A. Wittig, Chaotic lensing aroundboson stars and Kerr black holes with scalar hair, Phys.Rev. D , 104023 (2016), arXiv:1609.01340 [gr-qc].[27] A. Bohn, W. Throwe, F. Hébert, K. Henriksson,D. Bunandar, et al. , What does a binary black holemerger look like?, Class. Quant. Grav. , 065002 (2015),arXiv:1410.7775 [gr-qc].[28] P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F.Runarsson, Shadows of Kerr black holes with scalar hair,Phys. Rev. Lett. , 211102 (2015), arXiv:1509.00021[gr-qc].[29] A. Einstein, Lens-Like Action of a Star by the Devia-tion of Light in the Gravitational Field, Science , 506(1936).[30] B. Carter, Global structure of the Kerr family of gravi-tational fields, Phys. Rev. , 1559 (1968). [31] R. Shaikh, Black hole shadow in a general rotating space-time obtained through Newman-Janis algorithm, Phys.Rev. D , 024028 (2019), arXiv:1904.08322 [gr-qc].[32] H. C. L. Junior, L. C. Crispino, P. V. Cunha, andC. A. Herdeiro, Spinning black holes with a separa-ble Hamilton–Jacobi equation from a modified New-man–Janis algorithm, Eur. Phys. J. C , 1036 (2020),arXiv:2011.07301 [gr-qc].[33] S. Benenti and M. Francaviglia, Remarks on certain sepa-rability structures and their applications to General Rel-ativity , Gen. Rel. Grav. , 79 (1979).[34] G. O. Papadopoulos and K. D. Kokkotas, Preservingkerr symmetries in deformed spacetimes, Class. Quan-tum Gravity , 185014 (2018).[35] E. Poisson, A relativist’s toolkit: the mathematicsof black-hole mechanics. (Cambridge University Press,Cambridge, 2004).[36] D. C. Wilkins, Bound Geodesics in the Kerr Metric, Phys.Rev. D , 814 (1972).[37] P. V. P. Cunha and C. A. R. Herdeiro, Shadows andstrong gravitational lensing: a brief review, Gen. Rel.Grav. , 42 (2018), arXiv:1801.00860 [gr-qc].[38] J. Mazza, E. Franzin, and S. Liberati, A novel family ofrotating black hole mimickers, (2021), arXiv:2102.01105[gr-qc].[39] R. Shaikh, K. Pal, K. Pal, and T. Sarkar, Con-straining alternatives to the Kerr black hole, (2021),arXiv:2102.04299 [gr-qc].[40] E. T. Newman, R. Couch, K. Chinnapared, A. Exton,A. Prakash, and R. Torrence, Metric of a Rotating,Charged Mass, J. Math. Phys. , 918 (1965).[41] A. Sen, Rotating charged black hole solution in het-erotic string theory, Phys. Rev. Lett. , 1006 (1992),arXiv:hep-th/9204046.[42] D. Rasheed, The rotating dyonic black holes of Kaluza-Klein theory , Nucl. Phys. B. , 379 (1995).[43] P. V. P. Cunha, Herdeiro, C. A. R., and E. Radu, Isolatedblack holes without Z isometry, Phys. Rev. D , 104060(2018), arXiv:1808.06692 [gr-qc].[44] L. Smarr, Surface geometry of charged rotating blackholes, Phys. Rev. D , 289 (1973).[45] G. W. Gibbons, C. A. R. Herdeiro, and C. Rebelo,Global embedding of the Kerr black hole event horizoninto hyperbolic 3-space, Phys. Rev. D80