JHEP07(2020)054 Springer July 8, 2020 June 8, 2020 : June 15, 2020 April 29, 2020 : : : ed to a special Revised Published Received Accepted symmetry breaking and its hat the propagation of light endent of the polarization of otation of black hole. In the n bumblebee vector field and antum Structures Republic of China asted by polarized lights could n which the black hole shadow enon. The dependence of black ations, ience and Technology, cience and Technology, [email protected] , a,c Published for SISSA by https://doi.org/10.1007/JHEP07(2020)054 and Jiliang Jing [email protected] b , . 3 Mingzhi Wang 1 2004.08857 The Authors. a,c, c Black Holes, Classical Theories of Gravity

We present firstly the equation of motion for the photon coupl , [email protected] Corresponding author. 1 Yangzhou 225009, People’s Republic of China E-mail: School of Mathematics andQingdao, Physics, Shandong Qingdao 266061, University People’s of RepublicCenter Sc of for China Gravitation andYangzhou Cosmology, University, College of Physical S Synergetic Innovation Center forHunan Quantum Normal Effects University, and Changsha, Applic Hunan 410081, People’s Department of Physics, Keyand Laboratory of Quantum Low Control Dimensional of Qu Ministry of Education, b c a Open Access Article funded by SCOAP Abstract: Songbai Chen, Polarization effects in Kerrthe black coupling hole between shadow photon due and to bumblebee field ArXiv ePrint: electromagnetic field. These featureshelp of us black hole to shadow understand c theinteraction bumblebee with vector electromagnetic field field. with Lorentz Keywords: depends on its polarization duehole to the shadow birefringence on phenom thenon-rotating light’s case, polarization we is find dominated thatlight. by the the However, black the r hole status shadow is isdepends indep changed on in the the light’s rotating polarization case, and i the coupling betwee bumblebee vector field in a Kerr black hole spacetime and find t JHEP07(2020)054 1 2 6 12 into - ], the 9 – 5 e center of rk matter [ ortance in many fields defined in a higher scale the image. Generally, the ge has been applied exten- k hole, the propagation of ] is a simple effective theory fied gauge theories and the at Event Horizon Telescope nonzero vacuum expectation g of Lorentz symmetry can hole shadow depends also on f couplings. This interesting s under interactions between 15 ing effects could be observed in r higher energy scale. However, arameters and extra dimension nsion. In this theoretical model, tions between electromagnetic and o test such kind of physical theories ], which can be attributed to the bire- 11 ] imply that the spontaneous breaking of 13 , – 1 – 12 ]. Bumblebee model [ 14 ], and so on. Black hole shadow, caused by light rays that fall 10 ] released the first image of the supermassive black hole in th 2 , 1 ], and to probe some fundamental physics issues including da 4 , 3 It is well known that Lorentz invariance has been great of imp black hole spacetime blebee vector field of the fundamental physics.signals However, from the development high of energy uni cosmic rays [ it is possibleemerge that at sufficiently some low energy signals scalesexperiments associated and at their current with correspond energy scales the [ of breakin gravity with Lorentz violation in thethe standard spontaneous model breaking exte of Lorentz symmetry is induced by a Lorentz symmetry may emerge inof energy. the Generally, more it fundamental is currentlywith physics impossible Lorentz directly violation t through experimentation due to thei fringence phenomenon of lightfeature originating could from trigger such the kindselectromagnetic further o and study other of fields. black hole shadow size [ equivalence principle [ M87 galaxy at the lastsively year. to The study information the stored possibility in of the constraining ima black hole p One of the mostCollaboration exciting [ events in observing black holes is th 1 Introduction 3 Kerr black hole shadow due to coupling4 between the Summary photon and bum Contents 1 Introduction 2 Equation of motion for the photons coupled to bumblebee field in a Kerr the polarization of light itselfgravitational if fields, we consider such some as interac Weyl tensor coupling [ an event horizon, isblack one hole of shadow the inlight most the ray important and observer’s ingredients the sky position in is of determined observer. by It blac is shown that black JHEP07(2020)054 ] y 26 ]. In 41 d in a – ] and get 34 41 e solution is – ] are analyzed 34 tion of light and 32 ]. These analysis ] and the deflection 31 28 here chose bumblebee orresponding birefrin- ] and the cosmological err black hole spacetime proximation [ , we derive equations of 30 2 rent polarizations are not d some classical tests are he information on Lorentz oximation [ omagnetic field and gravi- ter candidate and provides cting the effects originating he propagation of photon in blebee vector field in a Kerr an axion cloud is studied in tion. We end the paper with lyze the propagation paths of e photon birefringence as pho- , we will consider the coupling igated in [ es of black hole shadow. More- ns. The polarization-dependent netic field coupled to bumblebee ck hole solution is generalized to . The recent investigation shows ] and the Hawking radiation [ 25 ]. An exact Schwarzschild-like black 24 ], accretion disk [ – 27 16 – 2 – ]. However, it is an open issue how the polarized problem. The axion dark matter distribution near 33 CP under a suitable potential. Recently, the bumblebee gravit µ B ] and in the protoplanetary disk around a young star [ 5 ] around the black hole are studied. The traversable wormhol 29 ]. The gravitational deflection angle of light [ , we study how Kerr black hole shadow change with the polariza 14 3 The plan of our paper is organized as follows: in section From the previous analysis, the interaction between electr Kerr black hole spacetime the equations of motion forblack hole the spacetime. photons The interacting simplest with action of bum the electromag section over, we also want tosymmetry see breaking. whether black hole shadow contains t motion for the photonsfrom coupled the to bumblebee modified vector Maxwell field equation in in a the K geometric optics ap between photon and bumblebeegence field on the to shadow study offield what a is Kerr effect that black of hole. thecoupled the The equation together c main of in reason motions this whythe model, for we different which the polarization is photons photons very and with critical to diffe probe to the ana properti by using the birefringence effectsbending of that the a polarization ray photo the of background light of experiences a bylight Kerr affect traveling black the through hole shadow of [ a rotating black hole. In this paper of particle [ implications of bumblebee gravitymay model be useful are in further testingfrom invest the the bumblebee spontaneous gravity Lorentz model symmetry and breaking. dete hole solution instudied this [ bumblebee gravityare model studied in is this obtained black holethe an spacetime. rotation Moreover, case this and bla the corresponding shadow [ value of a bumblebee field model has been extensively studied in literature [ M87 black hole [ tational field will modifythe Maxwell curved equation spacetime and depends lead on tothat its the that polarization coupling t direction between axion andton photon crosses also results over in axion th matter.an elegant Axion solution is to a compelling the dark strong mat the coupling parameter undera the summary. photon-bumblebee interac 2 Equation of motion for the photons coupled to bumblebee fiel also found in the framework of the bumblebee gravity theory [ In this section, we will make use of the geometric optics appr JHEP07(2020)054 ] 15 (2.1) (2.6) (2.2) (2.3) (2.4) (2.9) (2.7) (2.5) (2.8) . This , e ]  λ ) 24 µ – ]. With this B ] ( 14 41 24 – V . The condition – 2 is defined by [ 34 − b 14 ). In the geometric  U(1) symmetry, the ∓ µν ν ρ B 2.5 = F µ µν itational and electromag- ell equation is modified as , B is a dimensionless coupling F ion ( µ ρ α B B ropagation [ = 0 itten as a simpler form omagnetic field in background n, one can obtain equation of µ ,  . µ ρ αB = 0 4 F . ρ = 0 , , µ b , − ) is assumed to be much smaller than λµ µ , i.e., ν 2 µ B F λµ a b b λ , µν ν ν µν f ν αb ∂ iθ ν f k ± B = D is a rapidly varying phase. It implies that e k µ − > − µν + 2 θ + µν + ν B ν µ B f ν µ ρ a B νλ – 3 – νλ µ + µ F B = F f ( k ρ ∂ < B µ µ b µν V µν k µ = = D = F F µ = + αb + µν µν µν 2 B f V µν F B µν f −  F λ 1 4 λ k µν D − F , but larger than the electron Compton wavelength  is not dominated so that it can be neglected in this case. The L µ πG R λ ∇ ; , which can be treated as the usual photon momentum in quantum ) is supposed to own a minimum at 2 θ 16 is the usual electromagnetic tensor and µν b µ  f ), inducing Lorentz violation, can be expressed as [ ∂ g µν ± µ can be further expressed as − F = µ B ( . √ µ µν B 2 x k f V µ b 4 is satisfied when the vector field has a nonzero vacuum value [ B d ∓ 2 ( is a slowly varying amplitude and b is bumblebee field and the corresponding strength tensor = Z V is a positive constant. In order to ensure the breaking of the ∓ µ µ µν 2 = b f B b = µ S b µ B µ ensures that we cannetic neglect fields the with change the typical of curvature the scale background for grav the photon p a typical curvature scale where potential approximation, the electromagnetic field strength can be wr optics approximation, the wavelength of photon theories. With the help of the Bianchi identity where where The potential constant. With the variational method, one can find that Maxw B wave vector is vector field in the curved spacetime can be expressed as the derivative term with one can obtain the constraint for the amplitude motion for a coupled photon from the corrected Maxwell equat which means that the couplingspacetime. changes propagation Resorting of electr to the geometric optics approximatio It means that JHEP07(2020)054 ) d 2.6 (2.14) (2.10) (2.11) (2.12) (2.15) (2.13) (2.16) . . ∆ , θ 2 ) 2 = 0 ρ sin le, whose line-element ωdt a 2 µ a − = 0. Inserting eqs. ( k θ, ν 2 − µ k dφ a 2 ned by µ ( ) µ b cos θ 2 k , l set of orthonormal frames in ρ 2 2 a . s the propagation of the coupled a .       αb + sin + θ       2 = 0 2 2 θ 2 r + 2 θ ρ r Σ sin    ν ρ 0 0 t sin θ a φ = ( = Σ , + ρ a µ a a Σ sin b ν ω ρ Σ 2 2 2 k e ρ Σ − 1 ρ 0 0 a µ    dθ k e 2 0 0 ρ µ    ∆ ρ ab b ρ 0 0 0 0 0 η ρ √ ∆ 33 23 13 – 4 – ρ 0 0 0 0 + √ = αb K K K 2 ∆ ∆ Σ 2 0 0 0 Σ , √ √ ω ∆ dr µν 32 22 12 ρ ρ Σ φ 0 0 0 − 2 √ g = 0, one can find that the equation of motion of the a ∆ K K K ρ ρ ρ       µ a k µ + 31 21 11       , = µ k 2 2 a K K K µ µ a a = denote the mass and the angular momentum per unit mass , and satisfying the condition that e dt k θ a µ    a ν + 2 a µ e b ∆ ) can be simplified as a set of equations for three independent , a Σ ρ t 2 a , ρ Mr ρ and αb 2 2.10 2 − ), we can get that the equation of motion for the photon couple 2 − M − Σ = aMr 2.5 2 ν 2 r 2 a µ = ds k µ ω ∆ = k is the Minkowski metric and the vierbeins ) into eq. ( ab η 2.9 The Kerr metric describes the geometry of a rotation black ho photon coupling with ( with the inverse photon in the background spacetime. It is obvious that the interaction with bumblebee field affect polarisation components with and ( with bumblebee vector field Here the parameters Making use of the relationship with a polarization vector of the black hole,Kerr respectively. black In hole order spacetime, to one introduce can a use the loca vierbein fields defi where can be expressed as JHEP07(2020)054 ) ]. , 2 42 ) 2.17 φ ), i.e., = 0 (2.19) (2.22) (2.21) (2.20) (2.17) (2.18) k θ 2 3 φ = 0, the k e 2 = 0 [ r 2.16 dθ α k + dθ 2 ν θ ) t b k α k r αr µ 3 t b 2 tion ( k r, θ 2 e ρ ( ) + 2 µν 2 θ 1 + 2 e γ 2 W + ( ( ρ θ 2 ) recover to the usual + ) k 1 r + θ e owing a special case of k ( on vanishes because the 2 2.18 , i.e., 2 pacelike and has the form ) field yields that a coupled drdθ he polarizations of photon, 2 θ = 0 in a general case. Here, µν ) drdθ e oupling constant αb | γ α 2 , these photons can be looked ]( ∆ 8 of light, the bumblebee vector θ K θ r, θ | sponded to the equations ( − b ( √ ctions. ) only if the determinant of the ) and ( θ . 1 ) θ . b cos α W 2  + ) 0 2.16 2.17 ∆ cos 2 θ = 0 2 e ) αar √ = 0 for above bumblebee field and it ( θ, φ ) 4 =1 α k (1 + 2 µ α αar µ 3 φ = 0). However, we find that in the Kerr 2 8 B cos B e | ) actually indicate that the motion of the − µ ν − 2 ∂ B + K

, a | t (1 ν dr − 2.18 k ∆ – 5 – + [1 + 2 θ r 3 t ν ∂V − e ∂x r √ 2 B 2 k , µ r = + ( 0 )∆ cos ∂ k dr , , θ 2 2 α )  ) and ( 2 2 ) µν k ) ) = 1 r θ ) are very complicated and we do not list them here for αa = B e r, θ k 2 µ ( 2.17 2 ]( µ µν ωdt ρ ωdt = 1, and ) r ∇ (1 + 2 + 2 B 2 θ b B W 2 − . r − e 2 b t, θ, φ b ∆ ρ 2 dφ ) dφ = + ( = 0 ( + 1 r ( + r θ e θ θ 2 2 ( k k 2 is equal to zero (.i.e., 2 i, j r r dt α dt | k k 2 2 sin ,( 2 sin θ ∆ ∆ K ) Σ Σ b ij | 2 2 2 1 r 2 r 2 2 e ρ Σ ρ b Σ K ρ ρ 2 ) + − − + 0). We find that in this case there exist two solutions for equa 2 θ + ( + [1 + 2 , e t t θ = ( = k k 2 t t , b 2 I ) r k k 2 II 1 r 2 2 ds e ) is ) ) , b ( ds 0 t 0 t α e e ( ( The light cone conditions ( 2.18 = (0 − − + 4 µ satisfies the equation of motion It is obvious that the tensor or ( field and thebumblebee coupling vector constant. field with Here we focus on only the foll photon propagation is independent of its polarization dire coupled photons is non-geodesic inas the Kerr moving metric. along However Obviously, the the null effective metric geodesics depends of on the the polarization effective metric form in a Kerr spacetime in which the birefringence phenomen simplicity. There exists thecoefficient non-zero matrix solution of eq. ( or The coefficients We find that the effective metric for the polarized light corre black hole spacetime it is difficult to find a solution satisfied we focus on only ab special case in which the bumblebee field is s This means that thephoton effects propagation of in the a Kerr interactionwhich black with is hole bumblebee the spacetime so-called depend birefringence on t phenomenon. When the c interaction vanish and then the light-cone conditions ( JHEP07(2020)054 ) ). ) at , the 2.21 2.25 2 1 (2.27) (2.26) (2.25) (2.23) (2.28) (2.24) 2.22 − ic ( , d bum- , sual Kerr ) 2 = )–( ) L α − . θ ωdt 2.21 2 2 ) does not affect ,  −  L sin -component of the , which means that θ z dφ α 2.21 − ( 2 aE θ L θ θ ( 2 , 2 ) does not depend on its sin a sin ive effective metric ( sin − − sin = 0 ] s. Thus, in order to avoid 2 2.22 desics for the effective met- 2 arized light in the Kerr black ) reduces to that of the usual 2 ctive metrics ( ρ casted by the polarized lights. Σ ) aE aE ˙ . aL t parameter ve metric (  θ  + ω ve metric, the coupling constant . Obviously, there exist two con- 2.21 − tric ( oton does not depend on its four- 2 4 2 θ − − − ρ E ] 2 ˙ dθ 2 ) cos φ ) ] 2 2 ( 2 α ρ aL a θ and i a 2 aL 2 2 r U − + r ∆ − − 2 2 sin + E r α ) (1 E 2 2 2 ) ∆ ∆ ρ )[( 16 Σ 2 a 2 . For a sake of simplicity, we write the effective a – 6 – drdθ a
− + + 1 2 + 2 2 2 ∆ + ˙ , t r 2 ρ 2 1 2 √ 2 r [( r i ∆ Σ [( ( a − ) = 1 H 2 2 at arbitrary position. This means that the propagation ρ = = = ∈ r, θ ˙ − ˙ ˙ t − φ ( 2 i tends to zero, the metric ( φ ˙ α 2 θ 2 2 = ρ a 4 at the arbitrary spacetime point. Similarly, as W ρ ρ ν dr ˙ i θ and ˙ 2 x , which corresponds to the energy and the U ˙ ∆ t ρ µ L ˙ i x ) as a unified form + F and i µν ˙ θ are functions of coordinates r γ + i and ˙ i r 2.22 2 , one can find the null geodesic condition in the effective metr makes the effective metric more complicated than that of the u U 1 2 4 ∆ E dt ρ rθ 2 √ γ = ∆ i Σ 2 H α ) and ( , and ρ i 2 − H − 2.21 , ) can be expressed as 2 i i = . As ˙ r 2 1 F 2 i 4 ∆ ± ρ blebee vector field 2.25 ds i F = must be limited in the range angular momentum of the polarized photon moving in the effect served quantities α of polarized photonthese are singularities unphysical arising in from these the two coupling limited in the case effecti Schwarzschild black hole and is independent of the coupling With the help ofric these ( two conserved quantities, the null geo four-velocity components becomes α dispersion relation for arbitrary photon in the effective me the polarized light moving along the geodesics in the effecti where one. Moreover, we find that the singularity occurs for the effe metrics ( The cross term 3 Kerr black hole shadow due to coupling between the photon an With these equations, we can studyhole space the time propagation and of probe the the pol properties of black hole shadow velocity components ˙ which implies that the dispersion relation for arbitrary ph with When the rotation parameter JHEP07(2020)054 , ] ). 3, 2 . Q 54 E 0 – ) for (3.3) (3.1) (3.2) − 2.22 = n this 43 ect the = σ 2.28 α ween black and , 0 L E ton sphere for θ 2 = ξ 0 for the metric ( cot θ 2 ξ ), it is related to the α ξ same as that of usual sin − − 2.22 σ tion paths are different in e operation in refs. [ = rameters p marked with adjacent brown , ng of black hole. The white ) y different colors (green, blue, on of Einstein ring. The black 0 ± ght and the coupling parameter ,θ )∆ 0 2 = = 0, and the observer locates in the r ( L )

a 0 θ, r φ ,θ + 2 confirms that the black hole shadows p changes with the polarization of light 0 p r Q 1 ( 1 rr φφ

( cot g g r θ . This means that the black hole shadows 2 p − p √ √ α L 2 r rr θθ g g − E 4 √ √ – 7 – Q r →∞ r lim 0 = 0. The corresponding null geodesics ( r = = a − 2 2 →∞ 2. The figures from left to right correspond to ˙ ˙ r 0 θ r = 4 ) , we also plot the effect of the polarized light on the π/ r ) α 1 0 = lim = 4 ,θ ) 0 r 0 r ,θ ( obs

0 θ r (1 + 2 = 1 and rotation parameter ˆ ˆ r φ (

p p ˆ ˆ r θ r M p p and r in the case with . The distribution of these color regions and lines in can refl M →∞ , which means the birefringence phenomenon occurs exactly i ◦ lim 0 α α r are caused by lights from a reference spot lied in the line bet →∞ 0 r − 1 = 50 = = lim obs 2, respectively. . r y x is the Carter constant. It is obvious that in this case the pho . The change of black hole shadow with the coupling constant Q 2, 0, and 0 . 0 . However, the color region distribution in figure − the black hole shadow in this case. However, for the metric ( position with we can obtain the two celestial coordinates hole and observer, which couldpatch provide is a used direct to demonstrati denotein non-rotating black case hole do shadow. not depends Figure α on the polarization of li and the parameter black hole shadow inand the divide non-rotating the case. celestial Here, spherered, we into and four adopt yellow). quadrants th marked Thelines b grid separated of by longitude 10 and latitude lines is coupling parameter the polarized light becomes Here we set the black hole distortion of an imageregions in due figures to the strong gravitational lensi which are independent of the coupling constant where casted by the polarizedthe lights non-rotating are case. same In even figure if their propaga Figure 1 Schwarzschild case without the coupling. With two impact pa the polarized light does not depend on the coupling and is the JHEP07(2020)054 (3.8) (3.5) (3.4) (3.6) (3.7) eaking , ). , which is φφ ay-tracing” α rized lights. -component γ 2.19 θ tt γ φφ denote the lens- γ − . 2 tφ OL , where corresponds γ φφ 1 ) can not be variable- D γ s tt l system of two kind of } ] γ φ φφ bserver backward in time 2.28 = 56 is the metric of Minkowski , and γ , ∂ − – ˆ β θ umblebee field ( ˆ LS α 2 tφ 43 y assigned to a pixel in a final -component and LS ), can be expressed as η , ∂ γ ole shadow will change. Due to r D ole shadows do not contain any r OS ), one can expand the observer D ated with a reference frame with , ζ s ely. It is obvious that the radius D , ∂ OL . 2.22 t , tφ ∂ φφ D 2.25 φφ , and 1 OL { γ γ γ ˆ      2 β / ˆ α D φ √ 1 − η 0 0 γ ) A , i.e., the + ˙ α = = , , θ = M ν ˆ ˆ ν θ µ r rθ φ ˆ 4 ν β δ ∂ LS A e A ˆ ν µ ˆ D e µ α = r e (1 + 2 0 0 – 8 – ˆ ν A = = e µν s ˆ ˆ µ µ 0 0 0 0 0 0 ζ γ case. It is clear that eq. ( e e OS =      a ,A D E = θ θθ , γ γ 1 ˆ ν µ 2 rθ are real coefficients. From the Minkowski normalization e OL √ γ obeys to φ D . The radius of Einstein ring in figure ˆ µ ν − = A α e θθ ≃ θ γ θθ E γ R ,and rr γ rθ decreases with the increase of the coupling parameter A ,A r as a form in the coordinate basis . In a word, in the non-rotating case, the Lorentz symmetry br E , 1 ) without separable variables, we must resort to “backward r } rr rθ θθ θ R ˆ φ γ 1 γ γ A √ ] to study numerically the shadow casted by such kinds of pola , e , − 2.28 ˆ θ r 54 = = A , e – ˆ r r , rθ γ 43 A , e A ˆ t , e ζ { Let us now focus on the nonzero observed in figure source distance and theof observer-lens Einstein distance, respectiv ring do not decouple frompolarized each photons other. is non-integrable and Thisthe the means shape equation of that ( black the h dynamica where the transform matrix spactime. It is convenient tozero choose a axial decomposition angular associ momentum in relation to spatial infinity [ where the observer-source distance method [ In this method, theand light the rays information are carried assumed byimage each to in ray evolve would the from be observer’s the respectivel basis sky. o For the effective metric ( separable due to the presence of the cross term ˙ changes the propagation of polarizedinformation lights, on but Lorentz the black symmetry h breaking for the selected b to the non-rotating case related to the effective metric ( case without non-vanishing where one can obtain JHEP07(2020)054 ). 3, . 0 (3.9) − 2.21 (3.11) (3.10) , = θ squeezes , we find , we find α 5 5 α – – 2 4 Σ sin , i θ
˙ p r . Comparing the i , we present black a 2 rθ 5 , H γ – for the metric (
˙ 2 r − − i α or figures ˙ θθ -axial direction for the θ . γ H and the polarization of y θθ 3 can be further written as ν ∆ γ – p ˆ − µ α of a photon by the projec- 2 √ ν i rr ˆ p ˙ L i θ e ˆ γ µ F ∆ p ), but the positive ∆ i o depends on the polarization res = However, in the rotating case, r
i 5, and the observer locates in the case. From figures ˙ F . √ i r ˆ i ) i H rθ θθ p 2 i 2.22 F γ γ H = 0 + = H i ). In figures
a − i − ˆ ˙ H θ − ˙ r L. θ obs p 2 ∆ i + ∆ ) and ( , θ φφ F rr
√ and rotation parameter 1 ( ˙ γ i √ ], one can obtain the position of photon’s γ θ 1 i obs α √ H q √ ∆ F r 54 2.21 – √ − 2 = i i = ˙ – 9 – r 43 ˆ φ H ˆ r i H F − − , p elongates the shadow along 2 2 i ˙ , i ν r 2. The figures from left to right correspond to i ˆ µ p F α H , we also plot the change of the black hole shadow size e ˆ ν t F π/ 6 e q − 2 i = − 2 i H F onto = obs − ˆ t θ µ q = 1 and rotation parameter p 2 p i h in the both cases ( − , p F a M θ and obs γL, p ), the locally measured four-momentum p = q r ˆ t − θθ M − p 3.8 obs γ 1 r -axis. In figure = √ ζE y ˆ ˆ r = 50 = φ p p = = ˆ ˆ r θ ˆ t p p ˆ obs θ 2, respectively. . r p obs p r obs r − . The change of black hole shadow with the coupling constant = = y x 2, 0, and 0 . 0 − Therefore, one can get the locally measured four-momentum tion of its four-momentum With the help of eq. ( position with color regions distributions with the same parameters in figu the shadow along where the spatial position of observer is set to ( the black hole shadowlight, which also differs depends from on thosethat in the the the negative coupling previous coupling non-rotating constant constant hole shadow for the different coupling constant that the propagation of lightof in light, the which black is hole similar spacetime to als those in the non-rotating case. image in observer’s sky Here we set the black hole fixed rotation parameter Repeating the similar operations in refs. [ Figure 2 JHEP07(2020)054 ). ). ). 3, 3, 3, 0) . . . , 0 0 0 and r − − − 2.21 2.22 2.22 x s (3.12) R = = = α α α . Here, α for the metric ( for the metric ( for the metric ( α α α , left position of shadow l x s − R r x 5, and the observer locates in the . 998, and the observer locates in the 998, and the observer locates in the − . . = 0 = 0 = 0 = 2 a a a s with the coupling constant s δ – 10 – , δ 2 t y ) 2. The figures from left to right correspond to 2. The figures from left to right correspond to 2. The figures from left to right correspond to t + x π/ π/ π/ 2 ) − r = = = r x x obs obs obs − θ θ θ 2( t = 1 and rotation parameter ], can be expressed as = 1 and rotation parameter = 1 and rotation parameter x ( 57 M and and and M M = ) is located at the top position of the shadow. The points ( t s M M M , y R t x = 50 = 50 = 50 and the distortion parameter obs obs obs 2, respectively. 2, respectively. 2, respectively. . . . r r r s R . The change of black hole shadow with the coupling constant . The change of black hole shadow with the coupling constant . The change of black hole shadow with the coupling constant 0) lie, respectively, at the most right position and the most , l x 2, 0, and 0 2, 0, and 0 2, 0, and 0 . . . , firstly defined in ref. [ 0 0 0 s − δ position with − − parameter position with position with and ( Here we set the black hole Here we set the black hole Here we set the black hole where the point ( Figure 5 Figure 4 Figure 3 JHEP07(2020)054 in ht α with the olarized s δ than in the s δ w size decrease for fixed th the absolute value s obvious that the effect of w arising from the chaotic lensing . The left and right panels correspond a ck hole shadow. . It shows that there exist the eyebrow and the distortion parameter 5 s , we can find the eyebrow shape shadow R 5 , we find that the negative coupling constant 6 – 11 – ) has a smaller distortion parameter ), respectively. 2.21 2.22 = 0. From figure ) and ( y 2.21 . Moreover, we find that the shadow casted by the polarized lig ). With the increase of the rotation parameter, the effect of p for the different rotation parameter a α 2.22 . The enlargement of the left panel in figure . The change of the black hole shadow size make the shadow size increase and positive one make the shado lights on the blackLorentz hole symmetry shadow breaking becomes on morethis black case. distinctly. hole Moreover, It shadow from increases i theand left the wi panel self-similar in fractal structures figure of black hole shado along the horizontal line α shape shadow and the self-similar fractal structures in bla rotation parameter Figure 7 Figure 6 effective metric ( propagating in the effective metric ( coupling parameter to the effective metrics ( JHEP07(2020)054 998, . t the = 0 squeezes a α 2. ) has a smaller π/ 4 and . 0 = 2.21 − -axial direction, and obs = θ y ctor field with Lorentz α neration Event Horizon ith the quicker rotation . The positive and 4. These features of black se, we find that the black a . sponding interactions with e status is changed because stand deeply the bumblebee 0 r ack hole spacetime and then he coupling constant even if eans that Lorentz symmetry M ht and the coupling constant arized lights may be detected tion for the photon coupled to − ) with tive metric ( in shadow. Here we set the black c field in the nature. ). The effect of polarized lights = = 50 3, but it is disconnected from the 2.22 . α 0 2.22 obs r − ) does not affect the black hole shadow = with . We also note that the main eyebrow is 7 8 α 2.19 with – 12 – 5 elongates the shadow along α than in the effective metric ( s δ -axis, but the shadow size is decreased. Moreover, we find tha y . The black hole shadow for the effective metric ( = 1 and the observer locates in the position with M . The negative coupling constant of polarized lights, which is also shown in figure hole which shows that the main eyebrow is disconnected from the ma Telescope, these features of blackin hole the shadow future casted by andsymmetry pol breaking could and help its us interaction with to electromagneti understand the bumblebee ve distortion parameter on the black holeand shadow the becomes stronger more coupling. distinctly in With the the case development of w the new ge in these two specialthe cases. black hole However, shadow in alsoα depends the on rotating the case, polarization th of lig connected to the main shadow in figure breaking arising from bumblebee vector field ( In this paper, we havethe investigated firstly three the special equation kinds of ofprobed mo bumblebee further vector the fields in blackhole a hole shadow Kerr is shadow. bl independentthe In of polarized the the lights polarization non-rotating propagate of ca along light different and paths. t This m hole shadow casted by polarizedvector lights fields could with help Lorentz uselectromagnetic symmetry to field. under breaking and their corre 4 Summary main shadow by a bright vertical line in figure make the shadow size increase for the fixed rotation paramete the shadow along Figure 8 shadow casted by the polarized light propagating in the effec JHEP07(2020)054 ] , 4 ]. ]. ommons , SPIRE , SPIRE IN IN ][ JETP Lett. ][ , , (2019) L6 les arXiv:1909.09385 [ 875 , an Provincial Education pported by the National (2019) L1 ience Foundation of China ]. ]. 5. ]. with Shadow Images redited. 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