New test on the Einstein equivalence principle through the photon ring of black holes
NNew test on the Einstein equivalence principle through the photon ring of black holes
Chunlong Li,
1, 2, 3, ∗ Hongsheng Zhao,
1, 4, † and Yi-Fu Cai
1, 2, 3, ‡ Department of Astronomy, School of Physical Sciences,University of Science and Technology of China, Hefei, Anhui 230026, China CAS Key Laboratory for Research in Galaxies and Cosmology,University of Science and Technology of China, Hefei, Anhui 230026, China School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China Scottish Universities Physics Alliance, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK
Einstein equivalence principle (EEP), as one of the foundations of general relativity, is a fun-damental test of gravity theories. In this paper, we propose a new method to test the EEP ofelectromagnetic interactions through observations of black hole photon rings, which naturally ex-tends the scale of Newtonian and post-Newtoian gravity where the EEP violation through a variablefine structure constant has been well constrained to that of stronger gravity. We start from a generalform of Lagrangian that violates EEP, where a specific EEP violation model could be regarded asone of the cases of this Lagrangian. Within the geometrical optical approximation, we find thatthe dispersion relation of photons is modified: for photons moving in circular orbit, the dispersionrelation simplifies, and behaves such that photons with different linear polarizations perceive differ-ent gravitational potentials. This makes the size of black hole photon ring depend on polarization.Further assuming that the EEP violation is small, we derive an approximate analytic expressionfor spherical black holes showing that the change in size of the photon ring is proportional to theviolation parameters. We also discuss several cases of this analytic expression for specific models.Finally, we explore the effects of black hole rotation and derive a modified proportionality relationbetween the change in size of photon ring and the violation parameters. The numerical and analyticresults show that the influence of black hole rotation on the constraints of EEP violation is relativelyweak for small magnitude of EEP violation and small rotation speed of black holes.
I. INTRODUCTION
The Event Horizon Telescope (EHT) has captured thefirst image of a supermassive black hole [1–6]. This pro-vides us possibilities of probing new physics in a stronggravitational field. In the center of the galaxy M87, thecompact radio source is resolved as an asymmetric brightemission disk, which encompasses a central dark region.In the current literature, although the details remain tobe examined, it is pointed out that there could exit astrong lensing structure which is called “photon ring”behind the dominated direct emission profile from theaccretion disk [7, 8]. With the help of sufficient high-resolution imaging, complexities from astrophysical ef-fects can be mitigated, as the size and shape of the pho-ton ring are totally determined by the instabilities of pho-ton orbits predicted by geodesic equations, which makesblack hole photon ring become a potential probe to testgravity and related new physics [9–13].The high spatial resolution image taken by the EHTand the great potential of the photon ring on testing grav-ity have inspired a series of work. One type of the worksfocuses on the possible contamination of the perfect vac-uum environments around the black hole, which couldarise from the coupling of gravity to other backgroundfields and the accumulation of dark matter due to ac- ∗ [email protected] † [email protected] ‡ [email protected] cretion. Such novel physics modify the black hole met-ric and leave observable effects on the black hole photonring [14–20]. On the other side, the question is abouttesting gravity theories. Some modified gravity theoriescould have different black hole solutions from those of thegeneral relativity and thus lead to different patterns ofphoton motion [21–37]. We refer readers to [38] for con-straints on gravity theories under a parameterized post-Newtonian formalism. For these two types of works, inaddition to mass, spin and electric charge, more param-eters are introduced to describe the spacetime aroundblack holes. These lines of works can thus be effectivelyregraded as theories that violate the no-hair theorem ofblack holes.Besides violation of no-hair theorem, another mani-festation of new physics beyond general relativity is thebreakdown of Einstein equivalence principle (EEP). Instandard general relativity, the unique gravitational fielddescribed by the metric is minimally coupled to the cos-mic components including matter and interactions. Thismeans gravity plays the role of a geometrical background:in a local free-falling frame where geometrical effectsare canceled by the local transformations of referenceframes, the fundamental non-gravitational physics returnto those without gravity [39–41]. If the cosmic compo-nents are non-minimally coupled to gravity, or coupledto other unknown background fields, the coupling effectsgenerally cannot be canceled by local transformations ofthe reference frame, and thus the EEP is not valid any-more. Therefore, whether EEP is established or not con-tains the information about the coupling among differentcomponents of the Universe to gravity, which allowing us a r X i v : . [ g r- q c ] M a r to test possible new physics. We refer readers to [40] fora more precise introduction of the EEP and other kindsof equivalence principle.One of the main challenges of modern physics is theconfirmation of the EEP at different scales and differentcontexts both from the theoretical and the experimentalpoints of view [42–51]. However, the current experimentson testing EEP are mainly conducted on the scale ofNewtonian or post-Newtonian gravity [52–61]. As formore extreme gravitational field such as that around thehorizon of black holes, whether EEP holds or not is stillunknown.Black hole photon ring provides us with the opportu-nity to test EEP in the extremely strong gravitationalfiled. It is the lensed image of unstable bound orbits ofphotons around black hole. For Schwarzschild black hole,these orbits are circular with their radius equals to onlythree times the gravitational radius. Therefore, blackhole photon ring, as the observable of unstable boundorbits, could become a potential probe to detect newphysics in the gravitational field near the horizon scale.Possibility of violation of the EEP near a black holecould be suggested by the superradiance process of ro-tating black holes [62–66], which implies there might bea fruitful environment of light particles around rotat-ing black holes at horizon scale [67–73]. Moreover, ifthese particles are those beyond the standard model andare coupled to photons, there might be a phenomeno-logical violation of EEP. Furthermore, when we considerthe effects of quantum field theory in curved spacetime,the vacuum polarization of photons can introduce a non-minimal coupling of electromagnetic fields to the space-time curvature [74–77]. This could also lead to a violationof EEP.In this paper, we put forward a new method to test theEEP by using the observations of black hole photon ring.In Sec. II, we focus on the EEP violation occurring inelectromagnetic interactions and start from a general La-grangian that describes a new background field coupledwith the electromagnetic field. In Sec. III, we apply thegeometric optics approximation to derive a modified dis-persion relation of photons and obtain the correspondingphenomenological behavior by restricting our discussionsto a system with the static and spherical symmetry. Thenin Sec. IV and Sec. V, we derive and show a connectionbetween the size of the photon ring and the parametersof the EEP violation. We also give several specific exam-ples of the EEP violation in Sec. VI. Finally, in Sec. VII,we generalize our discussions to rotating black holes andstudy the influence of all kinds of black hole parameterson the constraints of violation parameters. Sec. VIII isa summary of the main results with a discussion and afuture outlook. We work in units where the gravitationalconstant G = 1 and the speed of light c = 1 and we adoptthe metric convention ( − , + , + , +). II. THE MODEL OF EEP VIOLATION
There are several characteristic scales for an electro-magnetic system in a curved spacetime. One is the vary-ing scale λ of the electromagnetic field, the other is thecharacteristic length L R of the spacetime curvature. Andif there exists some additional background fields that arecoupled to this electro-gravitational system, more lengthscales L Q characterizing these fields would be involved.The geometric optics approximation states that if λ ismuch smaller than another other characteristic scales ofthe system, i.e. λ (cid:28) min { L R , L Q } , the electromagneticvector A µ generally have the following solution [78] A µ ( x ) = a µ ( (cid:15), x ) e i(cid:15) S ( x ) , (1)where (cid:15) is a small quantity which represents a rapidlyvarying phase and the expression of a µ is a µ ( (cid:15), x ) = ∞ (cid:88) m =0 (cid:16) (cid:15)i (cid:17) m a m ( x ) . (2)At the lowest order of the geometric optics approxima-tion, i.e. the order of 1 /(cid:15) , the wave equation givesrise to the dispersion relation of the 4-wave vector k µ =(1 /(cid:15) ) ∂ µ S , which determines the equation of motion oftest photons. Intuitively, we could generally write thedispersion relation as b µ k µ + c µν k µ k ν + O ( k ) = 0 , (3)where b µ and c µν are vector and tensor fields. O ( k )represents the potential correction terms that are higherthan the second power of k µ . In the current literature,there exists the following three cases for O ( k ) = 0: • b µ = 0, c µν = g µν . This corresponds to the stan-dard case g µν k µ k ν = 0 in the general relativity,where g µν is spacetime metric. • b µ (cid:54) = 0, c µν = g µν . This case describes a back-ground vector field couples with the motion of pho-tons [79]. An example is the correction to the low-est order geometric approximation due to the largespacetime curvature, where b µ is constructed by thespacetime curvature tensor and the null tetrad ofphotons [80, 81]. • b µ = 0, c µν (cid:54) = g µν . A background field non-minimally coupled to the electromagnetic tensoroften leads to this kind of modification. An ex-ample is the non-minimal coupling of electromag-netic field to spacetime curvature induced by thevirtual electron loops in quantum electrodynamics[74–77], which could give a c µν different from themetric g µν .A modified dispersion relation given by Eq. (3) oftenmanifests itself by violation of the EEP. In this paper,we focus on the third cases, i.e. the quadratic correction c µν . Firstly, let us consider the below electromagneticLagrangian which preserves the diffeomorphism and U (1)gauge invariance L em = − F µν F µν − qQ µνρσ ( x ) F µν F ρσ − eJ µ A µ , (4)where F µν = ∇ µ A ν − ∇ ν A µ is the electromagnetic ten-sor and ∇ µ is the covariant derivative with respect to theLevi-Civta connection. The final term describes the cou-pling to the matter current J µ through the charge e andwe does not written down the corresponding matter ac-tion since it is not related to our discussions. The secondterm is beyond the standard physics, which describes anunknown background field Q µνρσ is non-minimally cou-pled to electromagnetic tensor through the coupling con-stant q . Because of the index symmetry of F µν F ρσ , thefield Q µνρσ should satisfy Q [ µν ] ρσ = Q µν [ ρσ ] = Q µνρσ andthe exchanging symmetry of µν , ρσ as a whole.In principle, Q µνρσ ( x ) could be a scalar, vector, ten-sor or a sum of these parts. The current work ontesting action (4) mainly focus on the scalar part of Q µνρσ ( x ). For example, if Q µνρσ is a scalar field φ , i.e. Q µνρσ = 2 f ( φ ) g ρ [ µ g ν ] σ [56, 57], varying the Lagrangian(4) with respect to A µ , one will obtain the modifiedMaxwell equations ∇ µ [(1 − qf ( φ )) F µν ] = eJ ν . (5)For the practical case, φ should only vary little over largedistances and times. Thus, qf ( φ ) could be taken outof the derivative, which is equivalent to replacing theelectric charge e with a field e (cid:48) = e/ (1 − qf ( φ )). Thisfact tells us that the fine structure constant will have avariable value over the spacetime, which is often calledthe violation of the local position invariance (LPI) in theliterature, as one of the elements of the EEP [39–41].Then for the experimental test of this new coupling, onecould detect whether atomic spectra at different locationsrepresent the same fine structure constant such as a givenkind of atom on Earth and the same kind of atom on starsin orbits of supermassive black holes. We refer readersto [52–54] for more details.As for the vector or tensor parts of Q µνρσ ( x ), the abovemethod could not give rise to a simple result character-izing by a varying fine structure constant. Another de-fect of the above method is that it is an indirect test onthe EEP violation Lagrangian (4), which depends on thecoupling to the matter field, i.e. − eJ µ A µ and might notexclude the influence of properties of matter itself. Inorder to explore whether there exists the EEP violationterm in the Lagrangian (4), one need to seek a directmethod to test this term. III. THE PHENOMENOLOGICAL BEHAVIOROF THE EEP VIOLATION MODEL
In this section, we apply the geometric optics approxi-mation to the first and second term of the Lagrangian (4). This leads to a modified dispersion relation c µν k µ k ν = 0under some conditions where c µν is no longer the space-time metric and depends on the polarizations of photons.Therefore, photons with different polarizations could fol-low different propagation paths, which thus violates theweak equivalence principle (WEP), as another element ofthe EEP [39–41]. In the following, we will explain howthis mechanism works.Let us neglect the last term in the Lagrangian (4) andvary this action with respect to 4-vector potential A µ ,which gives ∇ µ (cid:18) F µν − qQ µνσρ F ρσ (cid:19) = 0 . (6)After applying the geometric approximation (1) and onlyretaining the lowest order terms, the combination of theLorentz gauge ∇ µ A µ = 0 and the modified Maxwellequation (6) gives rise to k σ k µ ( qQ µνσρ + g σµ g ρν ) a ρ = 0 , (7)Eq. (7) implies a modified dispersion relation of pho-tons, which contains the information about the path ofphotons in spacetime. When q = 0, this equation givesrise to the null curve k µ k µ = 0 of the motion of photonsin the standard general relativity. The Lorentz gauge ∇ µ A µ = 0 gives rise to k µ a µ = 0 under the geometricoptics approximation. In order to take advantage of thisfeature to simplify the Eq. (7), one could introduce theantisymmetry basis [75] U µνab = e µa e νb − e νa e µb , (8)where e µa are the tetrad fields with a = 0 , , ,
3, whichsatisfy g µν = η ab e aµ e bν . The tetrad indices a, b... are raisedand lowered by η ab . In this paper, given the index sym-metry of tensor Q µνσρ , we consider Q µνσρ has the belowexpansion form Q µνσρ = C ( x ) U µν U σρ + C ( x ) U µν U σρ + C ( x ) U µν U σρ + C ( x ) U µν U σρ + C ( x ) U µν U σρ + C ( x ) U µν U σρ , (9)where the expansion coefficients C abcd ( x ) are functionsof the spacetime coordinate x . By introducing the pro-jection of k µ on the antisymmetry basis l ν ≡ k µ U µν , (10) m ν ≡ k µ U µν , (11) n ν ≡ k µ U µν , (12)together with the independent projections p ν ≡ k µ U µν = 1 k (cid:0) k m ν − k l ν (cid:1) , (13) q ν ≡ k µ U µν = 1 k (cid:0) k n ν − k l ν (cid:1) , (14) r ν ≡ k µ U µν = 1 k (cid:0) k n ν − k m ν (cid:1) , (15)the dispersion relation (7) could be written as K K K K K K K K K a · la · ma · n = 0 , (16)where the expressions of the matrix components are K = k · k + qC ( k k − k k ) + qC k k + qC k k , (17) K = − qC k k − qC k k , (18) K = − qC k k − qC k k , (19) K = − qC k k − qC k k , (20) K = k · k + qC ( k k − k k ) + qC k k + qC k k , (21) K = − qC k k − qC k k , (22) K = − qC k k − qC k k , (23) K = − qC k k − qC k k , (24) K = k · k + qC ( k k − k k ) + qC k k + qC k k . (25)In order to simplify the system (16), one could diag-onalize this matrix and the condition that the productof the eigenvalues (the determinant of the matrix) equalsto zero will give rise to the criterion of non-zero solu-tions of this system, which thus implies the dispersionrelation of photons’ motion. However, from the analysisof the actual physical situation, we could make severalassumptions on this system firstly.We focus on the static spherical spacetime and the met-ric is ds = g tt ( u ) dt + g rr ( u ) dr + r ( dθ + sin θdφ ) , (26)where u = M/r and M is the mass of the centralblack hole. The asymptotically flat condition requires g tt (0) = − g rr (0) = 1 in the limit of far distancefrom the black hole. The corresponding tetrad fieldscould be written as e aµ = diag (cid:16)(cid:112) − g tt ( r ) , (cid:112) g rr ( r ) , r, r sin θ (cid:17) . (27)Now we make two assumptions. The first one is thatthe motion of photons follow the same symmetry as thespacetime. Therefore, there should exist the integral con-stant of motion making the orbits of photons be boundin a plane, i.e. θ = π/ k = k θ e θ = 0. And thisassumption also accounts for the reason why we focuson the expansion (9) without the cross terms of U µνab such as C ( x ) U µν U σρ since these terms will breakthis assumption. The second assumption is that thereexits circular orbits in the motion plane which satisfy k = k r e r = 0. The reason of adopting this assump-tion is that in the numerical simulation, the presence ofthe photon ring is caused by the existence of the unsta-ble bound circular orbits, which makes it observationally interesting [7]. Then under these two assumptions, theonly non-zero matrix components for the photons movingin a plane circular orbits are K = (1 − qC ) g tt k t k t + (1 + qC ) r k φ k φ , (28) K = (1 − qC ) g tt k t k t + (1 + qC ) r k φ k φ , (29) K = (1 − qC ) g tt k t k t + (1 − qC ) r k φ k φ , (30)where we have applied g tt = − e t e t , g rr = e r e r and g φφ = e φ e φ . The condition of non-zero solutions is K K K = 0, which gives the below three roots andthe corresponding solution of a µ : • K = 0, a µ ∝ l µ , (cid:126)r direction polarization, • K = 0, a µ ∝ m µ , (cid:126)θ direction polarization, • K = 0, a µ ∝ n µ , unphysical polarization,where we also write down the corresponding directionof linear polarization, i.e. the direction of electric fieldaccording to E i = ∂ t A i − ∂ i A t . K = 0 correspondsto the unphysical polarization since n ν = k e ν − k e ν contains the non-transverse polarization component e ν .As a result, one could find that photons with differ-ent linear polarizations sense different two dimensional“effective metrics” on the circular orbits. For example,the photons with (cid:126)r polarization for K = 0 correspondsto g (cid:48) tt = (1 − qC ) g tt and g (cid:48) φφ = (1 + qC ) r . Inorder to be consistent with the motion in a planar cir-cular orbits, the expansion coefficients C abcd should onlydepend on the coordinate r , which makes the effectivemetric component g (cid:48) φφ could in principle be set to r bya redefinition of r , i.e. ds = g stt ( u ) dt + r dφ , (31)where the superscript s represents different linear polar-ized photons. This kind of redefinition will not influencethe observables according to the below Eq. (40). Eq.(31) means photons with different polarizations will feela different g tt component, i.e. the gravitational potential,which could therefore behave as the violation of WEP. Asa result, one could test whether the new coupling term inthe Lagrangian (4) exists by checking whether photonswith different linear polarizations sense different gravita-tional potentials. IV. BLACK HOLE PHOTON RING AS A NEWPROBE OF THE EEP
In the above discussions, we have pointed out for thephotons in circular orbits, different linear polarized pho-tons sense different gravitational potentials. In order todetect this novel phenomenon, one need to seek an ob-servable corresponding to the gravitational potential. Forthe planar motion, let us consider the below 2 + 1 planarmetric ds = g stt ( u ) dt + g srr ( u ) dr + r dφ , (32)where we have added the rr component g srr ( u ) that sat-isfies g srr (0) = 1 comparing with (31) since it is the directapproach to yield a circular orbit. Then the Lagrangianfor the motion of photons is L = 12 g sµν ( x ) ˙ x µ ˙ x ν , (33)where x is the spacetime coordinate and dot representsthe derivative with respect to the affine parameter λ .There are three conservation quantities corresponding tothe absence of λ , t and φ in the Lagrangian respectively,i.e. E ≡ g stt ( r ) ˙ t, (34) L ≡ r ˙ φ, (35) ζ ≡ g stt ( r ) ˙ t + g srr ( r ) ˙ r + r ˙ φ . (36)For the massless particle, ζ = 0. According to these threeconservation quantities, we could obtain the equation ofthe trajectory u ( φ ) g stt ( u ) g srr ( u ) (cid:18) dudφ (cid:19) + u g stt ( u ) = − E L M . (37)For the conditions of bound orbits du/dφ = 0 and d u/dφ = 0, Eq. (37) could be simplified as the be-low two equations of u , u g stt ( u ) = − X − , (38)2 g stt ( u ) + ug stt (cid:48) ( u ) = 0 , (39)where the prime represents the derivative with respect to u and we have defined X = L/ ( M E ). As for the necessityfor the existence of photon bound orbits, we refer readersto [87] for more details.The physical meaning of X is the radius d of the photonring on the image plane divided by the mass M , whoseexpression is [88] dM = 1 M lim r →∞ r dφdr = X = (cid:2) − u g stt ( u ) (cid:3) − , (40)where we have used Eq. (37) and g srr (0) = − g stt (0) = 1.Eq. (39) gives rise to the radius of the bound circularorbits and this radius will become the observed size ofthe photon ring by Eq. (38). Therefore, observations ofphoton ring would provide us with the direct informationon the values of the gravitational potential g stt at theradius of the circular bound orbits and thus could letus know if the EEP violation coupling (4) exists. Notethat the g srr component will not affect the test due to itsabsence in Eq. (40). V. METHOD AND RESULTS
In this section we will discuss the method to use blackhole photon ring to constrain the EEP violation. Accord-ing to the asymptotically flat condition g stt (0) = −
1, the metric component g stt ( u ) could be generally written as g stt ( u ) = − u (1 + β s ( u )) , (41)where β s ( u ) is the correction function for theSchwarzschild black hole induced by violation of the EEP.It could be formally expressed as β s ( u ) = (cid:88) n β sn u n , (42)where n ≥ − n does not have to be an integer. The above ex-pression effectively illustrates violation of the EEP couldstem from any order of M/r , i.e. any strength of gravi-tational fields in principle.If β s ( u ) is small, we could decompose u as the per-turbed part δu and the unperturbed part u , i.e. u = u + δu . And correspondingly, the size X of the photonring could be written as X = X + δX . The equation(39) up to the first order of β s and δu gives1 − u = 0 , (43) − δu = 3 u β s ( u ) + u β s (cid:48) ( u ) . (44)So according to Eq. (38), we could obtain δX = 3 √ β s ( u ) (45)and X = 3 √
3, where u = 1 / X has nothing to do withthe derivative of β s ( u ) and also the Eq. (44).By observing the deviation of the photon ring’s sizefrom that of the standard general relativity δX , one coulddirectly obtain the constraint on the magnitude of β s ( u )according to Eq. (45). However, in order to give a de-cisive criterion whether the EEP is violated, we need atleast two groups of photons with different linear polar-izations. The observable is the difference of the photonring’s size presented by these two kinds of photons. Afterusing l and m to label the corresponding (cid:126)r and (cid:126)θ polar-ization, this difference according to Eq. (45) is given by∆ X = δX l − δX m = 3 √ (cid:0) β l ( u ) − β m ( u ) (cid:1) ≡ √ β ( u ) , (46)where we can see that in the linear approximation, ∆ X only has dependence on the difference between β l and β m while the specific values of β l or β m does not haveinfluence. By observing the magnitude of ∆ X , onecould obtain the constraints on ∆ β ( u ). Now we choose β s ( u ) = β sn u n , which gives ∆ X = 3 − n ∆ β n by Eq. (46).In Fig. 1 we plot ∆ X from n = 0 to n = 3 for differentvalues of ∆ β n in dashed lines, where we can find that be-sides the proportional relation (46), the effects of ∆ β n issuppressed as n grows. This is caused by the suppressionof β s ( u ) in Eq. (41) for large values of n . Δβ n = Δβ n = Δβ n =- Δβ n =- - - Δ X FIG. 1. Plot showing the difference in the size of the photonring ∆ X given by polarized l and m photons as n changesfor various ∆ β n . The dashed lines correspond to the approx-imated expression (46) and sold lines are results from rigidlysolving Eq. (38) and Eq. (39). The precise results could be obtained by rigidly solvingEq. (38) and Eq. (39) for different values of β ln and β mn ,which are also shown in Fig. 1 through the solid line.We can find that the approximated expression (46) couldoverestimate the values of ∆ X and this overestimation issuppressed by large and small values of n . The reason ofthis suppression is that a larger n tends to give a smallervalue of β s ( u ) and β s ( u ) ≈ β sn when n approaches zero,which make the approximated equations (43) and (44)work better.In order to better estimate the error of the proportionalrelation (46), we define E to measure the deviation ratiofrom the precise results, whose expression is E ≡ ∆ X N − ∆ X ∆ X , (47)where ∆ X N is the precise result from rigidly solving Eq.(38) and Eq. (39). In Fig. 2 we show E for differentparameters so that one could intuitively obtain the con-ditions for the approximation (46). Note that differentfrom the approximated result, the specific value of β l or β m will matter in precise results. In this figure, besidesthe same information revealed by Fig. 1, we could findthat the positive and negative of β mn have opposite effectson E and as the absolute value of β mn and ∆ β n grows,the difference between precise results and approximatedresults is enlarged, which is caused by the fact that theapproximation (46) only works well for the small EEPviolation function β s ( u ).Finally, we emphasize that although it is difficulty toobtain a general solution of photon ring’s size X as afunction of any order of the EEP violation parameters β sn and n in Eq. (42), one could also obtain a generalexpression of photon ring’s size X through a matchingmethod by not specifying a form of the EEP violationfunction β s ( u ). This matching method is based on theassumption that the deviation of the radius of circularorbits from that of general relativity is relatively small. Specifically, one could firstly define a function B ( u ) = b p u + b + b n u − . After replacing β s ( u ) with B ( u ), Eq.(38) and Eq. (39) will have the below analytical solutionfor X − X − ( b p , b , b n ) = − − b n ) (cid:16) (cid:113) − b b ) b p (1+ b ) (cid:17) (1 + b ) (cid:16) (cid:113) − b b ) b p (1+ b ) (cid:17) . (48)For any target function β s ( u ), we could match the zero,first, third order derivative of B ( u ) to those of β s ( u ) at u = 1 /
3, which leads to the below expressions for thecoefficients of B ( u ) b p = β s (cid:48) (cid:18) (cid:19) + 16 β s (cid:48)(cid:48) (cid:18) (cid:19) , (49) b = β s (cid:18) (cid:19) − β s (cid:48) (cid:18) (cid:19) − β s (cid:48)(cid:48) (cid:18) (cid:19) , (50) b n = 154 β s (cid:48)(cid:48) (cid:18) (cid:19) . (51)Then by substituting the above coefficients into Eq.(48), one could obtain an approximated expression ofthe photon ring’s size X for the target function β ( u ).For example, β s ( u ) = β + β u + β u corresponds to X − ( β + β , β − / β , / β ). And if we choose β = 1 / β = − / β = 1 /
2, the fitting methodgives X = 5 .
375 while the rigid solution gives X = 5 . . β s ( u ) = β n u n , one couldverify that the maximum deviation ratio of the above fit-ting method from the rigid solutions is around 0 .
1% forthe case of a 50% change in photon ring’s size.
VI. EXAMPLES
In this section, we present several examples to illus-trate how to get the EEP violation parameters β sn froma specific model. A. Vector field
Now we consider a vector field V µ being coupled toelectromagnetic tensor F µν . Given that the index sym-metry of Q µνρσ , the corresponding expression of Q µνρσ is Q µνρσ = 12 g ν [ σ V ρ ] V µ + 12 g µ [ ρ V σ ] V ν . (52)As an example, we choose the Schwarzschild spacetime.If the vector field V µ has the same symmetry as thespacetime, i.e. the static spherical symmetry, one ofthe allowed forms of V µ is (0 , v ( r ) , , C = C = − C = f ( r ) , (53) β nm = β nm = β nm = β nm =- - - - - Δβ n E n = β nm = β nm = β nm = β nm =- - - - - Δβ n E n = β nm = β nm = β nm = β nm =- - - - - Δβ n E n = FIG. 2. Plot showing the deviation ratio E defined by Eq. (47) of the approximation (46) for different values of n , ∆ β n and β mn . where the definition of f ( r ) is f ( r ) = 1 / V V and V = V µ e µ . Now let us choose a specific expression f ( r ) =1 /r . For the planar circular orbits k θ = 0, k r = 0 and θ = π/
2, the dispersion relation (28) corresponding to (cid:126)r polarized photons gives the below gravitational potential g ltt = − Mr − q r + 3 q Mr , (54)where we have eliminated the g φφ component by the re-definition of r and only retained the first order termswith respect to q . For (cid:126)θ polarized photons, the gravita-tional potential is not modified, i.e. β mn = 0. Then theexpansion coefficients of Eq. (42) for (cid:126)r polarized photonsare β l = − q M , β l = 3 q M . (55)The existence of the planar circular orbits could be ver-ified by the matrix (16). The condition of nonzero so-lutions is reduced to K K K = 0 for the expansioncoefficients (53) and the expressions of K , K and K are K = (1 + qf )( g tt k t k t + g rr k r k r + g θθ k θ k θ + g φφ k φ k φ ) , (56) K = K = g tt k t k t + (1 + qf ) g rr k r k r + g θθ k θ k θ + g φφ k φφ . (57)Therefore, for each of the situations K = 0, K = 0or K = 0, photons will have a dispersion relation de-scribed by a modified Schwarzschild metric, which stillhas static spherical symmetry and thus leads to the ex-istence of the planar circular orbits. The planar motionis described by the metric (32). B. Tensor field
We choose Q µνσρ = R µνσρ , which describes the cor-rection of the virtual electron loops on the propagationof photons [82, 83]. For the Schwarzschild spacetime, the nonzero expansion coefficients of R µνσρ are C = − Mr , C = Mr , C = Mr ,C = − Mr , C = − Mr , C = 2 Mr . (58)Under the condition of planar circular orbits k θ = 0, k r = 0 and θ = π/
2, the dispersion relations (28) and(29) corresponding to the (cid:126)r and (cid:126)θ polarized photons re-spectively lead to g ltt = − Mr − q Mr + 3 q M r , (59) g mtt = − Mr + q Mr , (60)where we have eliminated the g φφ components by theredefinition of r and only retained the first order termswith respect to q . Therefore, the corresponding expan-sion coefficients of Eq. (42) are β l = − qM , β l = 3 q M , (61) β m = q M . (62)As for the existence of the planar circular orbits, onecould directly calculate the determinant of the matrix(16), the condition of nonzero solutions gives rise to K K (cid:48) K (cid:48) = 0 and the expressions of K , K (cid:48) and K (cid:48) are K = (cid:18) q Mr (cid:19) (cid:0) g tt k t k t + g rr k r k r (cid:1) + (cid:18) − q Mr (cid:19) (cid:0) g θθ k θ k θ + g φφ k φ k φ (cid:1) , (63) K (cid:48) = (cid:18) − q Mr (cid:19) (cid:0) g tt k t k t + g rr k r k r (cid:1) + (cid:18) q Mr (cid:19) (cid:0) g θθ k θ k θ + g φφ k φ k φ (cid:1) , (64) K (cid:48) = (cid:18) − q Mr (cid:19) ( g tt k t k t + g rr k r k r + g θθ k θ k θ + g φφ k φ k φ ) . (65)Similar to the vector situation, for each of the conditions K = 0, K (cid:48) = 0 or K (cid:48) = 0, the dispersion relationis described by a static metric with spherical symmetry,which leads to the existence of the planar circular orbitsand the planar motion is described by the metric (32). C. Scalar field
For the scalar coupling such as Q µνρσ =2 f ( φ ) g ρ [ µ g ν ] σ [56, 57] as discussed earlier, one coulddirectly verify that the standard dispersion relation k µ k µ = 0 is not modified according to Eq. (7). If φ is the axion field, when Q µνρσ = − φξ µνρσ and ξ µνρσ is the Levi-Civita symbol [86], the standard dispersionrelation k µ k µ = 0 is still preserved. In order to directlytest this kind of coupling, one could look for the higherorder geometric optics approximation, which leads to therotation of polarization vectors along the path of photonsand could be tested by the precise measurements ofpolarizations in principle [66]. For other forms of tensorcoupling, we refer readers to [19, 20] for more details. VII. THE INFLUENCE OF THE ROTATION OFBLACK HOLE
In the above, we have discussed the EEP violation forthe spacetime of static spherically symmetric black hole.The only metric component associated with the photonring observations is g stt , which thus could let us excludethe influence of other metric components and only fo-cus on the EEP violation manifested as different grav-itational potentials. However, when the black hole hasrotation, the situation will become complicated. Due tothe diminution of spacetime symmetries, one could ex-pect that apart from g tt , other metric components suchas g tφ ( r, θ ) will become relevance. This fact makes theabove model independent discussions based on the as-sumption of planar circular orbits become difficult tocarry on. Furthermore, for rotating black holes, the spinparameter and the inclination angle of rotation axis alsoneed to be determined in order to give a precise pre-diction of photon ring. All of these factors add moreparameters to the process of constraining the EEP viola-tion and thus increase the complexity. Fortunately, thedetailed numerical study based on the Kerr black holeshows that new parameters introduced by the rotation,which include the spin parameter and the inclination an-gle of rotation axis, mainly affect the horizontal displace-ment and the outline’s asymmetry of photon ring. Whilethe overall size of ring could hardly be changed by rota-tion effects [84, 85].Therefore, in this section we focus on the overall size ofthe black hole photon ring. What we are interested in ishow much the rotation of black holes affects the results ofconstraining the EEP violation. Our strategy is to char-acterize the black hole rotation by directly generalizing the effective metric (32). Then we derive the relationbetween the observable and the EEP violation param-eters similar to Eq. (46). And based on this we studythe influence of various parameters related to rotation onconstraint results. A. The model
According to the effective metric (32) and the expres-sion (41), we start from the following Schwarzschild-likemetric ds = − [1 − u (1 + β s ( u ))] dt + [1 − u (1 + β s ( u ))] − dr + r (cid:0) dθ + sin θφ (cid:1) . (66)A solution of rotating black hole could be obtained byapplying the Newman-Janis algorithm [89]. The abovemetric could correspond to the below rotating one in theBoyer-Lindquist coordinates ( t, r, θ, φ ) [90] ds = − (cid:18) − u (1 + β s ( u ))1 + A u cos θ (cid:19) dt + 1 + A u cos θ − u (1 + β s ( u )) + A u dr + M u (cid:0) A u cos θ (cid:1) dθ + M u (cid:18) A u + 2 A (1 + β s ( u )) u sin θ A u cos θ (cid:19) sin θdφ − M β s ( u )) Au sin θ A u cos θ dφdt, (67)where A = a/M and a is the angular momentum per unitmass of black hole, i.e. a = J/M . When β s vanishes, thiswill lead to the standard Kerr black hole.The equation of motion could be obtained by theHamilton-Jacobi equation: H + ∂S∂λ = 0 , (68)where S is the Hamilton principal function S ( λ, x µ ) and λ is the affine parameter. The Hamiltonian H is H = − g µν P µ P ν (69)and P µ in the Hamilton-Jacobi formalism is P µ = ∂S∂x µ . (70)The solution of Eq. (68) with the metric (67) is separableand we refer readers to [88, 91] for more details. Theconditions of bound orbits ˙ r = 0 and ¨ r = 0 with thesolution give rise to( u − + A − Ax ) − [ u − − u − (1 + β s ( u )) + A ][ y + ( x − A ) ] = 0 , (71)2 u − ( u − + A − Ax ) − ( u − + uβ s (cid:48) ( u ) − − β s ( u ))( y + ( x − A ) ) = 0 , (72)where the definitions of x and y are x = L z EM , (73) y = K E M . (74) E and L z are the integral constants corresponding to theabsences of t and φ in the Hamiltonian (69) respectively. K is Carter constant which is introduced by the separa-bility of the Hamilton-Jacobi equation (68). From Eq.(71) and Eq. (72), one could obtain the solutions of x and y with respect to u , i.e. x ( u ) and y ( u ).For the rotating black holes, we need two coordinates X and Y on the image plane to describe the appearanceof the black hole photon ring. These two parameters areequivalent to the initial conditions of light rays and inthe unit of mass are given by [88] X = − θ x, (75) Y = ± (cid:34) y + ( A − x ) − (cid:18) A sin θ − x sin θ (cid:19) (cid:35) , (76)where θ is the inclination angle between the rotationaxis of the black hole and the line of sight of the distantobserver. Substituting the solutions x ( u ) and y ( u ) of Eq.(71) and Eq. (72) into the expressions (75) and (76), onecould obtain the photon ring outline as the functions of u , i.e. X ( u ) and Y ( u ) in the domain having solutions. B. Method and results
Similar to the discussion of static spherically symmet-ric black holes, we decompose u as u = u + δu in thecondition that β s ( u ) is small. Then the first order equa-tions with respect to δu and β from Eq. (71) and Eq.(72) are u [2 u ( A − x ) + x ] δx + u (1 − u + A u ) y δy + (cid:8) (cid:2) u − + A ( A − x ) (cid:3) + ( u − (cid:2) ( A − x ) + y (cid:3)(cid:9) δu − u (cid:2) ( A − x ) + y (cid:3) β s ( u ) = 0 , (77)2 (cid:2) (1 − u − ) x − A (cid:3) δx + 2(1 − u − ) y δy − u − (cid:2) u ( A − x − y ) (cid:3) δu + (cid:2) ( A − x ) + y (cid:3) β s ( u ) − u (cid:2) ( A − x ) + y (cid:3) β s (cid:48) ( u ) = 0 , (78) where δx and δy represent the perturbed part of x and y . x , y and u satisfy the zeroth order equations:( u − + A − Ax ) − [ u − − u − + A ][ y + ( x − A ) ] = 0 , (79)2 u − ( u − + A − Ax ) − ( u − − y + ( x − A ) ] = 0 . (80)Solving Eq. (79) and Eq. (80) to obtain x ( u ) and y ( u ) and substituting them into Eq. (77) and Eq. (78),we could obtain δx and δy as the functions of u and δu ,i.e. δx ( u , δu ) and δy ( u , δu ).Because of the symmetry described by the metric (67),the outline of the photon ring always has a symmetryaxis on the image plane which is implied by the sign ofthe expression (76). Therefore, one important feature ofthe photon ring relevant to the observations is the twointersections of the ring contour with the symmetry axis,which is shown as p + and p − in Fig. (3) with ± bethe positive and negative values of X . These two pointsare determined by the solutions of Y = 0. Specifically,according to the expression (76), Y could be written as Y ( x, y ) = Y ( x , y ) + δY ( δx, δy ) . (81)Substituting δx ( u , δu ) and δy ( u , δu ) into δY and solv-ing δY = 0, we could obtain δu at the points of p + and p − respectively, i.e. δu ± ( u ± ) where u ± are the solutionsof the zeroth order equation Y = 0.According to Eq. (75) and the expressions of δu ± ( u ± ), the perturbed intercepts δX ± between thecontour of the photon ring and the symmetry axis onthe image plane are δX ± ( u ± ) = − θ δx ± ( u ± , δu ± )= f ( A, θ , u ± ) β s ( u ± ) , (82)where we have defined f ( A, θ , u ± ) ≡ − A sin θ u ± (cid:0) − u ± )(1 − u ± + A cos θ u ± + A cos θ u ± (cid:1) . (83)Similar to the black hole without rotation, δX ± is pro-portional to β s at the linear approximation. As for theunperturbed parts X ± , they could be obtained by re-placing x in Eq. (75) with the solution x ( u ) of Eqs.(79, 80) and let u equal to u ± .Now let us consider the observables that characterizethe photon ring contour. We use the method developedin [92] where two observables R S and D S are defined toquantify the size and the distortion in shape of the pho-ton ring respectively. This method is based on the as-sumption that the outline of the black hole photon ringis nearly circular, which is true for the standard Kerr0 XY R S D S p + p − FIG. 3. Plot illustrating the definition and the geometricalmeaning of the observables R S and D S , where the blue lineoutlines the contour of the photon ring. black hole and various of rotating black holes with sep-arable geodesics including the model (67). Accordingto the way in which we defined the coordinates α and β , the general shape and position of the photon ring onthe image plane are shown in Fig. 3 as the blue line.One can always draw a reference circle which is uniquelydefined by three points: the top, bottom and the right-most points as shown in Fig. 3 with black line. Thefirst observable R S is the radius of the reference circlewhich describes the apparent overall size of the photonring. The second observable D S is defined by the appar-ent distance between leftmost point of the ring contourand that of the reference circle, which thus measures thedegree of the ring contour deviating from a perfect circle.We denote the coordinate of the center of the referencecircle to ( C, β s ( u ), the centerof the reference circle will also be changed along the axisof α . Therefore, we use δC to label the changed part and C to label the original part. The expressions of R S and D S could be written as R S = X + − C = X + δX + − C − δC, (84) D S = X + − | X − | − C = X + δX + − | X − + δX − | − C − δC. (85)In this paper, we only focus on the overall size R S since itis a much more obvious observable than the deformation D S and the above approximation works well. For twogroups of photons l and m with different linear polariza-tions, we have∆ R S = R lS − R mS = δX l + − δX m + − δC l + δC m ≈ δX l + − δX m + = f ( A, θ, u )∆ β ( u ) , (86)where ∆ β ( u ) = β l ( u ) − β m ( u ) and we have ig- / M θ / π E - - FIG. 4. The figure shows that the variation of E defined byEq. (87) as different values of the spin parameter a and theinclination angle θ for n = 1. nored the difference of δC between l and m since thedetailed numerical study shows that small β s makes thetop, bottom and the rightmost points in Fig. 3 almosthave the same magnitude of displacement, which makesthe change of the circle center be sub-dominated. ∆ R S is proportional to ∆ β and does not depend on the spe-cific values of β l or β m , which is similar to the situationwithout rotation.In order to characterize the influence of the black holerotation, we define the fraction of the change in the over-all size caused by the rotation as E = ∆ R S − ∆ X ∆ X = f ( A, θ, u ) u n − √ u n √ u n , (87)where ∆ X is the result of no rotation situation, i.e. Eq.(46) and we have chosen β s ( u ) = β sn u n . In Fig. 4, weplot the value of E as the variation of the spin parameter a/M and the inclination angle θ for n = 1. We couldsee that the largest deviation ratio from the no rotationblack hole occurs in the largest a and in the nearly face-on or edge-on view corresponding to θ = 0 and θ = π/ n , weplot the corresponding contour lines of different values of E . The solid, dashed, dot lines correspond to n = 1 , , n tends to producea large deviation ratio from the no rotation case. Thereason is that for a given value of ∆ β n , a larger n meansa smaller effect of the EEP violation which will be morecomparable with the effects caused by the rotation.Finally, we compare the fully numerical results withthe approximated expression (86) by defining E = ∆ R NS − ∆ R S ∆ R S , (88)where ∆ R NS = R NlS − R NmS denotes the numerical results,i.e. R NlS and R NmS are obtained from the contour of pho-ton ring by numerically solving Eq. (71) and Eq. (72).1 / M θ / π E = E = E =- E =- FIG. 5. Similar to Fig. 4, this figure shows contour lines ofdifferent values of E . The solid, dashed, dot lines correspondto n = 1 , , E . The nonzero values of E have two sources. The first oneis the linear approximation (82) by assuming the EEPviolation function β ( u ) is small. The second one is theapproximated expression (86) of ∆ R S characterizing theoverall size of photon ring. In Fig. 6, we plot the devia-tion ratio E as the variation of all kinds of parametersincluding the spin parameter a , the inclination angle θ and the parameters related to the EEP violation ∆ β n , n and β mn . In each figure of Fig. 6, the parameters corre-sponding to the black curve are shown in below. Othercolored curves are results of different selections of the pa-rameter that are changed comparing with the black oneand the changed parameters are shown in the legends.A significant feature of Fig. 6 is that the deviation ra-tio E is approximately proportional to all parametersexcept for n and the proportionality coefficients are ap-proximately independent of any other parameters otherthan n . Therefore, in order to let readers better estimatethe magnitude of the deviation ratio E , we use propor-tional expressions to describe the trend of the black linesin Fig. 6(a), 6(b), 6(d) and 6(e) respectively, i.e. • (a): E = − . β n + 0 . • (b): E = − . β mn + 0 . • (d): E = 0 . a/M ) − . • (e): E = 0 . θ /π ) − . E increases with that of ∆ β n and β mn . As for another EEP violation parameter n , from Fig. 6(c) one could seethat E tends to vanish as n goes to zero. The reasonfor this is that the approximations (82) and (86) workwell for small n ( n (cid:46) . n ( n (cid:38) E asshown in Fig. 2, there is no tendency for E to convergeto zero as n increases. From Fig. 6(d), the small E tends to be given by the small spin parameter a . This iscaused by the fact that the small rotation speed of blackhole corresponds to a small distorsion of photon ring’sshape, which makes the approximation (86) work better.Finally from Fig. 6(e), one could find that the effect ofthe inclination angle θ is sub-dominated. VIII. CONCLUSIONS
In this paper, we have proposed a method to test theEinstein equivalence principle of the electromagnetic lawby observing the photon ring of black holes. Specifically,we start from a general Lagrangian (4) that character-izes violation of the EEP. By applying the geometric op-tics approximation, we obtain the Eq. (16) which impliesthe modified dispersion relation corresponding to the La-grangian. In order to simplify and manifest the physicalmeaning of this system, we focus on the situation that thespacetime and the motion of photons have the sphericalsymmetry. This fact tells us that for the planar circu-lar orbits, different polarized photons will sense differentstrength of gravitational potential and behaves as the vi-olation of WEP as a result. The observable is expressedas Eq. (46), which shows that the difference in the pho-ton ring’s size presented by two different linear polarizedphotons is proportionally connected to the difference ofthe corresponding EEP violation parameters. We alsoinvestigate the extend of the EEP violation to which theexpression (46) applies and display a few cases of specificEEP violation models.For rotating black holes, the discussions would becomemore complicated. Our strategy is to select a represen-tative model (67) to characterize the effects of black holerotation. We compare the outcomes of the approximatedexpression (46) having no rotation with those of the ap-proximated expression (86) having rotation. The resultsshow that the large deviation ratio only occurs when therotation of black hole is fast and the inclination angleof rotation axis is nearly edge-on or face-on. Similar tothe discussions without rotation, we also estimate the ac-curacy of the expression (86) by numerically solving thesystem, which is good for small magnitude of the EEPviolation and small rotation speed of black holes.In order to make the method in this paper workable,we need have the ability to distinguish photon ring in thephotos of the supermassive black hole, which cannot beachieved with current observational capabilities [13]. Re-cently, Johnson et al. show that the circular photon ring2 - - Δβ n E n = / M = θ / π = β nm = (a) a/M = 0 . θ = π/ n = 1, β cn = 0 - - - - β nm E Δβ n = = / M = θ / π = (b) a/M = 0 . θ = π/ n = 1, ∆ β n = 0 . - E Δβ n = / M = θ / π = β nm = (c) a/M = 0 . θ = π/
2, ∆ β n = 0 . β cn = 0 - / M E Δβ n = = θ / π = β nm = (d) θ = π/ n = 1, ∆ β n = 0 . β cn = 0 - - - θ / π E Δβ n = = / M = β nm = (e) a/M = 0 . n = 1, ∆ β n = 0 . β cn = 0 FIG. 6. Plot showing the deviation ratio E which is defined by Eq. (88) as the variation of all kinds of parameters. Theparameters corresponding to the black curve are shown in below of each figure. The legends represent different selections ofparameter that are changed comparing with the black curve. would manifest itself as a periodic visibility function onlong interferometric baselines [9], which thus makes thephoton ring become distinct in the accretion backgroundand the related lensing background. This work was sub-sequently extended to any shape of the photon ring by[10, 11] and the corresponding polarimetric signatures onlong interferometric baselines by [93]. The study of [12]made the forecast that a space-based interferometry ex-periment can reach a accuracy level where the photonring is remarkably insensitive to the astronomical sourceprofile and can therefore be used to precisely test gravity.Furthermore, besides the appearance, a recent work alsoshows that the two-point correlation function of intensityfluctuations on the photon ring could also become an ob-servable of the physics around black holes [94]. In thenear future, the next generation Event Horizon Telescope could have the ability to do the double band observationsas well as the corresponding dual-polarizations [95, 96].All these theoretical and experimental advances provideus with opportunities to explore possible new physics inthe strong gravitational field. ACKNOWLEDGMENTS
We are grateful to Can-Min Deng, Damien Easson, XinRen, Sheng-Feng Yan, Ye-Fei Yuan and Pierre Zhang forstimulating discussions. This work is supported in partby the NSFC (Nos. 11653002, 11961131007, 11722327,1201101448, 11421303), by the CAST (2016QNRC001),by the National Thousand Talents Program of China,by the Fundamental Research Funds for Central Uni-3versities, and by the USTC Fellowship for internationalcooperation. All numerics were operated on the com- puter clusters
LINDA & JUDY in the particle cosmologygroup at USTC. [1] K. Akiyama et al. [Event Horizon Telescope Collabora-tion], “First M87 Event Horizon Telescope Results. I. TheShadow of the Supermassive Black Hole,” Astrophys.J. , no. 1, L1 (2019) doi:10.3847/2041-8213/ab0ec7[arXiv:1906.11238 [astro-ph.GA]].[2] K. Akiyama et al. [Event Horizon Telescope Collabora-tion], “First M87 Event Horizon Telescope Results. II.Array and Instrumentation,” Astrophys. J. , no. 1, L2(2019) doi:10.3847/2041-8213/ab0c96 [arXiv:1906.11239[astro-ph.IM]].[3] K. Akiyama et al. [Event Horizon Telescope Collabo-ration], “First M87 Event Horizon Telescope Results.III. Data Processing and Calibration,” Astrophys. J. , no. 1, L3 (2019) doi:10.3847/2041-8213/ab0c57[arXiv:1906.11240 [astro-ph.GA]].[4] K. Akiyama et al. [Event Horizon Telescope Collabo-ration], “First M87 Event Horizon Telescope Results.IV. Imaging the Central Supermassive Black Hole,” As-trophys. J. , no. 1, L4 (2019) doi:10.3847/2041-8213/ab0e85 [arXiv:1906.11241 [astro-ph.GA]].[5] K. Akiyama et al. [Event Horizon Telescope Collabora-tion], “First M87 Event Horizon Telescope Results. V.Physical Origin of the Asymmetric Ring,” Astrophys.J. , no. 1, L5 (2019) doi:10.3847/2041-8213/ab0f43[arXiv:1906.11242 [astro-ph.GA]].[6] K. Akiyama et al. [Event Horizon Telescope Collabo-ration], “First M87 Event Horizon Telescope Results.VI. The Shadow and Mass of the Central Black Hole,”Astrophys. J. , no. 1, L6 (2019) doi:10.3847/2041-8213/ab1141 [arXiv:1906.11243 [astro-ph.GA]].[7] S. E. Gralla, D. E. Holz and R. M. Wald,“Black Hole Shadows, Photon Rings, and LensingRings,” Phys. Rev. D , no.2, 024018 (2019)doi:10.1103/PhysRevD.100.024018 [arXiv:1906.00873[astro-ph.HE]].[8] R. Narayan, M. D. Johnson and C. F. Gammie, “TheShadow of a Spherically Accreting Black Hole,” Astro-phys. J. Lett. , no.2, L33 (2019) doi:10.3847/2041-8213/ab518c [arXiv:1910.02957 [astro-ph.HE]].[9] M. D. Johnson, A. Lupsasca, A. Strominger, G. N. Wong,S. Hadar, D. Kapec, R. Narayan, A. Chael, C. F. Gam-mie, P. Galison, D. C. Palumbo, S. S. Doeleman,L. Blackburn, M. Wielgus, D. W. Pesce, J. R. Farah andJ. M. Moran, “Universal Interferometric Signatures of aBlack Hole’s Photon Ring,” doi:10.1126/sciadv.aaz1310[arXiv:1907.04329 [astro-ph.IM]].[10] S. E. Gralla, “Measuring the shape of a black hole photonring,” [arXiv:2005.03856 [astro-ph.HE]].[11] S. E. Gralla and A. Lupsasca, “On the Observable Shapeof Black Hole Photon Rings,” [arXiv:2007.10336 [gr-qc]].[12] S. E. Gralla, A. Lupsasca and D. P. Marrone, “The Shapeof the Black Hole Photon Ring: A Precise Test of Strong-Field General Relativity,” [arXiv:2008.03879 [gr-qc]].[13] S. E. Gralla, “Can the EHT M87 results be used to testgeneral relativity?,” [arXiv:2010.08557 [astro-ph.HE]]. [14] I. Banerjee, S. Sau and S. SenGupta, “Implications ofaxionic hair on shadow of M87*,” Phys. Rev. D ,no.10, 104057 (2020) doi:10.1103/PhysRevD.101.104057[arXiv:1911.05385 [gr-qc]].[15] R. C. Pantig and E. T. Rodulfo, “Rotating dirty blackhole and its shadow,” [arXiv:2003.06829 [gr-qc]].[16] R. Konoplya, “Shadow of a black hole surroundedby dark matter,” Phys. Lett. B , 1-6 (2019)doi:10.1016/j.physletb.2019.05.043 [arXiv:1905.00064[gr-qc]].[17] K. Jusufi, M. Jamil and T. Zhu, “Shadows of Sgr A ∗ blackhole surrounded by superfluid dark matter halo,” Eur.Phys. J. C , no.5, 354 (2020) doi:10.1140/epjc/s10052-020-7899-5 [arXiv:2005.05299 [gr-qc]].[18] K. Jusufi, M. Jamil, P. Salucci, T. Zhu and S. Ha-roon, “Black Hole Surrounded by a Dark MatterHalo in the M87 Galactic Center and its Identi-fication with Shadow Images,” Phys. Rev. D ,no.4, 044012 (2019) doi:10.1103/PhysRevD.100.044012[arXiv:1905.11803 [physics.gen-ph]].[19] S. Chen, M. Wang and J. Jing, “Polarization effects inKerr black hole shadow due to the coupling between pho-ton and bumblebee field,” [arXiv:2004.08857 [gr-qc]].[20] Y. Huang, S. Chen and J. Jing, “Double shadowof a regular phantom black hole as photons cou-ple to the Weyl tensor,” Eur. Phys. J. C ,no.11, 594 (2016) doi:10.1140/epjc/s10052-016-4442-9[arXiv:1606.04634 [gr-qc]].[21] A. Held, R. Gold and A. Eichhorn, “Asymptoticsafety casts its shadow,” JCAP , 029 (2019)doi:10.1088/1475-7516/2019/06/029 [arXiv:1904.07133[gr-qc]].[22] R. Kumar, B. P. Singh and S. G. Ghosh, “Rotat-ing black hole shadow in asymptotically safe gravity,”[arXiv:1904.07652 [gr-qc]].[23] Y. F. Cai and D. A. Easson, “Black holes inan asymptotically safe gravity theory with higherderivatives,” JCAP , 002 (2010) doi:10.1088/1475-7516/2010/09/002 [arXiv:1007.1317 [hep-th]].[24] Y. F. Cai, D. A. Easson, C. Gao and E. N. Sari-dakis, “Charged black holes in nonlinear mas-sive gravity,” Phys. Rev. D , 064001 (2013)doi:10.1103/PhysRevD.87.064001 [arXiv:1211.0563 [hep-th]].[25] Y. F. Cai, G. Cheng, J. Liu, M. Wang and H. Zhang,“Features and stability analysis of non-Schwarzschildblack hole in quadratic gravity,” JHEP , 108 (2016)doi:10.1007/JHEP01(2016)108 [arXiv:1508.04776 [hep-th]].[26] T. Zhu, Q. Wu, M. Jamil and K. Jusufi, “Shadows anddeflection angle of charged and slowly rotating blackholes in Einstein-Æther theory,” Phys. Rev. D ,no.4, 044055 (2019) doi:10.1103/PhysRevD.100.044055[arXiv:1906.05673 [gr-qc]].[27] L. Amarilla and E. F. Eiroa, “Shadows ofrotating black holes in alternative theories,” doi:10.1142/9789813226609 0459 [arXiv:1512.08956[gr-qc]].[28] A. ¨Ovg¨un, ˙I. Sakallı, J. Saavedra and C. Leiva,“Shadow cast of non-commutative black holes inRastall gravity,” Mod. Phys. Lett. A , 2020doi:10.1142/S0217732320501631 [arXiv:1906.05954 [hep-th]].[29] A. ¨Ovg¨un and ˙I. Sakalli, “Testing General-ized Einstein-Cartan-Kibble-Sciama Gravity Us-ing Weak Deflection Angle and Shadow Cast,”doi:10.20944/preprints202005.0032.v1 [arXiv:2005.00982[gr-qc]].[30] S. W. Wei and Y. X. Liu, “Testing the nature of Gauss-Bonnet gravity by four-dimensional rotating black holeshadow,” [arXiv:2003.07769 [gr-qc]].[31] R. Kumar, S. G. Ghosh and A. Wang, “Gravita-tional deflection of light and shadow cast by rotat-ing Kalb-Ramond black holes,” Phys. Rev. D ,no.10, 104001 (2020) doi:10.1103/PhysRevD.101.104001[arXiv:2001.00460 [gr-qc]].[32] S. Tian and Z. H. Zhu, “Testing the Schwarzschildmetric in a strong field region with the Event HorizonTelescope,” Phys. Rev. D , no.6, 064011 (2019)doi:10.1103/PhysRevD.100.064011 [arXiv:1908.11794[gr-qc]].[33] M. Guo and P. C. Li, “The innermost stable circularorbit and shadow in the novel 4 D Einstein-Gauss-Bonnetgravity,” [arXiv:2003.02523 [gr-qc]].[34] S. Vagnozzi and L. Visinelli, “Hunting for extra di-mensions in the shadow of M87*,” Phys. Rev. D ,no.2, 024020 (2019) doi:10.1103/PhysRevD.100.024020[arXiv:1905.12421 [gr-qc]].[35] M. Khodadi, A. Allahyari, S. Vagnozzi and D. F. Mota,“Black holes with scalar hair in light of the Event Hori-zon Telescope,” JCAP , 026 (2020) doi:10.1088/1475-7516/2020/09/026 [arXiv:2005.05992 [gr-qc]].[36] F. Long, J. Wang, S. Chen and J. Jing, “Shadow of a ro-tating squashed Kaluza-Klein black hole,” JHEP , 269(2019) doi:10.1007/JHEP10(2019)269 [arXiv:1906.04456[gr-qc]].[37] I. Banerjee, S. Chakraborty and S. SenGupta, “Sil-houette of M87*: A New Window to Peek into theWorld of Hidden Dimensions,” Phys. Rev. D ,no.4, 041301 (2020) doi:10.1103/PhysRevD.101.041301[arXiv:1909.09385 [gr-qc]].[38] D. Psaltis et al. [Event Horizon Telescope], “Grav-itational Test Beyond the First Post-NewtonianOrder with the Shadow of the M87 Black Hole,”Phys. Rev. Lett. , no.14, 141104 (2020)doi:10.1103/PhysRevLett.125.141104 [arXiv:2010.01055[gr-qc]].[39] C. M. Will, “The Confrontation between General Rela-tivity and Experiment,” Living Rev. Rel. , 4 (2014)doi:10.12942/lrr-2014-4 [arXiv:1403.7377 [gr-qc]].[40] G. Tino, L. Cacciapuoti, S. Capozziello, G. Lambiaseand F. Sorrentino, “Precision Gravity Tests and the Ein-stein Equivalence Principle,” Prog. Part. Nucl. Phys. , 103772 (2020) doi:10.1016/j.ppnp.2020.103772[arXiv:2002.02907 [gr-qc]].[41] E. Di Casola, S. Liberati and S. Sonego, “Nonequivalenceof equivalence principles,” Am. J. Phys. , 39 (2015)doi:10.1119/1.4895342 [arXiv:1310.7426 [gr-qc]]. [42] C. Wetterich, “Probing quintessence with time variationof couplings,” JCAP , 002 (2003) doi:10.1088/1475-7516/2003/10/002 [arXiv:hep-ph/0203266 [hep-ph]].[43] R. Peccei, J. Sola and C. Wetterich, “Adjusting the Cos-mological Constant Dynamically: Cosmons and a NewForce Weaker Than Gravity,” Phys. Lett. B , 183-190 (1987) doi:10.1016/0370-2693(87)91191-9[44] L. Hui, A. Nicolis and C. Stubbs, “Equivalence PrincipleImplications of Modified Gravity Models,” Phys. Rev.D , 104002 (2009) doi:10.1103/PhysRevD.80.104002[arXiv:0905.2966 [astro-ph.CO]].[45] L. Kraiselburd, S. J. Landau, M. Salgado, D. Sudarskyand H. Vucetich, “Equivalence Principle in ChameleonModels,” Phys. Rev. D , no.10, 104044 (2018)doi:10.1103/PhysRevD.97.104044 [arXiv:1511.06307 [gr-qc]].[46] S. J. Landau, P. D. Sisterna and H. Vucetich, “Chargeconservation and equivalence principle,” [arXiv:gr-qc/0105025 [gr-qc]].[47] W. T. Ni, “Equivalence Principles and Electro-magnetism,” Phys. Rev. Lett. , 301-304 (1977)doi:10.1103/PhysRevLett.38.301[48] J. F. Donoghue, B. R. Holstein and R. Robinett,“Renormalization of the Energy Momentum Tensorand the Validity of the Equivalence Principle at Fi-nite Temperature,” Phys. Rev. D , 2561 (1984)doi:10.1103/PhysRevD.30.2561[49] J. F. Donoghue, B. R. Holstein and R. Robinett, “ThePrinciple of Equivalence at Finite Temperature,” Gen.Rel. Grav. , 207 (1985) doi:10.1007/BF00760243[50] A. V. Kostelecky and J. D. Tasson, “Matter-gravitycouplings and Lorentz violation,” Phys. Rev. D , 016013 (2011) doi:10.1103/PhysRevD.83.016013[arXiv:1006.4106 [gr-qc]].[51] M. A. Hohensee, S. Chu, A. Peters and H. Muller,“Equivalence Principle and Gravitational Red-shift,” Phys. Rev. Lett. , 151102 (2011)doi:10.1103/PhysRevLett.106.151102 [arXiv:1102.4362[gr-qc]].[52] R. Angelil and P. Saha, “Galactic-center S-Stars as aprospective test of the Einstein Equivalence Principle,”Astrophys. J. Lett. , L19 (2011) doi:10.1088/2041-8205/734/1/L19 [arXiv:1105.0918 [astro-ph.GA]].[53] A. Amorim et al. [GRAVITY], “Test of the EinsteinEquivalence Principle near the Galactic Center Su-permassive Black Hole,” Phys. Rev. Lett. , no.10,101102 (2019) doi:10.1103/PhysRevLett.122.101102[arXiv:1902.04193 [astro-ph.GA]].[54] A. Hees, T. Do, B. Roberts, A. Ghez, S. Nishiyama,R. Bentley, A. Gautam, S. Jia, T. Kara, J. Lu,H. Saida, S. Sakai, M. Takahashi and Y. Takamori,“Search for a Variation of the Fine Structure Constantaround the Supermassive Black Hole in Our GalacticCenter,” Phys. Rev. Lett. , no.8, 081101 (2020)doi:10.1103/PhysRevLett.124.081101 [arXiv:2002.11567[astro-ph.GA]].[55] A. Hees and O. Minazzoli, “Post-Newtonian phenomenol-ogy of a massless dilaton,” [arXiv:1512.05233 [gr-qc]].[56] H. B. Sandvik, J. D. Barrow and J. Magueijo, “Asimple cosmology with a varying fine structureconstant,” Phys. Rev. Lett. , 031302 (2002)doi:10.1103/PhysRevLett.88.031302 [arXiv:astro-ph/0107512 [astro-ph]]. [57] J. D. Bekenstein, “Fine Structure Constant: Is It Re-ally a Constant?,” Phys. Rev. D , 1527-1539 (1982)doi:10.1103/PhysRevD.25.1527[58] J. J. Wei, B. B. Zhang, L. Shao, H. Gao, Y. Li,Q. Q. Yin, X. F. Wu, X. Y. Wang, B. Zhangand Z. G. Dai, “Multimessenger tests of Einstein’sweak equivalence principle and Lorentz invariancewith a high-energy neutrino from a flaring blazar,”JHEAp , 1-4 (2019) doi:10.1016/j.jheap.2019.01.002[arXiv:1807.06504 [astro-ph.HE]].[59] L. Giani and E. Frion, “Testing the Equivalence Prin-ciple with Strong Lensing Time Delay Variations,”[arXiv:2005.07533 [astro-ph.CO]].[60] M. P¨ossel, “The Shapiro time delay and the equivalenceprinciple,” [arXiv:2001.00229 [gr-qc]].[61] O. Bertolami and R. G. Landim, Phys. DarkUniv. , 16-20 (2018) doi:10.1016/j.dark.2018.05.002[arXiv:1712.04226 [gr-qc]].[62] R. Roy and U. A. Yajnik, “Evolution of blackhole shadow in the presence of ultralight bosons,”arXiv:1906.03190 [gr-qc].[63] N. Bar, K. Blum, T. Lacroix and P. Panci, “Look-ing for ultralight dark matter near supermassive blackholes,” JCAP , 045 (2019) doi:10.1088/1475-7516/2019/07/045 [arXiv:1905.11745 [astro-ph.CO]].[64] H. Davoudiasl and P. B. Denton, “Ultralight Boson DarkMatter and Event Horizon Telescope Observations ofM87*,” Phys. Rev. Lett. , no. 2, 021102 (2019)doi:10.1103/PhysRevLett.123.021102 [arXiv:1904.09242[astro-ph.CO]].[65] P. V. P. Cunha, C. A. R. Herdeiro and E. Radu, “EHTconstraint on the ultralight scalar hair of the M87 super-massive black hole,” arXiv:1909.08039 [gr-qc].[66] Y. Chen, J. Shu, X. Xue, Q. Yuan and Y. Zhao, “Prob-ing Axions with Event Horizon Telescope PolarimetricMeasurements,” arXiv:1905.02213 [hep-ph].[67] Ya. B. Zel’Dovich, ”Amplification of Cylindrical Elec-tromagnetic Waves Reflected from a Rotating Body.”Journal of Experimental and Theoretical Physics35.6(1972):2076-2081.[68] S. L. Detweiler, “Klein-gordon Equation And Rotat-ing Black Holes,” Phys. Rev. D , 2323 (1980).doi:10.1103/PhysRevD.22.2323[69] N. G. Nielsen, A. Palessandro and M. S. Sloth, “Gravi-tational Atoms,” Phys. Rev. D , no. 12, 123011 (2019)doi:10.1103/PhysRevD.99.123011 [arXiv:1903.12168[hep-ph]].[70] D. Baumann, H. S. Chia, J. Stout and L. ter Haar, “TheSpectra of Gravitational Atoms,” arXiv:1908.10370 [gr-qc].[71] J. H. Huang, W. X. Chen, Z. Y. Huang andZ. F. Mai, “Superradiant stability of the Kerrblack holes,” Phys. Lett. B , 135026 (2019)doi:10.1016/j.physletb.2019.135026 [arXiv:1907.09118[gr-qc]].[72] A. Pawl, “The Timescale for loss of massive vec-tor hair by a black hole and its consequences forproton decay,” Phys. Rev. D , 124005 (2004)doi:10.1103/PhysRevD.70.124005 [hep-th/0411175].[73] A. Arvanitaki, M. Baryakhtar and X. Huang, “Discov-ering the QCD Axion with Black Holes and Gravita-tional Waves,” Phys. Rev. D , no. 8, 084011 (2015)doi:10.1103/PhysRevD.91.084011 [arXiv:1411.2263 [hep-ph]]. [74] F. A. Berends and R. Gastmans, “Quantum Elec-trodynamical Corrections to Graviton-Matter Ver-tices,” Annals Phys. , 225 (1976) doi:10.1016/0003-4916(76)90245-1[75] I. T. Drummond and S. J. Hathrell, “QED Vacuum Po-larization in a Background Gravitational Field and ItsEffect on the Velocity of Photons,” Phys. Rev. D , 343(1980) doi:10.1103/PhysRevD.22.343[76] K. A. Milton, “Quantum Electrodynamic Corrections tothe Gravitational Interaction of the Photon,” Phys. Rev.D , 2149-2155 (1977) doi:10.1103/PhysRevD.15.2149[77] R. G. Cai, “Propagation of vacuum polarized photonsin topological black hole space-times,” Nucl. Phys. B , 639-657 (1998) doi:10.1016/S0550-3213(98)00274-0[arXiv:gr-qc/9801098 [gr-qc]].[78] C. W. Misner, K. Thorne and J. Wheeler, Gravitation ,(W.H. Freeman and Co., San Francisco, 1974).[79] S. F. Yan, C. Li, L. Xue, X. Ren, Y. F. Cai,D. A. Easson, Y. F. Yuan and H. Zhao, “Test-ing the equivalence principle via the shadow ofblack holes,” Phys. Rev. Res. , no.2, 023164(2020) doi:10.1103/PhysRevResearch.2.023164[arXiv:1912.12629 [astro-ph.CO]].[80] V. P. Frolov, “Maxwell equations in a curved spacetime:Spin optics approximation,” [arXiv:2007.03743 [gr-qc]].[81] V. P. Frolov and A. A. Shoom, “Scattering of circularlypolarized light by a rotating black hole,” Phys. Rev.D , 024010 (2012) doi:10.1103/PhysRevD.86.024010[arXiv:1205.4479 [gr-qc]].[82] R. D. Daniels and G. M. Shore, “’Faster than light’ pho-tons and rotating black holes,” Phys. Lett. B , 75-83 (1996) doi:10.1016/0370-2693(95)01468-3 [arXiv:gr-qc/9508048 [gr-qc]].[83] R. Lafrance and R. C. Myers, “Gravity’s rain-bow,” Phys. Rev. D , 2584-2590 (1995)doi:10.1103/PhysRevD.51.2584 [arXiv:hep-th/9411018[hep-th]].[84] C. k. Chan, D. Psaltis and F. Ozel, “GRay: a Mas-sively Parallel GPU-Based Code for Ray Tracing inRelativistic Spacetimes,” Astrophys. J. , 13 (2013)doi:10.1088/0004-637X/777/1/13 [arXiv:1303.5057[astro-ph.IM]].[85] T. Johannsen and D. Psaltis, “Testing the No-HairTheorem with Observations in the ElectromagneticSpectrum: II. Black-Hole Images,” Astrophys. J. , 446-454 (2010) doi:10.1088/0004-637X/718/1/446[arXiv:1005.1931 [astro-ph.HE]].[86] D. J. Schwarz, J. Goswami and A. Basu, “Geometric op-tics in the presence of axion-like particles in curved space-time,” [arXiv:2003.10205 [hep-ph]].[87] C. M. Claudel, K. S. Virbhadra and G. F. R. Ellis, J.Math. Phys. , 818-838 (2001) doi:10.1063/1.1308507[arXiv:gr-qc/0005050 [gr-qc]].[88] S. Chandrasekhar, The mathematical theory of blackholes , (Clarendon, Oxford, 1985).[89] E. Newman and A. Janis, “Note on the Kerr spin-ning particle metric,” J. Math. Phys. , 915-917 (1965)doi:10.1063/1.1704350[90] C. Bambi and L. Modesto, “Rotating regularblack holes,” Phys. Lett. B , 329-334 (2013)doi:10.1016/j.physletb.2013.03.025 [arXiv:1302.6075[gr-qc]].[91] A. Abdujabbarov, M. Amir, B. Ahmedov andS. G. Ghosh, “Shadow of rotating regular black holes,” Phys. Rev. D , no.10, 104004 (2016)doi:10.1103/PhysRevD.93.104004 [arXiv:1604.03809 [gr-qc]].[92] K. Hioki and K. i. Maeda, “Measurement of theKerr Spin Parameter by Observation of a CompactObject’s Shadow,” Phys. Rev. D , 024042 (2009)doi:10.1103/PhysRevD.80.024042 [arXiv:0904.3575[astro-ph.HE]].[93] E. Himwich, M. D. Johnson, A. Lupsasca andA. Strominger, “Universal polarimetric signatures ofthe black hole photon ring,” Phys. Rev. D101