Shadows of rotating Hayward-de Sitter black holes with astrometric observables
SShadows of rotating Hayward-de Sitter black holes withastrometric observables
Peng-Zhang He, Qi-Qi Fan, Hao-Ran Zhang, and Jian-Bo Deng ∗ Institute of Theoretical Physics & Research Center of Gravitation,Lanzhou University, Lanzhou 730000, China (Dated: September 16, 2020)
Abstract
Motivated by recent work on rotating black hole shadow [Phys. Rev. D101, 084029 (2020)], weinvestigate the shadow behaviors of rotating Hayward-de Sitter black hole for static observers at afinite distance in terms of astronomical observables. This paper uses the newly introduced distortionparameter in [arXiv:2006.00685] to describe the shadow’s shape quantitatively. We show that the spinparameter would distort shadows and the magnetic monopole charge would increase the degree ofdeformation. At the same time, the distortion could be relieved because of the cosmological constantand the distortion would increase with the distance from the black hole. Besides, the spin parameter,magnetic monopole charge and cosmological constant increase will cause the shadow to shrink.
Keywords: Rotating Hayward-de Sitter black holes, Shadows of black holes, Observable ∗ Jian-Bo Deng: [email protected] a r X i v : . [ g r- q c ] S e p . INTRODUCTION According to General Relativity (GR), the most interesting celestial body predicted maybe black holes. There is a strong gravitational field in the region near a black hole that canbend light rays. Due to the highly bending light rays in the strong gravity field, shadows castby black holes usually appear in the observer’s sky [1]. The light rays received come from theblack hole’s unstable photon orbits, or the photon region [2, 3]. The first image of a black holetaken by Event Horizon Telescope (EHT) [4] confirmed the existence of black holes, attractingmore researchers to study the observable effects of the black holes, e.g., the shadows of blackholes, gravitational deflection of light or massive particles and the like.For the simplest black hole, spherical black hole, the shadow’s boundary is a perfect circle.In the sixties of the last century, Synge considered a static observer to calculate the angularradius of the Schwarzschild black hole’s shadow in his seminal paper [5]. For rotating blackholes, the shadow’s shape is no longer circular but somewhat flattened on one side because ofthe “dragging” of null geodesics by black holes. Bardeen first gave the shadow’s shape of theKerr black hole for a distant observer; one can find the results in Chandrasekhar’s book [6]and in [7]. Since those pioneer works, shadows of objects have been extensively studied; onecan refer to the Refs. [8–27].Very recently, the authors of Refs. [28, 29] proposed a new method for calculating thesize and shape of shadow in terms of astrometric observables for finite-distant observers andintroduced a new distortion parameter to describe the shadow’s deviation from circularity. Theshadows of the Kerr-de Sitter black hole for static observers were revisited in this way withoutintroducing tetrads [28]. Furthermore, the appearance of the shadow of a static spherical blackhole and the Kerr black hole was discussed in a unified framework [29].This paper aims to apply this method to study the shadows of rotating Hayward-de Sitterblack holes and examine the parameters’ effects on the shadow’s size and distortion.We organize this article as follows. In Sec. II, we review the method of calculating blackhole shadows using astronomical observables briefly. In sec. III, we apply this approach torotating Hayward-de Sitter black holes to analyze the influences of parameters on the shadow’sshape and size. We conclude our results in Sec. IV. In this paper we set G = c = 1.2 I. SHADOWS OF ROTATING BLACK HOLES
In order to make this article self-sufficient, we briefly introduce some basics in this section.One can read Refs. [28, 29] for details.In astrometry, the angle (cid:15) between two incident light rays can be expressed by the followingformula [31]: cos (cid:15) ≡ γ ∗ w · γ ∗ k | γ ∗ w (cid:107) γ ∗ k | = w · k ( u · w )( u · k ) + 1 . (1)Here, k and w are tangent vectors of the two light rays, respectively. γ ∗ is the projectionoperator, γ µν = δ µν + u µ u ν , for a given observer, whose 4-velocity is denoted by vector u .Generally speaking, the metric of a rotating black hole can be written asd s = g d t + g d r + g d θ + g d φ + 2 g d t d φ. (2)The 4-velocity of a static observer is u = √ g ∂ t . For the asymptotically de Sitter spacetime,there is a cosmological horizon. The observer is fixed at the so-called domain of outer com-munication that is the region between the event horizon and the cosmological horizon. Whenthe observer located at θ = 0, it will find that the shadow is a disk and the angular radius iscot ψ = sgn (cid:16) π − ψ (cid:17) (cid:115) g g (cid:0) l l (cid:1) + (cid:16) g − g g (cid:17) (cid:0) l l (cid:1) . (3)Here, we have choose a light ray l = ( l , l , l , l ) comes from the photon region. “sgn”represents the sign function. For a observer located at θ >
0, the shadow’s silhouette is nota perfect circle as a consequence of the frame dragging effect. As an example, assume theobserver located at θ = π/
2. Let k = ( k , k , , k ) represent a light ray from a progradeorbit which moves in the same direction as the black hole’s rotation, and w = ( w , w , , w )represent a light ray from a retrograde orbit that moving against the black hole’s rotation.One can get the angle of the two light rays, in such a way thatcot γ = sgn( k, w ) (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) (cid:16) g K−W + (cid:16) g − g g (cid:17) W − K (cid:17) g (cid:16) g − g g (cid:17) , (4)where K ≡ k /k , W ≡ w /w , and sgn( k, w ) = sgn (cos γ ) = sgn ( g + ( g − g /g ) KW ).Similarly, the angle α between a light ray l = ( l , l , l , l ) from the photon region and k
3s cot α = sgn( k, l ) (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) (cid:18) g
11 1
K−L + (cid:16) g − g g (cid:17) L − K (cid:19) g (cid:32) g (cid:16) L K−L (cid:17) + (cid:16) g − g g (cid:17) (cid:18) L − L K (cid:19) (cid:33) + g (cid:16) g − g g (cid:17) ; (5)and the angle β between the light ray l and w iscot β = sgn( w, l ) (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) (cid:18) g
11 1
W−L + (cid:16) g − g g (cid:17) L − W (cid:19) g (cid:32) g (cid:16) L W−L (cid:17) + (cid:16) g − g g (cid:17) (cid:18) L − L W (cid:19) (cid:33) + g (cid:16) g − g g (cid:17) . (6)In above equations, L ≡ l /l , L ≡ l /l , sgn( k, l ) = sgn (cos α ) = sgn ( g + ( g − g /g ) KL ),and sgn( w, l ) = sgn (cos β ) = sgn ( g + ( g − g /g ) WL ). γ , α and β can provide us the shadow of black hole in the celestial sphere. For the sakeof convince in researching the shadow, one can use the following stereographic projection forthe celestial coordinates to describe the shape of shadow in a 2 D -plane [28]. Y sh = 2 sin Φ sin Ψ1 + cos Φ sin Ψ= 2 cos β sin γ − γ (cid:112) sin γ sin β + (cos( β + γ ) − cos α )(cos( β − γ ) − cos α )1 + cos β cos γ + (cid:112) sin γ sin β + (cos( β + γ ) − cos α )(cos( β − γ ) − cos α ) , (7) Z sh = 2 cos Ψ1 + cos Φ sin Ψ= 2 csc γ (cid:112) (cos α − cos( β + γ ))(cos( β − γ ) − cos α )1 + cos β cos γ + (cid:112) sin γ sin β + (cos( β + γ ) − cos α )(cos( β − γ ) − cos α ) . (8)Here, Φ and Ψ are azimuth angle and polar angle in celestial coordinate system.In order to quantitatively describe shadow’s shape, a distortion parameter Ξ in terms of α , β and γ is introduced, cos Ξ ≡ γ − cos α − cos β (cid:112) (1 − cos α )(1 − cos β ) , (9)where Ξ ranges from 0 to π . The cos Ξ = 0 for that shadow’s shape is circular in the celestialsphere. For the non-vanished cos Ξ, it can quantify the deviation from circularity. The authorsof Ref. [29] first proposed this kind of quantity for the shadow. Now, we can use γ and Ξ torepresent the sizes and shapes of shadows without confusion.4 II. APPLICATION IN ROTATING HAYWARD-DE SITTER BLACK HOLES
In this section, we will apply this method described in the previous section to obtain theshadows of rotating Hayward-de Sitter black holes without introducing tetrads.The metric of rotating Hayward-ds Sitter black holes in the Boyer-Lindquist coordinates( t, r, θ, φ ) is [30] ds = − ∆ r Σ (cid:18) dt − a sin θρ dφ (cid:19) + Σ∆ r dr + Σ∆ θ dθ + ∆ θ sin θ Σ (cid:18) adt − r + a ρ dφ (cid:19) , (10)where Σ = r + a cos θ, ρ = 1 + Λ3 a , (11)∆ r = (cid:0) r + a (cid:1) (cid:18) − Λ3 r (cid:19) − m ( r ) r, ∆ θ = 1 + Λ3 a cos θ, (12)˜ m ( r ) = M (cid:18) r r + g (cid:19) . (13)Here, M represents the mass of black hole, a is the black hole spin parameter, Λ is cosmologicalconstant, and the parameter g is the magnetic monopole charge arising from the nonlinearelectrodynamics. A. Null geodesic equations and photon regions
The motion equations of photons in the spacetime, determined by the metric (10), canbe given by the Lagrangian, L = 12 g µν ˙ x µ ˙ x ν , (14)where an overdot denotes the partial derivative with respect to an affine parameter. For themetric (10), one can obtain the momenta ( p µ = g µλ ˙ x λ ) are p t = (cid:18) a ∆ θ sin θ Σ − ∆ r Σ (cid:19) ˙ t + (cid:18) a ∆ r sin θρ Σ − a ( a + r ) ∆ θ sin θρ Σ (cid:19) ˙ φ, (15) p φ = (cid:18) a ∆ r sin θρ Σ − a ( a + r ) ∆ θ sin θρ Σ (cid:19) ˙ t + (cid:32) ( a + r ) ∆ θ sin θρ Σ − a ∆ r sin θρ Σ (cid:33) ˙ φ, (16) p r = Σ∆ r ˙ r, (17) p θ = Σ∆ θ ˙ θ, (18)5here p t = − E , p φ = L φ denote energy and angular momentum, respectively. Combining themomenta and Hamilton-Jacobi equation, we can get null geodesics equations.The Hamilton-Jacobi equation takes the following general form: − ∂S∂λ = 12 g µν ∂S∂x µ ∂S∂x ν , (19)where λ is an affine parameter and S is the Jacobi action which can be decomposed as a sum, S = − m λ − Et + L φ φ + S θ ( θ ) + S r ( r ) , (20)if S is separable. m is the mass of particle, which is zero for photons. From (19) and (20),one can get ∆ θ (cid:18) ∂S θ ∂θ (cid:19) + ( L φ ρ csc θ − aE sin θ ) ∆ θ = Q , (21)and ∆ r (cid:18) ∂S r ∂r (cid:19) − (( a + r ) E − aρL φ ) ∆ r = −Q , (22)where Q is a constant of separation called Carter constant, and ∂S/∂x µ = p µ . With theHamilton-Jacobi equation, it is not difficult to get the null geodesic equations as(Σ ˙ r ) = R, (23)(Σ ˙ θ ) = Θ , (24)Σ ˙ t = E (cid:32) ( a + r ) ( a + r − aλρ )∆ r + a (cid:0) λρ − a sin θ (cid:1) ∆ θ (cid:33) , (25)Σ ˙ φ = ρE (cid:32) a ( a + r ) − a λρ ∆ r + (cid:0) λρ − a sin θ (cid:1) ∆ θ sin θ (cid:33) , (26)where R = E (cid:16)(cid:0) a + r − aλρ (cid:1) − η ∆ r (cid:17) , (27)Θ = E (cid:0) ∆ θ η − ( λρ csc θ − a sin θ ) (cid:1) , (28)and λ ≡ L φ E , η ≡ Q E . (29)For spherical orbits, R ( r c ) = 0 (30)and dR ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) r = r c = 0 (31)6ust be satisfied, which lead to λ = − r ∆ r + ( a + r ) ∆ (cid:48) r aρ ∆ (cid:48) r (cid:12)(cid:12)(cid:12)(cid:12) r = r c , (32) η = 16 r ∆ r ∆ (cid:48) r (cid:12)(cid:12)(cid:12)(cid:12) r = r c , (33)where ∆ (cid:48) r denotes the derivative of ∆ r with respect to r , and r c is the location of photonsphere. Furthermore, we can rewrite R (cid:48)(cid:48) ( r c ) as R (cid:48)(cid:48) ( r c ) = 8 E (cid:18) r + 2 r ∆ r (∆ (cid:48) r − r ∆ (cid:48)(cid:48) r )∆ (cid:48) r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = r c . (34)A spherical null geodesic at r = r c is unstable with respect to radial perturbations if R (cid:48)(cid:48) ( r c ) >
0, and stable if R (cid:48)(cid:48) ( r c ) < . Unstable photon orbits determine the contour of shadow. Therange of r c (photon region) can be determined by Θ ≥ (cid:16) (4 r ∆ r − Σ∆ (cid:48) r ) − a r ∆ r ∆ θ sin θ (cid:17) r = r c ≤ . (35)From (34) and (35), we can get r c − ≤ r c ≤ r c + , where r c − and r c + are the minimum andmaximum radial position of the photon region. If limiting the light rays from the photonregion, one can regard p µ = ˙ x µ as functions of x µ , E , and r c . B. Sizes of shadow
For θ = 0, one can rewrite (35) as (cid:0) r ∆ r − (cid:0) r + a (cid:1) ∆ (cid:48) r (cid:1) r = r c = 0 . (36)This means that the photon region becomes photon sphere, and r c = r c − = r c + . Substitutingthe metric (10) and geodesic equations into (3), one can calculate the angular radius of theshadow in the following form,cot ψ = (cid:115) ( a + r − aλρ ) − η ∆ r ∆ r η + aλρ (2 a + 2 r − aλρ ) , (37)where λ and η are function of r c . Here, we only consider shadow in the view of observerslocated outside of the photon region. 7 =
0, g =
0, a = Λ = = = Λ = = = Λ = = = r ° ° ° ψ (a) g = = Λ = = = Λ = = = Λ = = = Λ = = = Λ = r ° ° ° ψ (b) a = Λ =
0, g = = Λ =
0, g = = Λ = = = Λ = = = Λ = = r ° ° ° ψ (c) Figure 1. The angular radius ψ of shadow as a function of the distance from the rotating Hayward-deSitter black holes for selected parameters, and the observers are located at inclination angle θ = 0.The vertical dotted lines are the outer boundaries and the cosmological horizons. Here we set M = 1. In Fig. 1, we plot the shadow’s angular radius as a function of the distance from the blackhole. The figures reflect that the photon sphere radius of the Schwarzschild black hole is thelargest, and it’s shadow has the largest size among the shadows observed at the same position.Besides, no matter which of a , g , and l increases, the size of shadow will become smaller.The situation of the observer locates at the equatorial plane ( θ = π/
2) will be morecomplicated. In this case, (35) can be rewritten as (cid:16)(cid:0) r ∆ r − r ∆ (cid:48) r (cid:1) − a r ∆ r (cid:17) r = r c ≤ . (38)Then one can obtain r c − ≤ r c ≤ r c + . From (4), we get the angular diameter γ ,cot γ = sgn (cid:18) r ρ (∆ r − a ) KW (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ρ √ ∆ r − a ∆ r K − W + ∆ r ρ √ ∆ r − a W − K (cid:12)(cid:12)(cid:12)(cid:12) , (39)where K = p φ p r (cid:12)(cid:12)(cid:12)(cid:12) r c = r c − , (40) W = p φ p r (cid:12)(cid:12)(cid:12)(cid:12) r c = r c + . (41)From (24) and (26), we get p φ p r = ˙ φ ˙ r = ρ ( a + a ( r − ∆ r ) − a λρ + ∆ r λρ )∆ r (cid:113) ( a + r − aλρ ) − ∆ r η . (42)It is worth noting that λ and η can be regarded as functions of r c , so (42) is a function of r and r c . Fig. 2 shows that the angular parameter γ changes with the increase of the distance8rom the black hole. It is not difficult to find that the angular parameter γ decreases with theincrease of a , Λ and g , and the outer boundary of the photon region r c + is larger than theradius of the photon sphere in Schwarzschild spacetime. Therefore, the size of the black holeshadow will decrease with the increase of a , Λ and g , and the shadow of the Schwarzschildblack hole has the largest size. Λ =
0, g =
0, a = Λ = = = Λ = = = Λ = = = r ° ° ° γ (a) g = = Λ = = = Λ = = = Λ = = = Λ = r ° ° ° γ (b) Λ = = = Λ = = = Λ = = = Λ = = = r ° ° ° γ (c) Figure 2. The angular diameter γ of shadow as a function of the distance from the rotating Hayward-de Sitter black holes for selected parameters, and the observers are located at inclination angle θ = 0.The vertical dotted lines are the outer boundaries and the cosmological horizons. Here we set M = 1 . C. Shadow’s shape
In this part, we will consider the shadow’s shape in different situations. The observerslocated at inclination angle θ = 0 would see the shadows as a perfect circle, while the observerslocated at θ = π would find that the shadows are distorted. according to (5) and (6) ,theangular distances α and β can be read ascot α = sgn (cid:18) r (∆ r − a ) ρ KL (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∆ r ρ √ ∆ r − a L − K + ρ √ ∆ r − a ∆ r K−L (cid:12)(cid:12)(cid:12)(cid:12)(cid:115) (cid:18) L − L K (cid:19) ∆ r + (∆ r − a ) ρ ∆ r (cid:16) L K−L (cid:17) , (43)and cot β = sgn (cid:18) r (∆ r − a ) ρ WL (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∆ r ρ √ ∆ r − a L − W + ρ √ ∆ r − a ∆ r W−L (cid:12)(cid:12)(cid:12)(cid:12)(cid:115) (cid:18) L − L W (cid:19) ∆ r + (∆ r − a ) ρ ∆ r (cid:16) L W−L (cid:17) , (44)9here K and W are given by (40), (41) and L ≡ p θ p r (cid:12)(cid:12)(cid:12)(cid:12) r c , (45) L ≡ p φ p r (cid:12)(cid:12)(cid:12)(cid:12) r c , (46)with p θ p r = ˙ θ ˙ r = ± (cid:115) η − ( λρ − a ) ( a + r − aλρ ) − ∆ r η . (47)In Fig. 3, we set g = 0 and get the same results as the Kerr(-de Sitter) black holes inRef. [28]. In Figs. 4 and 5, we scale the shadows appropriately so that the degree of distortionof these shadows can be compared qualitatively. The upper parts of Figs. 4 and 5 are theshadows after scaling, with different parameters selected, and the lower parts are the images ofthe corresponding distortion parameters vary with Φ /γ , which are the quantitative descriptionof the shadow’s distortion. In Fig. 4, the observers are not far from the photon regions of theblack holes, and in Fig. 5, the observers are far away from the black holes. It is not difficultto find that the shadow’s distortion will decrease as the cosmological constant increases. Incontrast, the distortion will increase with an increase of g or a .In Fig. 6, we plot the shapes and distortion parameters of the shadows for observersat different distances from the center of the black hole. It can be seen that the distortionparameter would increase with distance. Through the above discussion, we know that whenthe parameters g and a of rotating Hayward-de Sitter black holes are maximum, and thecosmological constant is zero, the distortion of the shadow is the largest. IV. CONCLUSIONS AND DISCUSSIONS
In this article, we calculated the size and shape of rotating Hayward-de Sitter black holeshadow for static observers at a finite distance in terms of astronomical observables. For θ = 0, the shadow’s boundary is a perfect circle, but for θ = π , the shadow’s boundarywill be distorted. To quantitatively describe the distortion of the shadows, we plotted thedistortion parameter affected by the black hole’s parameters, quantifying the distortion of theshape from circularity. We found that no matter which parameter increases, the size of theshadow will shrink. At the same distance, a Schwarzschild black hole has the largest shadow.10 =
0, a = - Λ =
0, a = - - - - Y Z r = Λ = = Λ = = - - - - Y r = (a) Λ =
0, a = - Λ = = Λ = = - - - - Y Z r = a = - , Λ = = - - , Λ = = - - , Λ = - - Y r = (b) Figure 3. Shadows of rotating Hayward-de Sitter black holes with g = 0 on projective plane ( Y, Z ) forselected parameters. r is the distance from the observer to the black holes. Here we set M = 1. (a)Shadows of rotating Kerr(-de Sitter) black holes for selected spin parameters for distant observers.(b) Shadows of rotating Kerr-de Sitter black holes for observers located at r = 4. Furthermore, the parameters g and a of rotating Hayward-de Sitter black holes are maximum,and the cosmological constant is zero, the distortion of the black hole shadow is the largest,and the distortion parameter would increase with distance.11 = = Λ = = = Λ = = = Λ = - - Y Ymax Z Y m ax r = a = = Λ = = = Λ = = = Λ = - Φγ c o s Ξ (a) Λ = = = Λ = = = Λ = = = - - Y Ymax r = Λ = = = Λ = = = Λ = = = - Φγ (b) g = Λ = = - g = Λ = = = Λ = = - - Y Ymax r = g = Λ = = - g = Λ = = = Λ = = - Φγ (c) Figure 4. The shape of shadows and corresponding distortion parameters Ξ as function of Φ γ forselected different parameters for observers located at r = 4. Here we set M = 1. (a) Shadows anddistortion parameters of rotating Hayward-de Sitter black holes of selected different cosmologicalconstants. (b) Shadows and distortion parameters of rotating Hayward-de Sitter black holes ofselected different magnetic monopole charges. (c) Shadows and distortion parameters of rotatingHayward-de Sitter black holes of selected different the spin parameters. We only considered static observers fixed at inclination angle θ = 0 and θ = π , but thismethod is suitable for arbitrary observers. Studying the shadows of black holes is an importantway for studying the properties of black holes, from which one can obtain rich informationabout space-time geometry. CONFLICTS OF INTEREST
The authors declare that there are no conflicts of interest regarding the publication ofthis paper. 12 = = = - Λ = = = Λ = = = - - - Y Ymax Z Y m ax r = Λ =
0, g = = - Λ =
0, g = = Λ =
0, g = = - Φγ c o s Ξ (a) Λ = = = Λ = = = Λ = = = - - - Y Ymax r = Λ =
0, a = = Λ =
0, a = = Λ =
0, a = = - Φγ (b) Λ = = = - Λ = = = Λ = = = - - - Y Ymax r = Λ = = = - Λ = = = Λ = = = - Φγ (c) Λ = = = Λ = = = Λ = = = - - - Y Ymax r = Λ = = = Λ = = = Λ = = = - Φγ (d) Figure 5. The shapes of shadows and corresponding distortion parameters Ξ as function of Φ γ forselected different parameters for distant observers. Here we set M = 1. (a) Shadows and distortionparameters of rotating Hayward black holes of selected different spin parameters for observers locatedat r = 40. (b) Shadows and distortion parameters of rotating Hayward black holes of selected differentmagnetic monopole charges for observers located at r = 40. (c) Shadows and distortion parametersof rotating Hayward-de black holes of selected different spin parameters for observers located at r = 6 .
31. (d) Shadows and distortion parameters of rotating Hayward-de black holes of selecteddifferent the magnetic monopole charges for observers located at r = 6 . ACKNOWLEDGMENTS
We would like to thank the National Natural Science Foundation of China (GrantNo.11571342) for supporting us on this work.13 = = = = - - ZY m ax Λ =
0, a = = r = = = = - Φγ c o s Ξ (a) r = = = = - - Λ = = = r = = = = - Φγ (b) Figure 6. The shapes of shadows and corresponding distortion parameters Ξ as function of Φ γ forobservers at selected position r . REFERENCES [1] P. V. P. Cunha and C. A. R. Herdeiro,
Shadows and Strong Gravitational Lensing: A BriefReview , Gen Relativ Gravit , 42 (2018).
2] A. Grenzebach, V. Perlick, and C. Lmmerzahl,
Photon Regions and Shadows of Kerr-Newman-NUT Black Holes with a Cosmological Constant , Physical Review D , (2014).[3] A. Grenzebach, V. Perlick, and C. Lmmerzahl, Photon Regions and Shadows of AcceleratedBlack Holes , International Journal of Modern Physics D , 1542024 (2015).[4] Event Horizon Telescope Collaboration , Astrophys. J. Lett. , L6 (2019).[5] J. L. Synge,
The Escape of Photons from Gravitationally Intense Stars , Mon Not R Astron Soc , 463 (1966).[6] Chandrasekhar S.,
The mathematical theory of black holes , (Oxford University Press, New York,1998).[7] J. M. Bardeen, in
Black Holes (Les Astres Occlus) , edited by C. Dewitt and B. S. Dewitt (Gordonand Breach,New York, 1973), pp. 215239.[8] V. Perlick, O. Yu. Tsupko, and G. S. Bisnovatyi-Kogan,
Black Hole Shadow in an ExpandingUniverse with a Cosmological Constant , Physical Review D , (2018).[9] Z. Chang and Q.-H. Zhu, Black Hole Shadow in the View of Freely Falling Observers , Journalof Cosmology and Astroparticle Physics , 055 (2020).[10] P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Rnarsson,
Shadows of Kerr Black Holeswith Scalar Hair , Phys. Rev. Lett. , 211102 (2015).[11] J. Bada and E. F. Eiroa,
Influence of an Anisotropic Matter Field on the Shadow of a RotatingBlack Hole , Physical Review D , (2020).[12] R. A. Konoplya,
Shadow of a Black Hole Surrounded by Dark Matter , Physics Letters B , 1(2019).[13] A. Abdujabbarov, M. Amir, B. Ahmedov, and S. G. Ghosh,
Shadow of Rotating Regular BlackHoles , Phys. Rev. D , 104004 (2016).[14] F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Shadow of Rotating Non-Kerr Black Hole ,Physical Review D , (2013).[15] G. S. Bisnovatyi-Kogan and O. Yu. Tsupko, Shadow of a Black Hole at Cosmological Distances ,Physical Review D , (2018).[16] Z. Younsi, A. Zhidenko, L. Rezzolla, R. Konoplya, and Y. Mizuno, New Method for ShadowCalculations: Application to Parametrized Axisymmetric Black Holes , Phys. Rev. D , 084025(2016).
17] A. A. Abdujabbarov, L. Rezzolla, and B. J. Ahmedov,
A Coordinate-Independent Characteri-zation of a Black Hole Shadow , Monthly Notices of the Royal Astronomical Society , 2423(2015).[18] V. Perlick, O. Yu. Tsupko, and G. S. Bisnovatyi-Kogan,
Influence of a Plasma on the Shadowof a Spherically Symmetric Black Hole , Physical Review D , (2015).[19] U. Papnoi, F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Shadow of Five-Dimensional RotatingMyers-Perry Black Hole , Phys. Rev. D , 024073 (2014).[20] P. V. P. Cunha, C. A. R. Herdeiro, B. Kleihaus, J. Kunz, and E. Radu, Shadows of EinsteinDila-tonGaussBonnet Black Holes , Physics Letters B , 373 (2017).[21] P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu,
Fundamental Photon Orbits: Black HoleShadows and Spacetime Instabilities , Phys. Rev. D , 024039 (2017).[22] F. Atamurotov, B. Ahmedov, and A. Abdujabbarov, Optical Properties of Black Holes in thePresence of a Plasma: The Shadow , Phys. Rev. D , 084005 (2015).[23] M. Amir and S. G. Ghosh, Shapes of Rotating Nonsingular Black Hole Shadows , Phys. Rev. D , 024054 (2016).[24] M. Sharif and S. Iftikhar, Shadow of a Charged Rotating Non-Commutative Black Hole , Eur.Phys. J. C , 630 (2016).[25] G. Z. Babar, A. Z. Babar, and F. Atamurotov, Optical Properties of KerrNewman Spacetime inthe Presence of Plasma , Eur. Phys. J. C , 761 (2020).[26] R. Kumar and S. G. Ghosh, Rotating Black Holes in 4D Einstein-Gauss-Bonnet Gravity and ItsShadow , J. Cosmol. Astropart. Phys. 2020, 053 (2020).[27] B. P. Singh and S. G. Ghosh,
Shadow of SchwarzschildTangherlini Black Holes , Annals of Physics , 127 (2018).[28] Z. Chang and Q.-H. Zhu,
Revisiting a Rotating Black Hole Shadow with Astrometric Observables ,Physical Review D , (2020).[29] Z. Chang and Q.-H. Zhu,
Does the Shape of the Shadow of a Black Hole Depend on MotionalStatus of an Observer? , arXiv:2006.00685 [Gr-Qc] (2020).[30] M. S. Ali and S. G. Ghosh,
Thermodynamics and Phase Transition of Rotating Hayward-deSitter Black Holes , arXiv:1906.11284 [Gr-Qc] (2019).[31] D. Lebedev and K. Lake,
On the Influence of the Cosmological Constant on Trajectories of Lightand Associated Measurements in Schwarzschild de Sitter Space , arXiv:1308.4931 [Gr-Qc] (2013)., arXiv:1308.4931 [Gr-Qc] (2013).