Strong field gravitational lensing by hairy Kerr black holes
aa r X i v : . [ g r- q c ] F e b Strong field gravitational lensing by hairy Kerr black holes
Shafqat Ul Islam a ∗ and Sushant G. Ghosh a, b † a Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India and b Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science,University of KwaZulu-Natal, Private Bag 54001, Durban 4000, South Africa
The recent time witnessed a surge of interest in strong gravitational lensing by black holes is due tothe Event Horizon Telescope (EHT) results, suggesting comparing the black hole lensing in generalrelativity and modified gravity theories. That may help us to assess the phenomenological differencesbetween these models. A Kerr black hole is also a solution to some alternative theories of gravity,while recently obtained modified Kerr black holes (hairy Kerr black holes) are due to additionalsources or surrounding fluid, like dark matter, having conserved energy-momentum tensor. Thishairy Kerr black holes may be solutions to an alternative theory of gravity. We generalize previouswork, on gravitational lensing by a Kerr black hole, in the strong deflection limits to the hairy Kerrblack holes, with deviation parameter α and a primary hair ℓ . Interestingly, the deflection coefficient¯ a , respectively, increases and decreases with increasing ℓ and α . ¯ b shows opposite behaviour with ℓ and α . We also find that the deflection angle α D , angular position θ ∞ and u m decreases, butangular separation s increases with α . We compare our results with those for Kerr black holes,and also the formalism is applied to discuss the astrophysical consequences in the context of thesupermassive black holes Sgr A* and M87*. I. INTRODUCTION
In Einstein’s general relativity there is only one uncharged rotating black hole solution given by the Kerr metric [1].The no-hair theorem states that black holes are uniquely characterized by their mass M and spin J and are describedby the Kerr metric [2–4]. Consequently, all astrophysical black holes are expected to be Kerr black holes. That thegravitational collapse of a sufficiently massive star is a Kerr black hole and astronomers have discovered several goodastrophysical candidates. While there is some indirect evidence suggesting that the latter have an event horizon, theyare black holes, a proof that the Kerr geometry describes the space-time around these objects is still lacking and it maybe difficult to rule out non-Kerr black holes [5, 6]. A rotating non-Kerr metric black hole has an additional deviationparameter from modifying gravity or matters, apart from mass and rotation parameters and encompasses the Kerrblack hole as a particular case. The important question arises: Are such black holes candidates for testing the no-hairtheorem or the Kerr hypothesis? The Kerr hypothesis, a strong-field prediction of general relativity, may not hold forthe non-Kerr black holes. Here, we use the gravitational lensing as a tool to investigate the constraints when rotatingnon-Kerr can be considered as astrophysical black hole candidates. The Gravitational Decoupling approach (GD)[7, 8] is precisely designed for describing deformations of known spherically symmetric solutions of General Relativityinduced by additional sources. The method is useful for generating new and more complex solutions from known(seed) solutions of the Einstein field equations and finding solutions of modified gravitational theories. Recently, it isshown that the GD could be used to obtain axially symmetric systems [9]. Indeed the black hole solution containsa source satisfying the SEC and provides an extension of the Kerr metric termed as Kerr black holes with primaryhair[9].Deflection of a light ray in a gravitational field is referred to as gravitational lensing, and the object causing adeflection is called a gravitational lens. Gravitational lensing by black holes is one of the most powerful astrophysicaltools for investigating the strong field features of gravity. It could provide a profound test of modified theories ofgravity in the strong field regimes [10–15] and also the cosmic censorship hypothesis [16, 17]. The strong field limitgravitational lensing studies due to black holes have received considerable attention in recent years, indicating thatone can extract the black hole information from it. Gravitational lensing by black holes began to be observationallycrucial in the 1990s, which motivated several quantitative studies of the Kerr metrics caustics [18–20]. Since then,gravitational deflection of light by rotating black holes has received significant attention due to the tremendousadvancement of current observational facilities [21–27]The recent time witnessed a flurry of interest in strong gravitational lensing by black holes due to the Event Horizon ∗ Electronic address: [email protected] † Electronic address: [email protected], [email protected]
Telescope (EHT) observations [28–33]. This paper aims to investigate the black hole lensing theory of recently derivedhairy Kerr black holes [9] and assess the phenomenological differences with the Kerr black holes. So the purpose ofthis paper is to study the effects of the primary hair ℓ and deviation parameter α on the strong gravitational lensingby the rotating hairy Kerr black holes in the context of supermassive black holes Sgr A* and M87*. Our results showthat there is a significant effect of the primary hair on the strong gravitational lensing.The paper is organized as follows: In the Sect. II, we briefly review the recently obtained hairy Kerr black holes. Aformalism for gravitational deflection of light in the strong-field limit is the subject in Sect. III. In Sec IV, we discussthe strong-lensing observables, numerical estimations of deflection angle, and lensing by supermassive black holes SgrA* and M 87*. We conclude with our significant results in Sect. V. Throughout this paper, unless otherwise stated,we adopt natural units ( G = c = 1) II. THE GD APPROACH FOR HAIRY KERR BLACK HOLES
Recently, Ovalle et.al. [34], (see also [7, 8]), proposed a simple approach to generate spherically symmetric hairyblack holes by requiring a well-defined event horizon and the SEC or DEC for the hair outside the horizon, which theyextended to rotating case [9]. Throughout the paper we shall call the procedure to generate the deformed solutionsas the
GD approach . We briefly review the GD approach to generate hairy rotating black holes. The straightforwardmethod is designed to create deformed solutions, to the known general relativity solution, because of the additionalsources. Thus, using the GD approach, one has a systematic and straightforward strategy to extensions of axially-symmetric black holes as well [9]. Therefore, one can without much effort obtain the Kerr black hole’s nontrivialextensions that can support primary hair [9]. To outline the GD approach, let us start with the Einstein fieldequations ˜ G µν = k ˜ T µν = k ( T µν + S µν ) (1)where T µν correspond to the energy momentum tensor (EMT) of the known solution in general relativity and S µν isthe EMT of the additional source [7, 8]. Since we are concerned with only black hole solution, so T µν = 0. Considera generic extension of the Kerr black hole which in the Boyer-Lindquist coordinates is given by [35, 36] ds = − (cid:20) − r ˜ m ( r )Σ (cid:21) dt + Σ∆ dr + Σ dθ − ar ˜ m ( r )Σ sin θ dt dφ + (cid:20) r + a + 2 a r ˜ m ( r )Σ sin θ (cid:21) sin θ dφ , (2)where Σ = r + a cos θ , ∆ = r + a − r ˜ m ( r ), a = J/M , and J is the angular momentum. Eq. (2) canbe used to describe rotating compact objects like black holes, which encompasses well known Kerr black holes when˜ m ( r ) = M . When a = 0, we obtain the following spherically symmetric static metric ds = − (cid:20) − m ( r ) r (cid:21) dt + (cid:20) − m ( r ) r (cid:21) − dr + r d Ω . (3)In the GD approach, by deforming the spherically symmetric static black hole solution of general relativity, one cangenerate rotating black hole spacetimes, e.g., one can obtain non-trivial extensions of the Kerr black holes, or the hairyKerr black hole. Let us suppose that the ˜ m ( r ) = m ( r ) correspond to the EMT T µν alone and adding the additionalsources S µν leads to ˜ m ( r ) = m ( r ) + αm s ( r ) , (4)where α is deformation parameter. Thus, the mass functions m and m s are, respectively, generated by the energy-momentum tensor T µν and S µν . The S µν representing additional sources surrounding black hole which could be EMTof dark matter or dark energy. This is because the Einstein tensor has only linear derivatives of the mass function˜ m ( r ), and hence we also have a linear decomposition of the Einstein tensor˜ G σγ ( ˜ m, a ) = G σγ ( m, a ) + αG σγ ( m s , a ) . (5)provided the rotational parameter a does not change. The Eq. (5), is the requirement for generating axially symmetricblack hole solutions.As an immediate consequence of the GD approach, one can generate the well-known Kerr-Newman solution of theEinstein-Maxwell system. For this we have to choose T µν = 0 and S µν = 14 π (cid:18) F µα F αν + 14 g µν F αβ F αβ (cid:19) (6)Solving the Einstein equations in the vacuum T µν = 0, we find the Schwarzschild solution with mass ˜ m ( r ) = M andfor the source S µν , one gets the Reissner-Nordstr¨om solution, whose mass function is identified as m s ( r ) = C − Q r , (7)where C and Q are integration constants, Q identified as the electric charge. Then Eq. (4) gives the total massfunction ˜ m ( r ) = M − Q r , (8)with M = C + M . Finally, for rotating solution we substitute Eq. (8) in metric (2), which gives∆ = r − r M + a + Q . (9)The metric (3) with mass function (8) and the above ∆, is the Kerr-Newman solution black hole solution.Next, we consider the Schwarzschild black hole surrounded by a spherically symmetric matter with a conservedenergy-momentum tensor S µν satisfying SEC. It leads to the hairy the Schwarzschild black hole [9] − (cid:20) − M r + αe − r/ ( M− ℓ / (cid:21) dt + (cid:20) − M r + αe − r/ ( M− ℓ / (cid:21) − + r d Ω , (10)where ℓ = α ℓ measures the increase of entropy caused by the hair, and must satisfy ℓ ≤ M = ℓ K to ensureasymptotic flatness. Eq. (10) can be used as seed metric to generate rotating black holes. As above identifying massfrom the metric (10), we have [9] ˜ m ( r ) = M − α r e − r/ ( M− ℓ / , (11)which implies ∆ = r + a − r M + αr e − r/ ( M− ℓ / . (12)Then metric (2) with mass (11) and the above ∆ represents hairy Kerr black holes. Finally, the spherically symmetricmetric with components (10) satisfies the SEC [34] which is also obeyed by the rotating metrics. III. DEFLECTION ANGLE BY ROTATING BLACK HOLES
In this section, we will study the gravitational lensing by the hairy Kerr black holes to investigate how the deformedparameter α affects the coefficients and the lensing observables in the strong field limit. For this purpose, definingthe distance, rotational parameter, primary hair and time in terms of Schwarzschild radius [37] x → r M , a → a M , t → t M , ℓ → ℓ M (13)we obtain ds = − (cid:20) − x ˜ m ( x )Σ (cid:21) dt + Σ∆ dx + Σ dθ − a x ˜ m ( x )Σ sin θ dt dφ + (cid:20) x + a + a x ˜ m ( x )Σ sin θ (cid:21) sin θ dφ , (14)where Σ = x + a cos θ , ∆ = x + a − x ˜ m ( x ) and ˜ m ( x ) = 1 − α x e − x/ (1 − ℓ ) . The metric (14) has an additionalsingularity when Σ = 0 and ∆ = 0 corresponding to the event horizon, which are the zeroes of g rr = ∆ = 0, i.e., x + a − x h − α x e − x/ (1 − ℓ ) i = 0 (15) - - x D a = and Α= l = l K l = l = l = - - x D a = and Α= l = l K l = l = l = - x D Α= and l = a = a = a = a = - x D Α= and l = a = a = a = a = FIG. 1: Plot showing the behaviour of horizon (∆ vs x ), with varying black hole parameter a , α and ℓ . a x ± Α = l = l K l = l = l = a x ± Α = l = l K l = l = l = FIG. 2: Plot showing the behaviour of event horizon (solid lines) and Cauchy horizon (dotted lines), with varying black holeparameter a , for different values of ℓ and α . Points on the horizontal axis correspond to extremal values of a . The case ℓ = ℓ K corresponds to Kerr black hole. One can find that there exists non-zero values of a , α and ℓ for which Eq. (15) admits two positive roots ( x ± )corresponding to Cauchy ( x − ) and event horizons ( x + ). The black hole rotates at less extreme value of spin whenthey have hair (cf. Fig. 2). The function (15) is plotted in Fig. 1 for a different values of ℓ , α and a . We see thatthe horizon shifts to larger radii when ℓ increases, reaching a maximum value corresponding to the Kerr horizon for ℓ = ℓ K = 1.Next, to discuss the strong deflection of light by hairy Kerr black holes we shall consider light rays strictly in the - - a x m Α = l = l K l = l = l = - - a x m Α = l = l K l = l = l = FIG. 3: Plot showing the behaviour of radius of photon sphere ( x m vs a ) and impact parameter ( u m vs a ), with varying blackhole parameter a , for different values of ℓ and α . ℓ = ℓ K corresponds to Kerr black hole. equatorial plane ( θ = π/
2) and the metric (14) simplifies tods = − A ( x ) dt + B ( x ) dx + C ( x ) dφ − D ( x ) dt dφ, (16)where A ( x ) = 1 − − α x e − x/ (1 − ℓ ) x , B ( x ) = x ∆ ,C ( x ) = (cid:0) x + a (cid:1) − a ∆ x , D ( x ) = 2 a (cid:2) − α x e − x/ (1 − ℓ ) (cid:3) x . (17)The black hole metric (16) admits two linearly independent killing vectors, η µ ( t ) = δ µt and η µ ( φ ) = δ µφ associated withthe time translation and rotational invariance [38]. The photon’s trajectory is determined by two conserved quantitiesadmitting the Killing vectors, i.e., angular momentum L and total energy E . The null geodesic equations for themetric can be derived using the Hamilton-Jacobi method, and we get the following differential systems˙ t = 4 C − LD AC + D , (18)˙ θ = 0 , (19)˙ φ = 2 D + 4 AL AC + D , (20)˙ x = ± s C − DL − AL B (4 AC + D ) . (21)where the dot indicates derivative with respect to affine parameter. Using Eq. (21) the effective potential V eff forradial motion can be obtained as V eff = 4( C − DL − AL ) B (4 AC + D ) , (22)which characterizes the different types of possible orbits. In the asymptotic limit, depending on the effective potential,a light ray from the source at infinity approaches the black hole and may turn at some radius x , only to escape towardsthe observer at infinity. It is a well-known fact that deflection angle diverges as light approaches the photon sphere,and as such, there can be an infinite number of images just outside the photon sphere. Circular photon orbitssimultaneously satisfy ˙ x = ¨ x = 0 and photon sphere which constitutes the unstable photon orbits additionally satisfy... x >
0. Photon sphere radius is plotted in Fig. 3. The photons are allowed allowed to get closer to the black hole forpositive a . The conditions for the unstable photon sphere are [39] dV eff dx (cid:12)(cid:12)(cid:12) ( x = x m ) = 0 , d V eff dx (cid:12)(cid:12)(cid:12) ( x = x m ) < , (23)where x m is the radius of the photon sphere and x is the distance of the light’s minimum approach towards the blackhole. The impact parameter, which is the perpendicular distance from the center of mass of the lens to the tangentof the null geodesics and remains constant throughout the trajectory, coincides with the angular momentum L in theequatorial plane. The turning point is marked by ˙ x = 0 thereby effective potential vanishes. Then V eff = 0 implies L = u = − D + p D + 4 A C A , = aP ( x ) + x p a + x [ x + P ( x x + P ( x (24) P ( x ) = αx e x / (1 − ℓ ) − A ( x ) = A ( x ). Eq. (24) relates the impact parameter u and minimum distance x .Photons winding in the same sense (prograde or direct photons) as that of black hole rotation form different orbitsthan those winding in opposite direction (retrograde photons). We fix the counterclockwise winding of light raysby choosing the positive sign before the square bracket in Eq. (24). For a >
0, the black hole also rotates in thecounterclockwise direction, while for a <
0, the black rotates in the opposite direction of photon winding.The deflection angle in rotating stationary spacetime described by the line element (16), at closest distance approach x , is given by [37] α D ( x ) = I ( x ) − π, (26)where I ( x ) = 2 Z ∞ x dφdx dx = 2 Z ∞ x √ A B (2 AL + D ) √ AC + D p A C − AC + L ( AD − A D ) dx, (27)However, the integral in Eq. (27) can not be solved in an explicit form. The deflection angle is very small inweak deflection limit (WDL), and an approximate solution can be obtained. But the classical WDL is invalid whendealing with lensing in strong gravitational field. For solving this problem one could seek particular function toreplace the integral as done in [40, 41] but a much effective way to handle the integral in Eq. (27) is to expandthe deflection angle in the strong deflection limit (SDL) near the photon sphere [37, 42]. This method provides ananalytical representation of the deflection angle and a straightforward and efficient connection between the coefficientsand observables as discussed in Sect. (IV).Introducing the variable z = 1 − x /x [21], we can rewrite the integral (27) I ( x ) = Z R ( z, x ) f ( z, x ) dz, (28)where R ( z, x ) = 2 x x √ B (2 A AL + A D ) √ CA √ AC + D , (29) f ( z, x ) = 1 q A − A C C + LC ( AD − A D ) . (30)The function R ( z, x ) is a regular term, where as f ( z, x ) diverges when z →
0. The function f ( z, x ) can beapproximated as f ( z, x ) ∼ f ( z, x ) = 1 √ c z + c z , (31)We get the radius of the photon sphere by solving c = 0 (cf. Fig. 3). By assuming the closest approach distance x not to be too large than x m , the deflection angle can be expanded as [37, 43] α D ( θ ) = − ¯ a log (cid:16) θD OL u m − (cid:17) + ¯ b + O ( u − u m ) . (32) - - a a - Α = l = l K l = l = l = - - a a - Α = l = l K l = l = l = - - - - - - - - - a b - Α = l = l K l = l = l = - - - - - - - - - a b - Α = l = l K l = l = l = - - a u m Α = l = l K l = l = l = - - a u m Α = l = l K l = l = l = FIG. 4: Plot showing the behaviour of lensing coefficients, ¯ a and ¯ b and u m as a function of black hole spin a , for different valuesof ℓ and α . The case ℓ = ℓ K corresponds to Kerr black hole. The coefficients ¯ a and ¯ b of strong field limit are given by¯ a = R (0 , x m )2 √ c m , and ¯ b = − π + I R ( x m ) + ¯ a log cx m u m (33) I R ( x m ) = Z [ R ( z, x m ) f ( z, x m ) − R (0 , x m ) f ( z, x m )] dz, (34) u − u m = c ( x − x m ) (35)The subscript m means that the functions are evaluated at x = x m . The strong-field deflection coefficients, ¯ a , ¯ b and u Α D H u L a = Α= l = l K l = l = l = u Α D H u L a = Α= l = l K l = l = l = FIG. 5: Plot showing the variation of deflection angle as a function of impact parameter u , for different values of a , ℓ and α .Points on the horizontal axis represents the values of impact parameter u = u m at which deflection angle diverges. The case ℓ = ℓ K corresponds to Kerr black hole. u m , are plotted against the angular spin in Fig. 4, which shows that ¯ a and ¯ b , increase and decrease, respectively with a . The minimum impact parameter u m decrases in a similar fashion as x m . For the retrograde photons, the lensingcoefficients are very close to the Kerr black hole. As can be seen in Fig. 4 that lensing coefficients ¯ a and ¯ b in hairyKerr black hole diverge at lower values of spin in comparison to standard Kerr black holes. The deflection angle is alsoplotted as a function of impact parameter (cf. Fig. 5) for different values of a , α and ℓ and it diverges as u → u m .For fixed values of parameters, the deflection angle diverges at lower impact parameter for larger α . Moreover, thevalues of u m gets smaller when the Kerr black holes have hair. It can also be infered from Fig. 5, that strong lensingis valid only when impact parameter is very close to u m . IV. OBSERVABLES AND RELATIVISTIC IMAGES
The gravitational lensing system consists of a light source ( S ), an observer ( O ) and a black hole ( L ) that act as alens and lies in between the source and observer. The unlensed source and the images are viewed by the observer atan angular separation of β and θ from the optical axis. α D is the total angle by which the photon is deviated fromits path by the lens’s gravitational field while travelling from the source to the observer. D OL , D LS and D OS are theobserver-lens, lens-source and observer-source distances. As long as the deflection angle is not vanishing the closestapproach distance x is different than the impact parameter u .In the lens equation, we consider the asymptotic approximation, i.e., both the source and the observer are notaffected by the lens’s curvature and lie in flat space-time. This allows us to use Euclidean geometry relations to relatethe various quantities and restore all the relativistic information in the deflection angle without the loss of generality.As discussed by the authors in [44], the lens equation used in [45] would be adequate, without the need to resort to theexact lens equation. Accordingly, we introduce here a coordinate independent Ohanian lens equation [45] connectingthe source and observer positions as ξ = D OL + D LS D LS θ − α D ( θ ) , (36) ξ is the angle between the direction of source and optical axis as viewed from the lens. The angle ξ and β are relatedby [44] D OL sin( ξ − β ) = D LS sin β (37)We use the hypothesis of small angles α D , β and θ in the classical lens equation so it makes sense to perform smallangle approximation, however rude, in all the trigonometric functions and reconsider the exact lens equation here.Also the lensing effects are prominent when all the objects are almost aligned. Rewriting the Eq. (36) for small valuesof β , ξ and θ using Eq. (37) we get [44], β = θ − D LS D OL + D LS α D ( θ ) . (38) - -
10 0 10 20 - - X Y FIG. 6: Plot showing the angular radius of the outermost Einstein Ring for supermassive black holes at the center of Milkyway (Black) and M87 (Blue).
It is interesting to see that for strong field lensing, where a ray of light emitted by the source S follows multiple loopsaround the black hole before reaching the observer, a similar expression is obtained. However α D ( θ ) is replaced by α D ( θ ) − nπ = ∆ α n ,with n ∈ N and 0 < ∆ α n ≪ β and the distancesof observer and source from the black hole. As the photons approach the event horizon, the deflection angle becomesgreater than 2 π such that at critical impact parameter, it diverges. Using Eq. (32) and Eq. (38), the angular seperationbetween the optical axis and n -loop relativistic image can be written as combination of two parts [37] θ n = θ n + ∆ θ n . (39)where θ n = u m D OL (1 + e n ) , (40)∆ θ n = D OL + D LS D LS u m e n D OL p ( β − θ n ) , (41) e n = e ¯ b − nπ ¯ a . (42)Here θ n is the corresponding value of θ when α D ( θ ) = 2 nπ . ∆ θ n is the correction term which is smaller than themain term θ n . The Eq. (39), gives images only on the same side of the source. One can solve the same equation for β < A. Einstein Ring
Einstein rings, though not a physical structure in space but just a play of light and gravity, are in fact the mostvisually striking and spectacular effects of gravitational lensing. A ring shaped image is produced when the point lensis in the line of sight of source [46], such that it spreads the light equally in all directions. In complex lens systems[47–50] light from two or more sources situated at different distances from the lens, form multiple Einstein rings. Ifthe deflection angle is larger that 2 π the rings are said to be relativistic. The angular radius of the Einstein rings canbe obtained by solving Eq. (39) for source, lens and the observer being perfectly aligned ( β = 0). The equation (39)accordingly gives [37, 51] θ En = (cid:18) − D OL + D LS D LS u m e n D OL p (cid:19) θ n , (43)When the lens is exactly halfway between observer and source, Eq. (43) gives θ En = (cid:18) − u m e n D OL p (cid:19) (cid:18) u m D OL (1 + e n ) (cid:19) . (44)0 Sgr A* M87* Lensing Observables a α ℓ θ ∞ ( µ as) s ( µ as) θ ∞ ( µ as) s ( µ as) r m ¯ a ¯ b u m -0.25 0.0 0.00 30.1611 0.0143267 23.3683 0.0111001 7.95972 0.85705 -0.275121 3.069082.0 0.40 29.8852 0.0190843 23.1545 0.0147862 7.54368 0.904318 -0.369204 3.0412.0 0.60 30.1477 0.014725 23.3579 0.0114087 7.91191 0.862229 -0.290723 3.067712.0 1.00 30.1611 0.0143267 23.3683 0.0111001 7.95972 0.85705 -0.275121 3.069083.0 0.40 29.7253 0.0227215 23.0307 0.0176042 7.27962 0.937121 -0.44201 3.024743.0 0.60 30.1409 0.0149321 23.3526 0.0115691 7.88741 0.864908 -0.298873 3.067023.0 1.00 30.1611 0.0143267 23.3683 0.0111001 7.95972 0.85705 -0.275121 3.06908-0.1 0.0 0.00 27.4463 0.0226273 21.2649 0.0175312 7.3151 0.932575 -0.338872 2.792832.0 0.40 27.0128 0.0330511 20.929 0.0256074 6.77644 1.00671 -0.467797 2.748722.0 0.60 27.4198 0.0236574 21.2443 0.0183293 7.23876 0.94241 -0.36584 2.790132.0 1.00 27.4463 0.0226273 21.2649 0.0175312 7.3151 0.932575 -0.338872 2.792833.0 0.40 26.744 0.042277 20.7208 0.0327555 6.40824 1.06455 -0.582964 2.721373.0 0.60 27.4062 0.0242065 21.2338 0.0187548 7.199 0.947615 -0.380349 2.788753.0 1.00 27.4463 0.0226273 21.2649 0.0175312 7.3151 0.932575 -0.338872 2.792830.0 0.0 0.00 25.5324 0.0319537 19.782 0.0247571 6.82188 1.0000 -0.40023 2.598082.0 0.40 24.9283 0.0507796 19.314 0.0393431 6.17328 1.10507 -0.564093 2.536612.0 0.60 25.489 0.0340225 19.7484 0.02636 6.71566 1.01582 -0.44049 2.593662.0 1.00 25.5324 0.0319537 19.782 0.0247571 6.82188 1. -0.40023 2.598083.0 0.40 24.5266 0.0698816 19.0028 0.054143 5.70217 1.19637 -0.728376 2.495743.0 0.60 25.4666 0.0351548 19.731 0.0272373 6.6594 1.0244 -0.462847 2.591383.0 1.00 25.5324 0.0319537 19.782 0.0247571 6.82188 1.0000 -0.40023 2.598080.1 0.0 0.00 23.5035 0.0471722 18.2101 0.0365481 6.25687 1.0903 -0.48879 2.391622.0 0.40 22.6305 0.0839756 17.5337 0.0650627 5.46158 1.24907 -0.707259 2.302792.0 0.60 23.4296 0.051641 18.1528 0.0400105 6.1056 1.11732 -0.551929 2.38412.0 1.00 23.5035 0.0471722 18.2101 0.0365481 6.25687 1.0903 -0.48879 2.391623.0 0.40 21.9846 0.129033 17.0333 0.0999726 4.84366 1.40842 -0.953293 2.237073.0 0.60 23.3905 0.054188 18.1225 0.0419839 6.02345 1.13255 -0.588726 2.380133.0 1.00 23.5035 0.0471722 18.2101 0.0365481 6.25687 1.0903 -0.48879 2.391620.25 0.0 0.00 20.1279 0.0959399 15.5947 0.0743325 5.19534 1.31308 -0.736706 2.048132.0 0.40 18.4142 0.224469 14.267 0.173915 4.029 1.6932 -1.17896 1.873752.0 0.60 19.9403 0.113604 15.4494 0.0880186 4.9162 1.38763 -0.88779 2.029042.0 1.00 20.1279 0.0959399 15.5947 0.0743325 5.19534 1.31308 -0.736706 2.048133.0 0.40 18.0944 0.297653 14.0192 0.230616 3.55085 1.92119 -1.60798 1.841213.0 0.60 19.8344 0.124905 15.3673 0.0967738 4.75394 1.43499 -0.988825 2.018273.0 1.00 20.1279 0.0959399 15.5947 0.0743325 5.19534 1.31308 -0.736706 2.04813 TABLE I: Estimates for the lensing observables for supermassive black holes at the center of nearby galaxies for different valuesof a , α and ℓ . The case a = α = 0, α = 0 respectively, correspond to Schwarzschild and Kerr black holes. Since D OL ≫ u m , the Eq. (44) gives [51] θ En = u m D OL (1 + e n ) , (45)which gives the radius of the n th relativistic Einstein ring. Here n = 1 gives the angular position of the outermostring, and rings become more packed as n increases. We have plotted Einstein rings for Sgr A* and M87 in Fig. 6. Itcan be deduced from Eq. (45) that radius of the Einstein ring decreases as the distance between the observer and lensincreases and increases with the mass of the black hole.Besides the position, magnification of the images can be another good source of information. The brightness ofthe relativistic images will be magnified by the lensing. Classically, magnification is the ratio of angular area elementof the image and corresponding area element of the unlensed source. For n -loop relativistic images magnification isgiven by [37, 52] µ n = 1 β " u m D OL (1 + e n ) D OS D LS u m e n D OL p ! . (46)Thus magnification decreases exponentially with n and the images become fainter as n increases.In practice, if the outermost image can be distinguished from the other inner packed ones, we can have three1 - - a Θ ¥ H Μ a s L Α = l = l K l = l = l = - - a Θ ¥ H Μ a s L Α = l = l K l = l = l = - - a s H Μ a s L Α = l = l K l = l = l = - - a s H Μ a s L Α= l = l K l = l = l = - - a r m a g Α = l = l K l = l = l = - - a r m a g Α = l = l K l = l = l = FIG. 7: Plot showing the behaviour of lensing Observables θ ∞ , s and r mag as a function of black hole spin a , with varyingparameters α , ℓ for Sagittarius Black hole. The case ℓ = ℓ K corresponds to Kerr black hole. distinguishable observables [37], defined as θ ∞ = u m D OL (47) s = θ − θ ∞ ≈ θ ∞ ( e ¯ b − π ¯ a ) (48) r mag = µ P ∞ n =2 µ n ≈ e π ¯ a (49)Here, θ ∞ is asymptotic position advanced by the set of images obtained in the limit n → ∞ , s is the angularseparation between the first image ( n = 1) and the inner packed ones at θ ∞ . r mag is the difference in magnitudeof flux of the first image and flux from all the other images. For numerical estimation of these observables we areconsidering a realistic case of black holes, namely, Sgr A* in our galactic center, and M87* in Messier 87 galaxy. ForSgr A*, we consider M = 4 . × M ⊙ and D OL = 8 .
35 Kpc [53], whereas for M87* the mass M = 6 . × M ⊙ and distance D OL = 16 . a , α ,and ℓ and comparison with Schwarzschild ( a = α = 0) and Kerr black hole ( α = 0). It is worth to note that, forKerr black hole with primary hair ℓ , the angular position and the relative magnification decrease with α and increase2 - - a Θ ¥ H Μ a s L Α = l = l K l = l = l = - - a Θ ¥ H Μ a s L Α = l = l K l = l = l = - - a s H Μ a s L Α = l = l K l = l = l = - - a s H Μ a s L Α = l = l K l = l = l = - - a r m a g Α = l = l K l = l = l = - - a r m a g Α = l = l K l = l = l = FIG. 8: Plot showing the behaviour of lensing Observables θ ∞ , s and r mag as a function of black hole spin a , with varyingparameters α , ℓ for M87 black hole. The case ℓ = ℓ K corresponds to Kerr black hole. with ℓ but is smaller than the Kerr black hole. The angular separation, on the other hand decreases with ℓ and butincreases with α . For retrograde orbits, s , is smaller than the prograde orbits. Further, for given values of parameters a , α and ℓ , the angular position and angular separation of the relativistic images for the Sgr A* black hole are largerthan the M87 black hole. We have plotted lensing observables in Fig. 7 and Fig. 8 for Sgr A* and M87 respectively. V. CONCLUSIONS
Experimental and observational test have been performed on the gravitational theories, some of which seem torule out theories alternative to the general relativity. Though implemented in a weak field, these tests did not showany defects in the general relativity proposed by Einstein. On the other hand, unlike the weak field regime tests,the observations in a strong field are restricted and are not enough to verify the gravitational theories. Practisinggravitational lensing can be an essential tool to investigate the supermassive black holes in the regions very close to theevent horizon and hence would give exceptional feedback on the correctness of the theory of gravitation in the strongfield. Unfortunately due to the bad influence of the accretion matter and the extinction of radiation in the black hole’svicinity, observing relativistic images is an uphill task. For these images to be very prominent, the lens components3(the source, the black hole and the observer) should be highly aligned, the probability of which is minimal. Despiteall these constraints, no doubt observing these images would have immense implications in astronomy. The EventHorizon Telescope has created a surge of interest in strong gravitational lensing and has opened up the possibilityof experimentally probing the gravity in its strongest regime. In addition, the strong field lensing can provide anopportunity to test it in the black hole’s vicinity by observing a set of infinite discrete relativistic images near thephoton sphere.Even though most of the available tests are consistent with general relativity, deviations from the Kerr black hole(or hairy black hole) due to additional sources or arising from modified theories of gravity can’t be igonored. Thesehairy black holes, in Boyer-Lindquist coordinates, are defined by the metric (3) with mass function m ( r ), and Kerrblack holes are included as a particular case when m ( r ) = M ( α = 0). The impact of the deviation parameter α ,arising due to surrounding matter, on gravitational lensing presents a good theoretical opportunity to distinguish thehairy rotating black holes from the Kerr black and test whether astrophysical black hole candidates are the blackholes as predicted by Einstein’s general relativity. Motivated by above aurguments, we have examined the effects of α and ℓ , in a strong-field observation, to the lensing observables due to rotating hairy black holes and compared withthose due to Kerr black holes. We have numerically calculated the strong lensing coefficients and lensing observablesas functions of α for relativistic images. In turn, we have applied our results to the supermassive black holes, Sgr A*and M87*, at the centre of galaxies. Our analysis shows that such hairy Kerr black holes’ properties are qualitativelydifferent from the Kerr black holes.We highlight results that are obtained by our analysis. The horizon radius of the hairy black hole increases with l and coincides with maximum horizon radius of the Kerr black hole in the limit l →
1, whereas the horizon radiusdecreases with α (cf. Fig 1). This is also true for the photon sphere. Intrestingly, the lensing coefficient, ¯ a , like Kerrblack hole case, increase with a whereas ¯ b decreases. ¯ a , always takes larger value when compared with the Kerr blackhole and ¯ b is smaller (cf. Fig. 4). Both ¯ a and ¯ b , diverges with opposite sign at critical values of a (e.g. for ℓ = 0 . α = 2 , a = 0 . α D , increaseswith u and diverges at impact parameter u = u m , u m decreases with with ℓ as well as α (cf. Fig. 5).We have also numerically calculated lensing observables θ ∞ , separation s and relative magnitude r mag . The θ ∞ and r mag decrease with the increasing a . They take smaller values compared to the Kerr black hole and both decreasewith deviation parameter α (cf. Fig. 7, Fig. 8 and Table I). Similarly the observable s , increases with a as well as with α , which means that separation between two relativistic images for hairy Kerr black hole is larger than analogousKerr black hole. Finally, considering the supermassive black holes Sgr A* and M87* as hairy Kerr black holes, wehave analysed the magnitude of lensing observables. It turns out that, for given values of parameters a , α and ℓ , theimages of Sgr A* are less packed than M87*.Therefore our results, in principle, could provide a possibility to test how hairy black holes deviates from the Kerrblack holes in the future astronomical observations. VI. ACKNOWLEDGMENTS
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