Gravitational lensing by Charged black hole in regularized $4D$ Einstein-Gauss-Bonnet gravity

Among the higher curvature gravities, the most extensively studied theory is the so-called Einstein-Gauss-Bonnet (EGB) gravity, whose Lagrangian contains Einstein term with the GB combination of quadratic curvature terms, and the GB term yields nontrivial gravitational dynamics in $ D\geq5$. Recently there has been a surge of interest in regularizing, a $ D \to 4 $ limit, the EGB gravity, and the resulting regularized $4D$ EGB gravity has nontrivial dynamics. We consider gravitational lensing by Charged black holes in the regularized $4D$ EGB gravity theory to calculate the deflection coefficients $\bar{a}$ and $\bar{b}$, while former increases with increasing GB parameter $\alpha$ and charge $q$, later decrease. We also find a decrease in the deflection angle $\alpha_D$, angular position $\theta_{\infty}$ decreases more slowly and impact parameter for photon orbits $u_{m}$ more quickly, but angular separation $s$ increases more rapidly with $\alpha$ and charge $q$. We compare our results with those analogous black holes in General Relativity (GR) and also formalism is applied to discuss the astrophysical consequences in the case of the supermassive black holes Sgr A*.


I. INTRODUCTION
General Relativity (GR) not only predicts the existence of black holes but also a mean to observe them through the gravitational impact on the electromagnetic radiation moving in the near vicinity of black holes. The advent of horizon-scale observations of astrophysical black holes [1,2] offer an unprecedented opportunity to understand the intricate details of photon geodesics in the black hole spacetimes [3,4], which has become of physical relevance in present-day astronomy. Photons passing in the gravitational field of a compact astrophysical object get deviate from their original path and the phenomenon is known as the gravitational lensing [5][6][7]. Though the theory of gravitational lensing was primarily developed in the weak-field thin-lens approximation for small deflection angles [8,9], but in black hole spacetimes, where photons can traveled close to the gravitational radius, a full treatment of lensing theory valid even in strong-field gravity regime is required. Darwin [7] led the study of strong gravitational lensing theory, and later Virbhadra and Ellis [10] numerically calculated the deflection angle due to the Schwarzschild black hole in an asymptotically flat background. Using an alternative formulation, Frittelli, Kling and Newman [11] analytically obtained an exact lens equation. A significant interest in the strong gravitational lensing is developed by the Bozza et al. [12], who gave a general and systematic investigation of light bending in the strong-gravity region, and exploiting the source-lens-observer geometry obtained the analytical * Electronic address: rahul.phy3@gmail.com † Electronic address: Shafphy@gmail.com ‡ Electronic address: sghosh2@jmi.ac.in, sgghosh@gmail.com expressions for the source's images positions.
One of the generic features of strong gravitational lensing is the logarithmic divergence of the deflection angle in the impact parameter and the existence of relativistic images, which are produced due to multiple winding of light around the black hole before emanating in observer's direction [13]. The strong gravitational lensing relevance for predicting the strong-field features of gravity, testing and comparing various theories of gravity in strong-field regime, estimating black hole parameters, and deducing any matter distributions in black hole background have resulted in a vast comprehensive literature [14][15][16][17][18]. Also, gravitational lensing from various modifications of Schwarzschild geometry arising due to modified gravities, e.g., regular black holes [19,20], massive gravity black holes [21], f (R) black holes [22] and also in Einstein-Gauss-Bonnet (EGB) gravity models [23].
EGB gravity theory is a natural extension of GR to higher dimensions D ≥ 5, in which Lagrangian density admits quadratic corrections constructed from the curvature tensors invariants [24,25]. The EGB gravity, which naturally appears in the low-energy limit of string theory [26], preserves the degrees of freedom and is free from gravitational instabilities and thereby leads to the ghost-free nontrivial gravitational self-interactions [27].
Due to much broader theoretical setup and consistency with the available astrophysical data, EGB gravity had been widely studied in varieties of context over the past decades [28][29][30].
The Gauss-Bonnet (GB) correction to the Einstein-Hilbert action is a topological invariant in D = 4 and therefore does not make any contribution to the gravitational dynamics. This issue has been addressed by Glavan and Lin [31] by re-scaling the GB coupling parameter as α → α/(D − 4) and defining the 4D theory as the limit of D → 4 at the level of field's equation. This kind of regularization was earlier proposed by Tomozawa [32] as quantum corrections to Einstein gravity, and they also found the spherically symmetric black hole solution. Later Cognola et al. [33] gave a simplified approach for Tomozawa [32] formulation, which mimic quantum corrections due to a GB invariant within a classical Lagrangian. Further, the static and spherically symmetric black hole solution [31][32][33] of 4D EGB gravity is identical as those found in semi-classical Einstein's equations with conformal anomaly [34], regularized Lovelock gravity [35,36], and the Horndeski scalar-tensor theory [37].
This motivated us to consider the gravitational lensing by Charged black hole in regularized 4D EGB gravity. Following the prescription of Bozza et al. [12], we determine the strong deflection coefficients and the resulting deflection angle, which becomes unboundedly large for smaller impact parameter values. Positions and magnifications of source's relativistic images are determined and the effect of charge is investigated. We also obtained the corrections in the deflection angle due to the GB coupling parameter in the supermassive black hole contexts.
The rest of paper is organized as follows. In the Sect. II, we discuss the static spherically symmetric Charged black hole in 4D EGB gravity. Formalism for gravitational bending of light in strong-field limit is setup in Sect. III, whereas strong-lensing observables, numerical estimations of deflection angle, and image positions and magnifications are presented in Sect. IV. Lensing by supermassive black hole Sgr A* is discussed in Sect. IV. Finally, we summarize our main findings in Sect. VI.

II. CHARGED BLACK HOLES IN 4D EGB GRAVITY
The EGB gravity action with re-scaled coupling constant α/(D − 4) and minimally coupled electromagnetic field in D dimensional spacetime reads [39] (1) with G is the GB term defined by g is the determinant of metric tensor g µν , R is the Ricci scalar, and F µν = ∂ µ A ν − ∂ ν A µ is the Maxwell tensor with A µ being the gauge potential. On varying the action (1) with respect to the metric tensor g µν , we obtain the field's equation Rg µν is the Einstein's tensor, and H µν is the Lanczos tensor [24] and is given by and T µν is the energy-momentum tensor for the electromagnetic field. Considering a static and spherically symmetric D-dimensional metric anstaz and where dΩ D−2 is the metric of a (D − 2)-dimensional spherical surface. Solving the field equations (3), in the limit D → 4, yields a solution [39] f ± (r) = 1 + Here, M and Q can be identified, respectively, as the mass and charge parameters of the black hole. In the limit of Q = 0, metric (5) with (6) corresponds to the static 4D EGB black hole [31]. Equation (6) corresponds to the two branches of solutions depending on the choice of "±", such that at large distances Eq. (6) reduces to In the vanishing limit of α only -ve branch smoothly recovers the Reissner-Nordstrom black hole [39]. Thus, we will limit our discussions to -ve branch only. The effect of GB coupling parameter faded at large distances, as the Charged black holes in 4D EGB gravity smoothly retrieve the Reissner-Nordstrom black hole. However, considerable departure can be expected in strong-field regime where usually full features of GB corrections come in to play. The Charged black holes in 4D EGB gravity are characterized uniquely by mass M , charge Q, and GB parameter α. To begin a discussion on the strong gravitational lensing, we adimensionlise the Charged black hole metric of EGB gravity (5) in terms of Schwarzschild radius 2M by defining x = r/2M , T = t/2M ,α = α/M 2 , and q = Q/2M . Then we have where (9) It is clear that metric (8) possess a coordinate singularity at which admits two real positive roots given by The two roots x ± correspond to the radii of black hole event horizon (x + ) and Cauchy horizon (x − ). It is clear from Eq. (11) that for the existence of black hole, the allowed values forα are given bỹ For a given value of q, there always exists an extremal value ofα =α e = 1 − q 2 , for which black hole possess degenerate horizons i.e., x − = x + = x e , such thatα <α e leads to two distinct horizons andα >α e leads to nohorizons (cf. Fig. 1). Similarly, one can find the extremal value of q e = √ 1 −α/2, for a given value ofα. The behavior of horizon radii with GB coupling parameterα and black hole charge q is shown in Fig. 2. Event horizon radius decreases whereas Cauchy horizon radius increases with increasing q orα, such that Charged black holes of 4D EGB gravity always possess smaller event horizon as compared to Schwarzschild and Reissner-Nordstrom black holes.

III. LIGHT DEFLECTION ANGLE
In this section, we investigate the strong gravitational lensing in the Charged black holes of 4D EGB gravity to compute the deflection angles, location of relativistic images, their magnifications and the effect ofα and q on them. We consider that light source S and observer O are sufficiently far from the black hole L, which acts as a lens, and they are nearly aligned. The light ray emanating from the source travel in straight path towards the black and only when it encounters the black hole gravitational field it suffers from the deflection (cf. Fig. 3). The amount of deflection suffered by light depends on the impact parameter u and distance of minimum approach x 0 , at which light suffer reflection and starts outward journey toward the observer [56]. Consider the propagation of light on the equatorial plane (θ = π/2), as due to spherical symmetry, the whole trajectory of the photon is limited on the same plane. The projection of photon four-momentum along the Killing vectors of isotmetries are conserved quantities, namely the energy E = −p µ ξ µ (t) and angular momentum L = p µ ξ µ (φ) are constant along the geodesics, where ξ µ (t) and ξ µ (φ) are, respectively, the Killing vectors due to timetranslational and rotational invariance [57].
Photons follow the null geodesics of metric (8), ds 2 = 0, which yields dx dτ where τ is the affine parameter along the geodesics. Photons traversing close to the black hole, experience radial turning pointsẋ = 0 and follows the unstable circular orbits, whose radii x m can be obtained from where prime corresponds to the derivative with respect to the x and the above admits at least one positive solution and then the largest real root is defined as the radius of the unstable circular photon orbits (cf. Fig. 2). A small radial perturbations drive these photons into the black hole or toward spatial infinity [57]. Due to spherical symmetry, these orbits generate a photon sphere around the black hole. The radii of photon orbits for Charged black holes of 4D EGB gravity decrease with increasing q andα (cf. Fig. 2). Further, at the distance of minimum approach, we have [56] Following the method developed by Bozza [13], the total deflection angle suffered by the light in its journey from source to observer is given by where The deflection angle increases as distance of minimum approach x 0 decreases and shows divergence as it approaches the photon sphere x m [13]. In the strong deflection limit, we can expand the deflection angle near the photon sphere, for the purpose we define a new variable z as [13] the integral (16) can be re-written as with the functions Making a Taylor series expansion of the function in Eq. (21), we get R(z, x 0 ) is regular for all values of z, however, for x 0 = x m , we have φ(x 0 ) = 0 and f 0 ≈ 1/z, which diverge as z → 0. Following the above definitions, the diverging part in the integral Eq. (19) can be identified as [13] whereas the regular part I R (x 0 ) is such that I D (x 0 ) has logarithmic divergence and I R (x 0 ) is regular with divergence subtracted from the complete integral (19). The deflection angle can be written in terms of x 0 as [13] The Eq. (27) is coordinate dependent, however, it can be written in terms of coordinate independent variable, impact parameter u, as follows whereā andb are called the strong deflection limit coefficients. In Fig. 4, we plotted the impact parameter u m for the photons moving on the unstable circular orbits around black hole as a functions of charge parameter q and GB coupling parameterα. It is clear that u m decrease with q andα. The behavior of lensing coefficients are shown in Fig. 5, which in the limits of α → 0 and q = 0, smoothly retain the values for the Schwarzschild black hole, viz.,ā = 1 andb = −0.4002 [12,13]. Coefficientā increases whereasb decreases with increasing q or α. The resulting deflection angle α D (u) is shown as a function of impact parameter u for various values of q andα in Fig. 6. For a fixed value of u, deflection angle decreases with increasing q orα, therefore, the deflection angle is higher for Schwarzschild and Reissner-Nordstrom black holes than those for the Charged black holes of 4D EGB gravity. Figure 6 infers that the α D (u) increases as impact parameter u approaches the u m and becomes unboundedly large for u = u m .

IV. STRONG LENSING OBSERVABLES
The deflection angle obtained in Eq. (30) is directly related to the positions and magnification of the relativistic images, which is given by lens equation [12] tan where θ and β, respectively, are the angular separations of image and source from the black hole as shown in Fig. (3). The D LS is the distance between the source and black hole and D OS is distance from the source to the observer; all distances are expressed in terms of the Schwarzschild radius x s = R s /2M . For nearly perfect alignment of source, black hole and observer, viz. small values of θ and β, Eq. (32) reduces to the following form In case of strong lensing, photons complete multiple circular orbits around black hole before escaping toward observer, therefore α D can be replaced by 2nπ + ∆α n in Eq. (32), with n ∈ N and 0 < ∆α n ≪ 1. The angular separation between the lens and nth relativistic images can be written as where where θ 0 n corresponds to the angular separation when photon winds completely 2nπ around the black hole and ∆θ n corresponds to the part exceeding 2nπ. Similarly, magnification of images is another good source of information, which is defined as the ratio between the solid angles subtended by the image and the source, and for small angles it is given by [13] Using Eqs. (33) and (35), the magnification (36) becomes: which can be simplified further by making Taylor series expansion, thus the magnification of nth image on both the sides is given by [13] Thus magnification decreases exponentially with winding number n and the higher-order images become fainter.
In order to relate the results obtained analytically with observations, Bozza defined the following observables [13] Since the outermost relativistic image is the brightest, the quantity s is the angular separation between the outermost image from the remaining bunch of relativistic images, r mag is the ratio of the received flux between the first image and all the others images clustered at θ ∞ . For a nearly perfect alignment of source, black hole and observer, these observables can be simplified to [13] Once these strong lensing observables are known from observations, one can estimate the strong lensing coefficientsā andb and compare with the theoretically calculated values.

V. LENSING BY GALACTIC SUPERMASSIVE BLACK HOLE
For numerical estimation, we consider a realistic case of a supermassive black hole Sgr A* at the center of our galaxy. Taking the distance to black hole D OL = 7.97 × 10 3 pc and mass of the black hole to be 3.98×10 6 M ⊙ [58], we have tabulated the observables for Sgr A* in Table (I). We compared the results with those for the Schwarzschild (α = 0, q = 0) and Reissner-Nordstrom black holes (α = 0). It is worth to notice that, for Charged black holes of 4D EGB gravity, the angular separation between images are higher whereas their magnification are lower than those for the Schwarzschild and Reissner-Nordstrom black holes.
For a given value of D OL , the limiting value of angular position is smaller than Schwarzschild case and decreases with q orα. On the other hand the separation between the images s, for the Charged black holes of 4D EGB gravity is larger than the Schwarzschild black hole and it goes on increasing with increasing q orα. So images are far away from the black hole and thereby less packed than in the Schwarzschild case. The observables are plotted againstα and q in Fig.(7).

VI. CONCLUSION
The regularized 4D EGB gravity proposed in [31][32][33] is characterized by the non-trivial contribution of the GB quadratic term to the gravitational dynamics in 4D spacetime. Thereby, this 4D EGB gravity with quadratic-curvatures bypasses the Lovelock's theorem and yields the diffeomorphism invariance and second order equations of motion.
The 4D EGB gravity possesses only the degrees of freedom of a massless graviton and thus free from the instabilities. Further, static and spherically symmetric black hole solutions of this 4D EGB gravity are also valid in other theories of gravity [34,35,37,51].
With this motivation, we have analysed the strong gravitational lensing of light due to static spherically symmetric Charged black holes to 4D EGB gravity which besides the mass M , depends also on two parameters q andα. We have examined the effects of q andα, in a strong-field observation, to the lensing observables due to Charged black holes to 4D EGB gravity and compared with those due to Schwarzschild and Reissner-Nordstrom black holes of GR. We have numerically calculated the strong lensing coefficientsā andb, and lensing observables θ ∞ , s, r mag , u m as functions ofα and q for relativistic images. In turn, we have applied our results to the supermassive black hole Sgr A* at the center of our galaxy. Interestingly, we find that, a increases when we increase q andα whereasb and deflection angle α D decrease, and observe the diverging behavior of deflection angle α D when u → u m . In addition, for a fixed value of impact parameter, Charged black holes to 4D EGB gravity cause smaller deflection angle as compared to their GR counterparts.
We have also estimated some properties of relativistic images, the variations of θ ∞ , s, and r mag , as function ofα and q which are depicted in the Figs. 7. We have shown that the angular position of outermost relativistic images θ ∞ and relative magnification of images r mag are   decreasing function of both q andα, but they decrease more sharply with q (cf. Fig. 7), while angular separation between images s increases with both q andα. To conclude, we find that Charged black holes to 4D EGB gravity cause higher angular separation between images but lower magnification than those for the Schwarzschild and Reissner-Nordstrom black holes.
The results presented here are the generalization of previous discussions, on the black holes in GR viz. Schwarzschild and Reissner-Nordstrom black holes and black holes to 4D EGB gravity, and they are encompassed, respectively in the limits, α, q → 0, α → 0, and q → 0.