# Gravitational Lensing of Charged Ayon-Beato-Garcia Black Holes and Nonlinear Effects of Maxwell Fields

Authors: H. Ghaffarnejad; M. Amirmojahedi; H. Niad
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

Nonsingular Ayon-Beato-Garcia (ABG) spherically symmetric static black hole (BH) with charge to mass ratioq=g/2m is metric solution of Born Infeld nonlinear Maxwell-Einstein theory. Central region of the BH behaves as (anti-)de Sitter for (|q|>1)|q|<1. In the case where |q|=1, the BH central region behaves as Minkowski flat metric. Nonlinear Electromagnetic (NEM) fields counterpart causes deviation of light geodesics and so light rays will be forced to move on from effective metric. In this paper we study weak and strong gravitational lensing of light rays by seeking effects of NEM fields counterpart on image locations and corresponding magnification. We set our calculations to experimentally observed Sgr A⁎ BH. In short we obtained the following: for large distances, the NEM counterpart is negligible and it reduces to linear Maxwell fields. The NEM field enlarges radius of the BH photon sphere linearly by raising |q|>1 but decreases by raising |q|≤1. Sign of deflection angle of bending light rays is changed in presence of NEM effects with respect to ones obtained in absence of NEM fields. Absolute value of deflection angle rises by increasing |q|→1. Image locations in weak deflection limit (WDL) decrease (increases) by raising 0<|q|<1 in presence (absence) of NEM fields. By raising the closest distance of the bending light rays image locations in WDL change from left (right) to right (left) in absence (presence) of NEM fields. In WDL, radius of Einstein rings and corresponding magnification centroid become larger (smaller) in presence (absence) of NEM fields. Angular separation called s between the innermost and outermost relativistic images increases (decreases) by increasing 0<|q|<1 in absence (presence) of NEM fields. Corresponding magnification r decreases (increases) by raising 0<|q|<1 in absence (presence) of NEM fields.


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## Body

## 1. Introduction

Since the advent of Einstein’s general relativity theory, black holes and the singularity problem of curved space-time become challenging subjects in modern (quantum) physics. Singularity is the intrinsic character of the most exact solutions of Einstein’s equations where Ricci and Kretschmann scalars reach to infinite value at singular point of the space-time [1]. Penrose cosmic censorship conjecture states that the causal singularities must be covered by the event horizon and so causes disconnecting of interior and exterior regions of the space-time [2, 3]. On the other side back-reaction corrections of quantum matter field cause shrinking of the quantum black hole horizon (see [4] and references therein) for which the Penrose cosmic censorship may become invalid for quantum regimes. At the moment one should infer that final state of evaporating quantum black holes is still open problem. This encourages one to seek nonsingular type of black hole solutions of the Einstein’s equation. However nonsingular metric solutions are also obtained from the Einstein field equation [5–26]. In the latter situations the Einstein field equation is coupled with suitable NEM fields for which the Ricci and the Kretschmann scalars become regular in whole space-time. A good classification of spherically symmetric static regular black holes is collected in [10]. Inspiring a physical central core idea, Bardeen (BAR) suggested the first spherically symmetric static regular black hole in 1968 containing a horizon without singularity [11]. After his work, other regular black holes were designed based on this model which we call here, for instance, ABG [12–15], Hayward (HAY) [16], and Neves-Saa (NS) [17, 18]. Nonsingular property of all of these solutions is controlled via dimensionless charge parameter q. HAY type of regular black hole is obtained by modifying the mass parameter of the BAR black hole. NS type of regular black hole is similar to a HAY type but its asymptotic behavior approaches to a vacuum de Sitter in presence of cosmological constant parameter. Regular black holes are studied also on brane words (see [18] and reference therein). The solutions of rotating regular black holes have been introduced in several articles [19–27]. A very important gravitational source is the Kerr-Newman-(anti) de Sitter (KNdS/KNAdS) black hole which is studied first by Stuchlík in [28] (see also [29, 30]). Kraniotis studied gravitational lensing of KNdS and KNAdS black hole in [31], where closed form analytic solutions of the null geodesics and the gravitational lens equations have been obtained versus the Appell-Lauricella generalized hypergeometric functions and the elliptic functions of Weierstrass. In these exact solutions all the fundamental parameters of the theory, namely, black hole mass, electric charge, rotation angular momentum, and the cosmological constant, enter on an equal footing while the electric charge effect on relativistic observable was also investigated. Rotating nonsingular black holes can be treat as natural particle accelerators [27]. Ultrahigh energy particle collisions are studied on the regular black holes [32] and backgrounds containing naked singularity [33]. Motion of test particles is studied in regular black hole space-time in [34]. Circular geodesics are obtained for BAR and ABG regular black holes in [35]. The optical effects related to Keplerian discs orbiting Kehagias-Sfetsos (KS) naked singularities were investigated in [36]. Authors of the latter work also mentioned the close similarity between circular geodesics in KS and properties of the circular geodesics of the RN naked singular space-time. Schee and Stuchlík studied also profiled spectral lines generated by Keplerian discs orbiting in the BAR and the ABG space-time in [37]. Correspondence between the black holes and the FRW geometries is studied for nonrelativistic gravity models in [38]. Reissner Nordström (RN) black hole gravitational lensing is studied in [39]. Gravitational lensing from regular black holes is studied in WDL (SDL) of light rays in [40–47]. Strong deflection limits of light rays can distinguish gravitational lensing between naked singularity and regular black holes background [47]. There is significant difference between optical phenomena characters of the singular space-times such as Schwarzschild (SCH), RN, and nonsingular space-times as HAY, BAR, and ABG [44]. It is related to the fact that central regions of the regular space-times reach to de Sitter (anti-de Sitter) for |q|<1(|q|>1) (see (7) and (9)). A detailed analysis of the null geodesic motion of the original regular space-time and comparison to the RN space-time is presented by Genzel et al. [48]. Furthermore we should point that the nonsingular charged black holes obtained from NEM models in curved space-time cause the photons to not move along null geodesics. As an applicable approach we should obtain corresponding effective metric for geodesics of moving photons [49–52] and then proceed to study their gravitational lensing. The black hole electric charge has also important effects on final state of the Hawking radiation and switching off effects of a quantum evaporating black hole (see, e.g., [53]). In this work we study gravitational lensing of light rays moving on the ABG nonsingular black hole in presence of NEM fields counterparts. The paper is organized as follows.Briefly, we introduce regular ABG black hole metric in Section2 and its asymptotically behavior for different values of q. In Section 3 we calculate effective metric of the ABG black hole for the moving photons by regarding the results of the original work [13]. We solve numerically the photon sphere equation of effective metric and obtain photon sphere radius against different charge values for q. In Section 4 we evaluate general formalism of deflection angle of bending light rays in weak and strong deflection limits. In weak deflection limits we apply the Ohanian lens equation [54] to determine nonrelativistic image locations versus source positions for observed Sgr A∗ black hole [48, 55–57]. Weak deflection angle of bending light rays and their magnifications are evaluated numerically point by point and they are plotted against source locations and also q. In the strong deflection limits we use Bozza’s formalism [43, 44] to obtain logarithmic form of the deflection angle. We calculate relative distance between innermost and outermost relativistic images and corresponding magnification and then plot their diagrams. Section 5 denotes the concluding remark.


## 2. ABG Space-Time

The ABG spherically symmetric black hole metric defined by Schwarzschild coordinates is given by [13](1)ds2=-Hrdt2+dr2Hr+r2dθ2+sin2⁡θdφ2with(2)Hr=1-2mr2r2+g23/2+g2r2r2+g22and associated electric field(3)Ftrr=Er=gr4r2-5g2r2+g24+152mr2+g27/2.m and g are total mass and electric charge parameters of the ABG BH, respectively. The line element (1) is nonsingular static solution of NEM-Einstein metric equation(4)Gμν=8πTμν=8πLFFμηFνη-Lgμν,LF=∂L∂Fwhich satisfies the action functional I=∫gdx4(R/16π-L(F)/4π) where R is Ricci scalar and L is a functional of F=(1/4)FμνFμν. This metric solution has only the coordinate singularity called horizon singularity without causal singularity called event horizon singularity because the Ricci Rμμ and the Kretschmann RμναλRμναλ scalars become regular at all points of the space-time 0≤r≤+∞. Defining mass and charge functions as(5)Mr=m1+g2r2-3/2,er=g1+g2r2-1one can show that the ABG metric (1) reduces apparently to a variable mass-charge RN type of BH as follows.(6)ds2=-1-2Mrr+e2rr2dt2+dr21-2Mr/r+e2r/r2+r2dθ2+sin2⁡θdφ2,where M(∞)=m and e2(∞)=g are ADM mass and electric charge viewed from observer located at infinity. Its central region 0<r<|g| behaves as vacuum de Sitter asymptotically:(7)ds2≈-1-Λ3r2dt2+dr21-Λ/3r2+r2dθ2+sin2⁡θdφ2for(8)q=g2m<1and anti-de Sitter(9)ds2≈-1+Λ3r2dt2+dr21+Λ/3r2+r2dθ2+sin2⁡θdφ2for(10)q=g2m>1,respectively, where we defined effective cosmological constant as(11)Λm,g=31-q4m2q3.In the particular case(12)q=g2m=1the effective cosmological parameter vanishes, Λ=0, and so, near the center r→0, the ABG black hole metric reduces to a flat Minkowski background asymptotically. Setting g=0, (5) read m=M, e=0 for which the metric solution (1) leads to singular charge-less Schwarzschild BH. Nonlinear counterpart of the Maxwell stress tensor causes deviation of the photon geodesics where the photons do not move along the null geodesics. Usually one uses an effective metric to study gravitational lensing of the light rays moving on such a charged black holes metric [47, 49–51, 58]. In the following section we seek effective metric of the ABG black hole for photon trajectories.


## 3. Effective Metric for Photon Trajectories

AssumingL(F)=F, (4) leads to the well known linear Einstein-Maxwell gravity where the photon propagates by the null equation (13)gμνkμkν=0,where kμ is corresponding four-momentum of the photon, but in general form where L(F)≠F the electric field given by (3) is self-interacting and so directly is reflected on the photon propagation. In the latter case the photons do not move along null geodesics of background geometry and so (13) becomes invalid. Instead, the photons will be propagate along null geodesics of an effective geometry which depends on the NEM source for which we will have [50, 51, 59]:(14)gμνeffkμkν=0,where(15)gμνeff=16LFFFμηFνη-LF+2FLFFgμνF2LFF2-16LF+FLFF2,(16)geffμν=LFFFημFνη+LFgμν.In absence of nonlinear counterpart of EM fields we can infer(17)LFF=0,LF=1,L=Ffor which gμνeff→gμν. Now we are in a position to obtain exact form for the effective metric of spherically symmetric static space-time (1). To do so we need to calculate all quantities defined by {L,LF,LFF} which satisfy the metric solution (1). We should first obtain corresponding Lagrangian density L(F). Authors of the original paper [13] used the following ansatz to solve (4) and to obtain (1).(18)FLF2=-122m2q2x4,where(19)Fx=-12m2q2x82x2-5q2x2+q24+1541x2+q27/22.One can obtain the above equation by substituting the following dimensionless parameters into the EM Lagrangian density F=(1/4)FμνFμν.(20)q=g2m,x=r2m.Equation (19) behaves asymptotically as F∞(x)≈-(1/2(2m)2)(q2/x4) for large distances x≫1. Comparing the latter result and (18) we can infer LF≈1 for large distances x≫1. One can integrate it to obtain linear Maxwell Lagrangian L→F. The latter result tells us NEM action functionals L(F) are negligible for regions of far from the black hole event horizon x≫xEH. Applying (18) and (19) we obtain parametric form of the Lagrangian density L(x) as follows.(21)Lx=-q22m2∫∞x1x2dx4x2-5q2x2+q24+1541x2+q27/2which has exact solution as(22)Lx=-q22m2x2x2-5q2x2+q24+154x2x2+q27/2+q22m212x2+q22-2q2x2+q23+321x2+q25/2.One can infer(23)LFx=L′xF′x,LFFx=1F′xL′F′′=L″F′-F″L′F′3,where prime ′ denotes differentiation with respect to x. If we need to obtain analytic form for L(F) we should substitute x from (19) into (22) but that form will be complicated. Hence we plot numerical diagram for {F,L} by substituting numerical values of Tables 1 and 2 into (19) and (22) and by computing their numerical values. Diagram for {F,L} given by Figure 1 shows that effects of the NEM fields are negligible for |q|>1 but not for |q|<1. However we will need to exact form of the functions {L(x),LF(x),LFF(x)} to study location of effective metric horizons, gravitational lensing images, and their magnifications. To do so we will use numerical method as follows. One can infer that effective metric of the background (1) defined by (15) reads(24)dseff2=-Ardt2+Brdr2+r2Crdθ2+sin2⁡θdφ2,where we defined(25)Ar=16HrLF16LF+FLFF2-F2LFF2,Br=1Hr16LF16LF+FLFF2-F2LFF2,Cr=82LF+4FLFF16LF+FLFF2-F2LFF2.Radius of the event horizon rH can be obtained by solving the equation A(r)=0 in the presence of the NEM field and by choosing its greatest positive root. According to study of black hole gravitational lensing, photon sphere construction is one of the important characters which must be considered here. It comes from energy condition [60] and is a particular hypersurface (r=constant) which does not evolve with time. In other words any null geodesic initially tangent to the photon sphere hypersurface will remain tangent to it. It is made from circulating photons turning around the black hole center. Radius of the photon sphere rps is the greatest positive solution of [48](26)1r2ArCrr=rpseff′=0.Setting (17), (25) reads(27)Ar=Hr,Br=1Hr,Cr=1which describe the original background space-time (1) without NEM field effects for which (26) reduces to the following form.(28)Hrr2r=rps′=0.Diagrams of (26) and (28) are plotted for larger roots in Figure 1. Linear branch of the right panel of the diagram in Figure 1 predicts large scale photon spheres for |q|>1 which are formed only in presence of NEM field. This linear branch of the effective photon sphere diagram can be approximated with the following equation.(29)xpseffq>1≈3.643q-0.796which rises by increasing |q|→∞. We calculated numerical values of the above photon sphere radius for 1<|q|<36 and collected them in Table 2. Corresponding diagram is given in Figure 1. One can deduct from Figure 1 that we have small scale photon sphere for |q|<1 from both of the effective metric (24) and the original background (1). Hence our obtained gravitational lensing results from (1) can be compared with ones obtained from (24) only for |q|<1. Thus we collect numerical solutions of the both photon sphere equations (26) and (28) for |q|<1 in Table 1. We will need them to evaluate numerical values of deflection angle, image locations, and corresponding magnifications. We will study gravitational lensing of the system separately for two different regimes |q|>1 and |q|<1, respectively, as follows. At first step we proceed to evaluate numerical values of the deflection angle of bending light rays.Table 1
Numerical major real roots of the photon sphere equations (26) and (28) for 0<|q|<1.
| q |,xps, xpseff | q |, xps, xpseff | q |, xps, xpseff | q |, xps, xpseff 0.00, 3.000, 3.000 0.25, 2.892, 2.874 0.50, 2.547, 2.645 0.75, -, 2.326 0.01, 2.999, 2.950 0.26, 2.887, 2.871 0.51, 2.528, 2.640 0.76, -, 2.320 0.02, 2.994, 2.947 0.27, 2.882, 2.865 0.52, 2.508, 2.630 0.77, -, 2.326 0.03, 2.989, 2.947 0.28, 2.872, 2.855 0.53, 2.475, 2.621 0.78, -, 2.329 0.04, 2.989, 2.947 0.29, 2.858, 2.852 0.54, 2.455, 2.609 0.79, -, 2.335 0.05, 2.989, 2.944 0.30, 2.848, 2.843 0.55, 2.431, 2.594 0.80, -, 2.344 0.06, 2.989, 2.943 0.31, 2.838, 2.831 0.56, 2.397, 2.578 0.81, -, 2.350 0.07, 2.984, 2.942 0.32, 2.828, 2.828 0.57, 2.378, 2.560 0.82, -, 2.362 0.08, 2.980, 2.941 0.33, 2.824, 2.819 0.58, 2.334, 2.530 0.83, -, 2.377 0.09, 2.980, 2.938 0.34, 2.809, 2.813 0.59, 2.300, 2.514 0.84, -, 2.393 0.10, 2.975, 2.935 0.35, 2.795, 2.804 0.60, 2.271, 2.508 0.85, -, 2.408 0.11, 2.974, 2.935 0.36, 2.785, 2.795 0.61, 2.227, 2.478 0.86, -, 2.429 0.12, 2.974, 2.935 0.37, 2.770, 2.785 0.62, 2.193, 2.469 0.87, -, 2.451 0.13, 2.964, 2.932 0.38, 2.756, 2.776 0.63, 2.150, 2.460 0.88, -, 2.475 0.14, 2.959, 2.928 0.39, 2.736, 2.767 0.64, 2.106, 2.441 0.89, -, 2.487 0.15, 2.955, 2.925 0.40, 2.722, 2.758 0.65, 2.058, 2.429 0.90, -, 2.514 0.16, 2.950, 2.919 0.41, 2.707, 2.749 0.66, 1.990, 2.411 0.91, -, 2.542 0.17, 2.945, 2.916 0.42, 2.698, 2.740 0.67, 1.902, 2.399 0.92, -, 2.566 0.18, 2.940, 2.913 0.43, 2.678, 2.731 0.68, 1.819, 2.384 0.93, -, 2.591 0.19, 2.935, 2.910 0.44, 2.659, 2.718 0.69, 1.645, 2.374 0.94, -, 2.612 0.20, 2.930, 2.904 0.45, 2.654, 2.706 0.70, -, 2.362 0.95, -, 2.636 0.21, 2.921, 2.989 0.46, 2.630, 2.694 0.71, -, 2.350 0.96, -, 2.670 0.22, 2.916, 2.895 0.47, 2.601, 2.685 0.72, -, 2.344 0.97, -, 2.703 0.23, 2.906, 2.889 0.48, 2.591, 2.670 0.73, -, 2.338 0.98, -, 2.734 0.24, 2.901, 2.883 0.49, 2.557, 2.658 0.74, -, 2.335 0.99, -, 2.764Table 2
Solutions of the effective photon sphere equation (29) for |q|>1.
| q |, xpseff | q |, xpseff | q |, xpseff | q |, xpseff | q |, xpseff | q |, xpseff 1, 2.847 7, 24.705 13, 46.563 19, 68.421 25, 90.279 31, 112.137 2, 6.490 8, 28.348 14, 50.206 20, 72.064 26, 93.922 32, 115.780 3, 10.133 9, 31.991 15, 53.849 21, 75.707 27, 97.565 33, 119.423 4, 13.776 10, 35.634 16, 57.492 22, 79.350 28, 101.208 34, 123.066 5, 17.419 11, 39.277 17, 61.135 23, 82.993 29, 104.851 35, 126.709 6, 21.062 12, 42.920 18, 64.778 24, 86.636 30, 108.494 36, 130.352Figure 1
Diagrams of photon sphere locationsxps and NEM field Lagrangian density L[F].


## 4. Deflection Angle

If the light beam passes through the ABG black hole without rotation, then weak lensing occurs. In the latter case closest approach distance of the bending light rays from the black hole centerr0 become larger than the photon sphere radius and two nonrelativistic images are usually formed. They are called primary and secondary images. In general, bending angle of light rays is obtained by solving the null geodesics equation defined by (13) as follows [61].(30)αeffr0=Ieffr0-π,where(31)Ieffx0=2∫x0>xps∞AxBx/C2xAx0/x02Cx0-Ax/x2Cxdxx2.Inserting(32)z=x0xthe integral equations (31) become(33)Ieffx0=2∫01Γx0/zΩx0-Ωx0/zz2dz,where we defined(34)Γx0z=ΩBA=LFLF+2FLFF,Ωx0z=AC=HΓ.According to method given in [59], we calculate Taylor series expansion of the functions Γ(x0/z) and Ω(x0)-Ω(x0/z)z2 versus powers of (1-z) as follows.(35)Γx0z=Γ0+Γ11-z+Γ21-z2+O3,Ωx0-Ωx0zz2=Ω11-z+Ω21-z2+O3,where we defined(36)Γ0=Γx0,Γ1=x0Γ′x0,Γ2=x0Γ′x0+x02Γ″x02,(37)Ω1=2Ωx0-x0Ω′x0,Ω2=x0Ω′x0-Ωx0-x02Ω″x02in which prime ′ denotes differentiation with respect to its argument x. Inserting (35) and neglecting their higher order terms, the integral equation (33) becomes(38)Ieffx0≈2∫01dzΓ0+Γ11-z+Γ21-z2Ω11-z+Ω21-z2where its solution reads(39)Ieffx0=1Ω2x01+Ω1x0Ω2x02Γ1x0+Γ2x0-32Γ2x0Ω1x0Ω2x0-1Ω2x02Γ0x0-Γ1x0Ω1x0Ω2x0+34Γ2x0Ω1x0Ω2x02×ln⁡1+2frac⁡1-1+Ω1x0Ω2x0Ω1x0Ω2x0.WDL and SDL of bending light rays are regimes where Ieff(x0)↛∞ and Ieff(x0)→∞, respectively. This restricts us to choose particular regimes of the ratio Ω1/Ω2 given by (39). Substituting (34) and (37) into the photon sphere equation (26) and setting x0=xpseff one can get xpsΩ′(xps)-2Ω(xps)=Ω1(xps)=0. The latter condition is valid for moving light rays near the photon sphere for which Ieff→∞. In other words one infers Ω1(x0≠xpseff)≠0 for WDL. Hence we can use asymptotic expansion of the solution (39) for x0>xps and ∀q as follows.


### 4.1. Weak Lensing Deflection Angles

One can obtain asymptotic expansion series of the functionsΩ1,2(x0) and Γ0,1,2(x0) which up to terms in order of O(x0-3) become, respectively,(40)Ω1x0≈2-3/8x0-32q2+345/8x02+4305q2/16+25875/128x03,Ω2x0≈-1+3/8x0+1035/16+48q2x02-4305q2/8+25875/64x03,Γ0x0≈1+15/8x0-9q2+225/32x02+495q2/16+3375/128x03,Γ1x0≈-15/8x0+18q2+225/16x02-1485q2/16+10125/128x03,Γ2x0≈-9q2+225/32x02+1485q2/16+10125/128x03.Substituting (40) into the integral solution (39) and using some simple calculations, one can infer(41)Ieffweakx0>xpseff≈π-33/8+471/128+9πq2/2x0+741πq2/32+64863π/2048-8841q/1024-997q2/16x03.Defining(42)y=x0xpseff>1and inserting (41) the deflection angle (30) becomes(43)αeffweaky0>1≈-My0+Ny03for WDL where we defined(44)Mxpseff,q=1xpseff338+471128+9πq22,Nxpseff,q=1xpseff3741πq232+64863π2048-8841q1024-997q216.Applying (17) and (34) we obtain(45)Ωx0=H,Ω1x0=2H-x0H′,Ω2x0=x0H′-H-x02H″2,Γx=Γ0x=1,Γ1x=0=Γ2xwhich are applicable for WDL in absence of the NEM field effects. In the latter case asymptotic behavior of the functions given by (45) for x0>xps reads(46)Ω1x0≈2-6x0+4q2x02+15q2x03,Ω2x0≈-1+6x0-6q2x02-30q2x03.Substituting (45) and (46) asymptotic series expansion of the solution (39) is obtained as follows.(47)Iweakx0≈π+3π-2x0+8q2-36-6πq2+27π/2x02+123q2-198-42q2π+135π/2x03where by substituting it into (30) one can obtain weak deflection angle (WDA) of bending light rays in absence of the NEM field such that(48)αweaky0>1≈Sy0+Ry02+Qy03,where we defined(49)Sxps,q=3π-2xps,Rxps,q=8q2-36-6πq2+27π/2xps2,Qxps,q=123q2-198-42q2π+135π/2xps3.Diagrams of (43) and (48) are plotted in Figure 2 for ansatz y0=10 by inserting numerical values given in Table 1. It is suitable to obtain the following averaged deflection angles.(50)σeffweak=α¯effweak≈-M¯y0+N¯y03,(51)σweak=α¯weak≈S¯y0+R¯y02+Q¯y03,where we defined all mean coefficients(52)C¯=1N∑i=1NCxpsi,qiin which C={M,N,S,R,Q}. Substituting numerical values of Table 1 we obtain(53)M¯=4.79,N¯=5.41,S¯=1.32,R¯=0.64,Q¯=0.80for which (50) and (51) become, respectively,(54)σeffweaky0>1≈-4.79y0+5.41y03,(55)σweaky0>1≈1.32y0+0.64y02+0.80y03.Diagrams of the above equations are given in Figure 2. They show that the sign of deflection angle is changed in presence of the NEM fields with respect to its sign without effects of the NEM field.Figure 2
Diagrams of weak lensing deflection angles for|q|<1.


### 4.2. Strong Lensing Deflection Angles

In case of SDL we write Taylor series expansion of the solution (39) at neighborhood of x0→xpseff or equivalently y0→1. In the latter case we must be obtain Taylor series expansion of the functions Ω1,2(x0) and Γ0,1,2(x0) which up to terms in order of O(3) become, respectively,(56)Ω1x0≈Ppsy0-1+Qpsy0-12,Ω2x0≈Ω2xps+Rpsy0-1-Qpsy0-12,Γ0x0≈Γ0xps+Upsy0-1+Vpsy0-12,Γ1x0≈Ups+Ups+2Vpsy0-1+2Vps+Wpsy0-12,Γ2x0≈Ups+Vps+Ups+4Vps+Wpsy0-1+3Vps+Wpsy0-12,where we defined(57)Pps=2Ω2xps,Ω2xps=Ωxps-xps2Ω″xps2,Rps=-xps2Ω″xps,Qxps=-xps3Ω′′′xps2,Ups=xpsΓ0′xps,Vps=xps2Γ0″xps2,Wps=xps3Γ0′′′xps2.Inserting (56) into the solution (39) one can obtain SDL of bending angle (30) as follows.(58)αeffstrongy0→1≈b+aln⁡y0-1,where we defined(59)b=-π+3Ups+2ln⁡2VpsΓ0xpsΩ2xps,a=-2Γ0xpsΩ2xps.Divergency of (58) for y0=1 can be described by Bozza formalism as follows.(60)αeffstrong=αn=Δαn+2nπ,where n=0,±1,±2,±3,…, means nth circulation of light rays around center of the lens to make nth relativistic image with deflection angle αn where differences between these angles should satisfy the condition 0<Δαn<1. Nonrelativistic images are determined by setting n=0 and relativistic images with positive (negative) parity are determined by setting n=1,2,…(n=-1,-2,…) where one can obtain Δα-n=2α-Δαn. In case of retrolensing where observer is located between source and lens, the light rays come back after turning around the lens (see Figure 1 at [46, 47]). In the latter case the parameter 2n given in the formula (60) must be replaced with 2n-1. In case of SDL in absence of NEM fields we should use (58) but by inserting(61)Pps=2Hxps-xps2H″xps,Ups=-2xpsH′H3,Vps=-xps2H″H3+3xps2H′2H4,Rps=-xps2H″,Qps=-xps3H′′′2.Now we study image locations for WDL and SDL of gravitational lensing.


## 4.1. Weak Lensing Deflection Angles

One can obtain asymptotic expansion series of the functionsΩ1,2(x0) and Γ0,1,2(x0) which up to terms in order of O(x0-3) become, respectively,(40)Ω1x0≈2-3/8x0-32q2+345/8x02+4305q2/16+25875/128x03,Ω2x0≈-1+3/8x0+1035/16+48q2x02-4305q2/8+25875/64x03,Γ0x0≈1+15/8x0-9q2+225/32x02+495q2/16+3375/128x03,Γ1x0≈-15/8x0+18q2+225/16x02-1485q2/16+10125/128x03,Γ2x0≈-9q2+225/32x02+1485q2/16+10125/128x03.Substituting (40) into the integral solution (39) and using some simple calculations, one can infer(41)Ieffweakx0>xpseff≈π-33/8+471/128+9πq2/2x0+741πq2/32+64863π/2048-8841q/1024-997q2/16x03.Defining(42)y=x0xpseff>1and inserting (41) the deflection angle (30) becomes(43)αeffweaky0>1≈-My0+Ny03for WDL where we defined(44)Mxpseff,q=1xpseff338+471128+9πq22,Nxpseff,q=1xpseff3741πq232+64863π2048-8841q1024-997q216.Applying (17) and (34) we obtain(45)Ωx0=H,Ω1x0=2H-x0H′,Ω2x0=x0H′-H-x02H″2,Γx=Γ0x=1,Γ1x=0=Γ2xwhich are applicable for WDL in absence of the NEM field effects. In the latter case asymptotic behavior of the functions given by (45) for x0>xps reads(46)Ω1x0≈2-6x0+4q2x02+15q2x03,Ω2x0≈-1+6x0-6q2x02-30q2x03.Substituting (45) and (46) asymptotic series expansion of the solution (39) is obtained as follows.(47)Iweakx0≈π+3π-2x0+8q2-36-6πq2+27π/2x02+123q2-198-42q2π+135π/2x03where by substituting it into (30) one can obtain weak deflection angle (WDA) of bending light rays in absence of the NEM field such that(48)αweaky0>1≈Sy0+Ry02+Qy03,where we defined(49)Sxps,q=3π-2xps,Rxps,q=8q2-36-6πq2+27π/2xps2,Qxps,q=123q2-198-42q2π+135π/2xps3.Diagrams of (43) and (48) are plotted in Figure 2 for ansatz y0=10 by inserting numerical values given in Table 1. It is suitable to obtain the following averaged deflection angles.(50)σeffweak=α¯effweak≈-M¯y0+N¯y03,(51)σweak=α¯weak≈S¯y0+R¯y02+Q¯y03,where we defined all mean coefficients(52)C¯=1N∑i=1NCxpsi,qiin which C={M,N,S,R,Q}. Substituting numerical values of Table 1 we obtain(53)M¯=4.79,N¯=5.41,S¯=1.32,R¯=0.64,Q¯=0.80for which (50) and (51) become, respectively,(54)σeffweaky0>1≈-4.79y0+5.41y03,(55)σweaky0>1≈1.32y0+0.64y02+0.80y03.Diagrams of the above equations are given in Figure 2. They show that the sign of deflection angle is changed in presence of the NEM fields with respect to its sign without effects of the NEM field.Figure 2
Diagrams of weak lensing deflection angles for|q|<1.


## 4.2. Strong Lensing Deflection Angles

In case of SDL we write Taylor series expansion of the solution (39) at neighborhood of x0→xpseff or equivalently y0→1. In the latter case we must be obtain Taylor series expansion of the functions Ω1,2(x0) and Γ0,1,2(x0) which up to terms in order of O(3) become, respectively,(56)Ω1x0≈Ppsy0-1+Qpsy0-12,Ω2x0≈Ω2xps+Rpsy0-1-Qpsy0-12,Γ0x0≈Γ0xps+Upsy0-1+Vpsy0-12,Γ1x0≈Ups+Ups+2Vpsy0-1+2Vps+Wpsy0-12,Γ2x0≈Ups+Vps+Ups+4Vps+Wpsy0-1+3Vps+Wpsy0-12,where we defined(57)Pps=2Ω2xps,Ω2xps=Ωxps-xps2Ω″xps2,Rps=-xps2Ω″xps,Qxps=-xps3Ω′′′xps2,Ups=xpsΓ0′xps,Vps=xps2Γ0″xps2,Wps=xps3Γ0′′′xps2.Inserting (56) into the solution (39) one can obtain SDL of bending angle (30) as follows.(58)αeffstrongy0→1≈b+aln⁡y0-1,where we defined(59)b=-π+3Ups+2ln⁡2VpsΓ0xpsΩ2xps,a=-2Γ0xpsΩ2xps.Divergency of (58) for y0=1 can be described by Bozza formalism as follows.(60)αeffstrong=αn=Δαn+2nπ,where n=0,±1,±2,±3,…, means nth circulation of light rays around center of the lens to make nth relativistic image with deflection angle αn where differences between these angles should satisfy the condition 0<Δαn<1. Nonrelativistic images are determined by setting n=0 and relativistic images with positive (negative) parity are determined by setting n=1,2,…(n=-1,-2,…) where one can obtain Δα-n=2α-Δαn. In case of retrolensing where observer is located between source and lens, the light rays come back after turning around the lens (see Figure 1 at [46, 47]). In the latter case the parameter 2n given in the formula (60) must be replaced with 2n-1. In case of SDL in absence of NEM fields we should use (58) but by inserting(61)Pps=2Hxps-xps2H″xps,Ups=-2xpsH′H3,Vps=-xps2H″H3+3xps2H′2H4,Rps=-xps2H″,Qps=-xps3H′′′2.Now we study image locations for WDL and SDL of gravitational lensing.


## 5. Images Locations

In order to calculate location of the images in WDL, we choose Ohanian lens equation [54] which has high accuracy and so lower errors with respect to other lens equations [62]. It has the advantage of being the closest relative of the exact lens equation, since it contains just the asymptotic approximation without any additional assumption. It can be rewritten versus observational coordinates, namely, the image position θ, the source position β, and the deflection angle of bending light rays αweak as follows (see [62] for more discussions).(62)arcsin⁡DLsin⁡θ-arcsin⁡DSsin⁡β=α-θin which we defined(63)DL=dOLdLS,DS=dOSdLS.In the above equations dOS is distance between the observer and the source, dOL is distance between the observer and the lens, and dLS is distance between the lens and the source. θ is formed when a line passing through the observer and the image crosses the optical axis (line passing through the observer and the lens). β is formed when a line passing through the observer and the source crosses the optical axis. One can obtain general solutions of the lens equation (62) as follows.(64)θKα,β=arctan⁡DScos⁡αsin⁡β+Ksin⁡α1-DS2sin2⁡βDL-DSsin⁡αsin⁡β+cos⁡α1-DS2sin2⁡β,where K=±1. It has some real solutions for(65)sin⁡β≤1DS.In the following we use (64) to obtain locations of nonrelativistic and relativistic image.


### 5.1. Weak Lensing Images

For WDL with large distances between the lens, the source, and the observer located in a straight line, one can infer [62](66)DS≈DL+1,where DL≫1 must be substituted via experimental date. As a realistic example of gravitational lens we consider a big black hole located in the center of galaxy and study image locations of a star located far from it. This black hole is called Sgr A∗ [48, 55–57]. Its mass is estimated as 3.6×106M⨀ and its distance from the earth is dOL=8kpc=2.47×1017m with corresponding Schwarzschild radius RSCH=1010m. We consider a source to be a star located at distance dLS=1.7×1013m from the black hole which is far from the margin of the accretion disk of the black hole, so it may not be falling toward the black hole center. For the latter black hole we will have(67)DL≈1.45×104≈DS.The relations (65) and (67) lead us to choose(68)sin⁡β≈β≤βM,for WDL of gravitational lensing where we defined(69)βM=1DS≈1DL≈7×10-5Rad≡40107μarc secin which the subscript M denotes the word “Maximum.” For critical source β=βM the lens equation (64) reads(70)θMα=arctan⁡cos⁡αDL-sin⁡αwhich for WDL α→0 leads to the following approximation.(71)θM≈1DL≈40107μarc sec.Defining normalized angular locations for the images and the source as(72)θ∗=θθM,β∗=ββMwe can obtain Taylor series expansion of the lens equation (64) at neighborhood of (|α|,|β∗|)≪(1,1) as follows.(73)θK∗≈KDL1+DLα+DL1+DLβ∗-K6DL2DL-11+DL3α3-DL2DL2-2KDL+2+DL-2K1+DL3β∗α2-DL2DL2K+DLK-2DL+2K-21+DL3αβ∗2+DL61+3DL1+DL3β∗3+KDL120DL4+11DL2-11DL3-DL1+DL5α5+⋯which in limits DL→∞ become(74)θK∗≈Kα+β∗-α2β∗2-Kαβ∗22-K6α3+K120α5+⋯.This is primary image location and by transforming β∗→-β∗ as θK∗(-β∗) one can obtain secondary image location for which K=+1(-1) describes right (left) handed bending angles of light rays. Setting K=+1, β∗=1, and y0=10 and computing numerical values of the deflection angles (43) and (48) via numerical values of Table 1 (see Figure 2) we plot numerical diagrams for θieff and θi against different values |qi| in Figure 3. Also we substitute (54) into the weak lens equation (74) by setting K=+1 and plot mean weak image locations in Figure 3 versus |q| in Figure 3. Now we are in a position to obtain location of relativistic image in the subsequent section.Figure 3
Diagrams of weak lensing primary image locations for|q|<1.


### 5.2. Strong Lensing Images

The Virbhadra-Ellis lens equation given by [63](75)tan⁡β=tan⁡θ-Dtan⁡α-θ+tan⁡θis useful to study gravitational lensing in SDL. In the above lens equation we have D=dls/dos for standard lensing in which the lens is located between the observer and the source, D=dos/dol for situations where the source is located between the observer and the lens, and D=dos/dls for situations where the observer is located between the source and the lens (the retrolensing). The lensing effects are more important when the objects are highly aligned, in which β, θ are small and α is close to 2nπ+Δαn for the standard lensing and (2n+1)π+Δαn for the retrolensing with 0<Δαn≪1, n=1,2,…. In the latter case we can use the approximations tan⁡θ≈θ and tan⁡β≈β for the lens equation (75) and substitute αn=2nπ+Δαn such that (see (32) in [59])(76)β=θ-DΔαn.Defining coordinate independent impact parameter(77)u=rΩr,one can obtain its Taylor series expansion by applying (56)-(57) as follows (see (28) in [59])(78)y0-1≈2Ω2xpsΩxps-1/2u0ups-11/2in which u0=dLOsin⁡θ for small θ reduces to u0≈dLOθ. Inserting the latter relation and (78), the strong deflection angle (58) becomes(79)αθ=c2-c1ln⁡dOLupsθ-1,where we defined(80)c1=-a2,c2=b+a2ln⁡2ΩxpsΩ2xps.One can obtain θα by inverting (79) as(81)θα=upsdOL1+ec2-α/c1and by substituting (60) the above equation can be rewritten as(82)θn=upsdOL1+ec2-2nπ-Δαn/c1.By making first order Taylor series expansion of the above equation around α=2nπ the angular position of nth relativistic image is obtained as(83)θn≈θn0-ζnΔαn,where we defined(84)θn0=upsdOL1+ec2-2nπ/c1,(85)ζn=upsc1dOLec2-2nπ/c1.Eliminating Δαn between (76) and (83) we obtain(86)θn=1+ζnD-1θn0+ζnDβin which 0<ζn/D≪1 and so we can use approximation (1+ζn/D)-1≈1-ζn/D. In the latter case the equation (86) reads(87)θn≈θn0+ζnDβ-θn0.The second term in the above lens equation is more smaller than the first term which means all relativistic image locations lie very close to θn0. There are other sets of relativistic images by changing θn(0)→-θn(0) into the above lens equation. In case of perfect aligned β=0 the above lens equation reaches to(88)θnE=1-ζnDθn0which describes angular location of nth relativistic Einstein ring.


#### 5.2.1. Magnifications

The magnificationμ of an image is defined as the ratio of flux of the image to flux of unlensed source. It has two components called tangential μt=sin⁡θ/sin⁡β and radial μr=dθ/dβ and their multiplication makes the magnification(89)μ=sin⁡βsin⁡θdβdθ-1.The above equation denotes primary image θp(β) magnifications with positive parity. Inserting the secondary image location θs(β)=θp(-β) into the magnification equation (89) one can obtain secondary image magnification μweaks(β)=μweakp(-β) with negative parity. In the microlensing state two images in WDL are not resolved and so the main observables that should be considered are the total magnification μtot and magnification-weighted-centroid μcent defined by, respectively, [64](90)μtot=μs+μp,μcent=θpμp+θsμsμp+μsand we calculate them for WDL and SDL in the next section.


### 5.3. Weak Lensing Magnifications

In case of WDL we can use approximationssin⁡θ≈θ and sin⁡β≈β to evaluate (89) as follows.(91)μweak≈βθdβdθ-1which by substituting (74) reads(92)μKweak≈Kα+β∗-α2β∗2-Kαβ∗22-Kα36+Kα5120×1β∗-α22β∗-Kα.Inserting numerical values given in Table 2 we calculate numerical values for μ, μtot, and μcent and plot their diagrams versus β∗ in Figure 4. Diagrams show that μ decreases by increasing |q| in presence and absence of NEM fields. μtot has minimum value for β>0(β<0) in presence (absence) of NEM fields, while corresponding μcent take s on a maximum value. Furthermore we obtain from the Figure 4 that magnification of the Einstein rings is major in presence of NEM fields with respect to situations where the NEM field effect is negligible. Now we proceeded to study magnifications of images for SDL.Figure 4
Diagrams of weak lensing image magnifications for|q|<1.


#### 5.3.1. Strong Lensing Magnifications

To calculatenth relativistic images magnification we should determine(93)μn≈βθndβdθn-1which by substituting (87) reads(94)μn≈ζnDθn0β.The above magnification is valid for a point source and for extended source there is obtained a different form of the magnification (see for instance (46) in [59]). Equation (94) shows that the first relativistic image is brightest one, and the magnifications decreases exponentially by raising “n”. Magnifications for retrolensing are obtained by (94) but by changing 2n→2n-1. Taking into account both sets of relativistic images, the total magnification is defined by μtotstrong=2∑n=1∞μn. Using the formula t/(1-t)=∑n=1∞tn and inserting (84), (85) and (94) we obtain(95)μtotstrong=1β2ups2ec2/c1Dc1dOL21+ec2/c1+e2π/c1e4π/c1-1for a point source where(96)upsxps=xpsΩxps,(97)c1xps=Γ0xpsΩ2xps,c2xps=-π+3Ups+Vps+2ln⁡2Γ0xpsΩ2xps-Γ0xpsΩ2xpsln⁡2ΩxpsΩ2xps.Equations (97) are obtained by substituting (59) into the relations (80).As an example we obtain the lensing observable defined by Bozza [43] as(98)s=θ1-θ∞,(99)r=μ1∑n=2∞μnwhich are useful when the outermost relativistic image can be resolved from the rest. “s” represents the angular separation between the first image and the limiting value of the images for n→∞. “r” is the ratio between the flux of the first image and sum of the fluxes of the other images. Applying (85) and (87) one can show exact form of the equation (98) which without the negligible term 0<ζ1/D≪1 becomes(100)s≈θ∞ec2-2π/c1in which we defined(101)θ∞=upsd~OLand numerical value of d~LO is used here for Sgr A∗ observed galactic black hole as follows.(102)d~OL=dOLRSCH=2.47×1017m1010m=2.47×107.One can infer that (99) can be rewritten as(103)r=2μ1μtotstrong-2μ1which by inserting (94) and (95) reads(104)r=e2π/c1+ec2/c1-e-2π/c1+ec2/c2e-4π/c11+e-2π/c1-ec2/c1e-4π/c1and its Taylor series expansion about e-2π/c1→0 becomes(105)r≈e2π/c1+ec2/c1-1.We plot diagrams of c1,2 against |q| in Figure 5 for |q|<1 and Figure 7 for |q|>1. They show increase of c1,2 to large negative values in absence of NEM fields. While in presence of NEM fields c1 takes on some positive values for |q|<0.6 and reaches to negative values for 0.6<|q|<1. c2 decreases by increasing |q|<1 in presence of NEM fields. For |q|>1 the diagrams of Figure 7 show increase of c1(c2) to some positive (negative) values by decreasing |q|>1 in absence (presence) of NEM fields. r(s) behaves as increasing (decreasing) function by raising |q|>1. In Figure 6 we see increase (decrease) for s by raising |q|<1 in absence (presence) of NEM fields. r decreases (increase) by raising |q|<1 in absence (presence) of the NEM fields.Figure 5
Diagrams forc1, c2 are plotted versus |q|<1.Figure 6
Diagrams fors, r are plotted against |q|<1.Figure 7
Diagrams forc1, c2, s, r are plotted versus |q|>1.


## 5.1. Weak Lensing Images

For WDL with large distances between the lens, the source, and the observer located in a straight line, one can infer [62](66)DS≈DL+1,where DL≫1 must be substituted via experimental date. As a realistic example of gravitational lens we consider a big black hole located in the center of galaxy and study image locations of a star located far from it. This black hole is called Sgr A∗ [48, 55–57]. Its mass is estimated as 3.6×106M⨀ and its distance from the earth is dOL=8kpc=2.47×1017m with corresponding Schwarzschild radius RSCH=1010m. We consider a source to be a star located at distance dLS=1.7×1013m from the black hole which is far from the margin of the accretion disk of the black hole, so it may not be falling toward the black hole center. For the latter black hole we will have(67)DL≈1.45×104≈DS.The relations (65) and (67) lead us to choose(68)sin⁡β≈β≤βM,for WDL of gravitational lensing where we defined(69)βM=1DS≈1DL≈7×10-5Rad≡40107μarc secin which the subscript M denotes the word “Maximum.” For critical source β=βM the lens equation (64) reads(70)θMα=arctan⁡cos⁡αDL-sin⁡αwhich for WDL α→0 leads to the following approximation.(71)θM≈1DL≈40107μarc sec.Defining normalized angular locations for the images and the source as(72)θ∗=θθM,β∗=ββMwe can obtain Taylor series expansion of the lens equation (64) at neighborhood of (|α|,|β∗|)≪(1,1) as follows.(73)θK∗≈KDL1+DLα+DL1+DLβ∗-K6DL2DL-11+DL3α3-DL2DL2-2KDL+2+DL-2K1+DL3β∗α2-DL2DL2K+DLK-2DL+2K-21+DL3αβ∗2+DL61+3DL1+DL3β∗3+KDL120DL4+11DL2-11DL3-DL1+DL5α5+⋯which in limits DL→∞ become(74)θK∗≈Kα+β∗-α2β∗2-Kαβ∗22-K6α3+K120α5+⋯.This is primary image location and by transforming β∗→-β∗ as θK∗(-β∗) one can obtain secondary image location for which K=+1(-1) describes right (left) handed bending angles of light rays. Setting K=+1, β∗=1, and y0=10 and computing numerical values of the deflection angles (43) and (48) via numerical values of Table 1 (see Figure 2) we plot numerical diagrams for θieff and θi against different values |qi| in Figure 3. Also we substitute (54) into the weak lens equation (74) by setting K=+1 and plot mean weak image locations in Figure 3 versus |q| in Figure 3. Now we are in a position to obtain location of relativistic image in the subsequent section.Figure 3
Diagrams of weak lensing primary image locations for|q|<1.


## 5.2. Strong Lensing Images

The Virbhadra-Ellis lens equation given by [63](75)tan⁡β=tan⁡θ-Dtan⁡α-θ+tan⁡θis useful to study gravitational lensing in SDL. In the above lens equation we have D=dls/dos for standard lensing in which the lens is located between the observer and the source, D=dos/dol for situations where the source is located between the observer and the lens, and D=dos/dls for situations where the observer is located between the source and the lens (the retrolensing). The lensing effects are more important when the objects are highly aligned, in which β, θ are small and α is close to 2nπ+Δαn for the standard lensing and (2n+1)π+Δαn for the retrolensing with 0<Δαn≪1, n=1,2,…. In the latter case we can use the approximations tan⁡θ≈θ and tan⁡β≈β for the lens equation (75) and substitute αn=2nπ+Δαn such that (see (32) in [59])(76)β=θ-DΔαn.Defining coordinate independent impact parameter(77)u=rΩr,one can obtain its Taylor series expansion by applying (56)-(57) as follows (see (28) in [59])(78)y0-1≈2Ω2xpsΩxps-1/2u0ups-11/2in which u0=dLOsin⁡θ for small θ reduces to u0≈dLOθ. Inserting the latter relation and (78), the strong deflection angle (58) becomes(79)αθ=c2-c1ln⁡dOLupsθ-1,where we defined(80)c1=-a2,c2=b+a2ln⁡2ΩxpsΩ2xps.One can obtain θα by inverting (79) as(81)θα=upsdOL1+ec2-α/c1and by substituting (60) the above equation can be rewritten as(82)θn=upsdOL1+ec2-2nπ-Δαn/c1.By making first order Taylor series expansion of the above equation around α=2nπ the angular position of nth relativistic image is obtained as(83)θn≈θn0-ζnΔαn,where we defined(84)θn0=upsdOL1+ec2-2nπ/c1,(85)ζn=upsc1dOLec2-2nπ/c1.Eliminating Δαn between (76) and (83) we obtain(86)θn=1+ζnD-1θn0+ζnDβin which 0<ζn/D≪1 and so we can use approximation (1+ζn/D)-1≈1-ζn/D. In the latter case the equation (86) reads(87)θn≈θn0+ζnDβ-θn0.The second term in the above lens equation is more smaller than the first term which means all relativistic image locations lie very close to θn0. There are other sets of relativistic images by changing θn(0)→-θn(0) into the above lens equation. In case of perfect aligned β=0 the above lens equation reaches to(88)θnE=1-ζnDθn0which describes angular location of nth relativistic Einstein ring.


### 5.2.1. Magnifications

The magnificationμ of an image is defined as the ratio of flux of the image to flux of unlensed source. It has two components called tangential μt=sin⁡θ/sin⁡β and radial μr=dθ/dβ and their multiplication makes the magnification(89)μ=sin⁡βsin⁡θdβdθ-1.The above equation denotes primary image θp(β) magnifications with positive parity. Inserting the secondary image location θs(β)=θp(-β) into the magnification equation (89) one can obtain secondary image magnification μweaks(β)=μweakp(-β) with negative parity. In the microlensing state two images in WDL are not resolved and so the main observables that should be considered are the total magnification μtot and magnification-weighted-centroid μcent defined by, respectively, [64](90)μtot=μs+μp,μcent=θpμp+θsμsμp+μsand we calculate them for WDL and SDL in the next section.


## 5.2.1. Magnifications

The magnificationμ of an image is defined as the ratio of flux of the image to flux of unlensed source. It has two components called tangential μt=sin⁡θ/sin⁡β and radial μr=dθ/dβ and their multiplication makes the magnification(89)μ=sin⁡βsin⁡θdβdθ-1.The above equation denotes primary image θp(β) magnifications with positive parity. Inserting the secondary image location θs(β)=θp(-β) into the magnification equation (89) one can obtain secondary image magnification μweaks(β)=μweakp(-β) with negative parity. In the microlensing state two images in WDL are not resolved and so the main observables that should be considered are the total magnification μtot and magnification-weighted-centroid μcent defined by, respectively, [64](90)μtot=μs+μp,μcent=θpμp+θsμsμp+μsand we calculate them for WDL and SDL in the next section.


## 5.3. Weak Lensing Magnifications

In case of WDL we can use approximationssin⁡θ≈θ and sin⁡β≈β to evaluate (89) as follows.(91)μweak≈βθdβdθ-1which by substituting (74) reads(92)μKweak≈Kα+β∗-α2β∗2-Kαβ∗22-Kα36+Kα5120×1β∗-α22β∗-Kα.Inserting numerical values given in Table 2 we calculate numerical values for μ, μtot, and μcent and plot their diagrams versus β∗ in Figure 4. Diagrams show that μ decreases by increasing |q| in presence and absence of NEM fields. μtot has minimum value for β>0(β<0) in presence (absence) of NEM fields, while corresponding μcent take s on a maximum value. Furthermore we obtain from the Figure 4 that magnification of the Einstein rings is major in presence of NEM fields with respect to situations where the NEM field effect is negligible. Now we proceeded to study magnifications of images for SDL.Figure 4
Diagrams of weak lensing image magnifications for|q|<1.


### 5.3.1. Strong Lensing Magnifications

To calculatenth relativistic images magnification we should determine(93)μn≈βθndβdθn-1which by substituting (87) reads(94)μn≈ζnDθn0β.The above magnification is valid for a point source and for extended source there is obtained a different form of the magnification (see for instance (46) in [59]). Equation (94) shows that the first relativistic image is brightest one, and the magnifications decreases exponentially by raising “n”. Magnifications for retrolensing are obtained by (94) but by changing 2n→2n-1. Taking into account both sets of relativistic images, the total magnification is defined by μtotstrong=2∑n=1∞μn. Using the formula t/(1-t)=∑n=1∞tn and inserting (84), (85) and (94) we obtain(95)μtotstrong=1β2ups2ec2/c1Dc1dOL21+ec2/c1+e2π/c1e4π/c1-1for a point source where(96)upsxps=xpsΩxps,(97)c1xps=Γ0xpsΩ2xps,c2xps=-π+3Ups+Vps+2ln⁡2Γ0xpsΩ2xps-Γ0xpsΩ2xpsln⁡2ΩxpsΩ2xps.Equations (97) are obtained by substituting (59) into the relations (80).As an example we obtain the lensing observable defined by Bozza [43] as(98)s=θ1-θ∞,(99)r=μ1∑n=2∞μnwhich are useful when the outermost relativistic image can be resolved from the rest. “s” represents the angular separation between the first image and the limiting value of the images for n→∞. “r” is the ratio between the flux of the first image and sum of the fluxes of the other images. Applying (85) and (87) one can show exact form of the equation (98) which without the negligible term 0<ζ1/D≪1 becomes(100)s≈θ∞ec2-2π/c1in which we defined(101)θ∞=upsd~OLand numerical value of d~LO is used here for Sgr A∗ observed galactic black hole as follows.(102)d~OL=dOLRSCH=2.47×1017m1010m=2.47×107.One can infer that (99) can be rewritten as(103)r=2μ1μtotstrong-2μ1which by inserting (94) and (95) reads(104)r=e2π/c1+ec2/c1-e-2π/c1+ec2/c2e-4π/c11+e-2π/c1-ec2/c1e-4π/c1and its Taylor series expansion about e-2π/c1→0 becomes(105)r≈e2π/c1+ec2/c1-1.We plot diagrams of c1,2 against |q| in Figure 5 for |q|<1 and Figure 7 for |q|>1. They show increase of c1,2 to large negative values in absence of NEM fields. While in presence of NEM fields c1 takes on some positive values for |q|<0.6 and reaches to negative values for 0.6<|q|<1. c2 decreases by increasing |q|<1 in presence of NEM fields. For |q|>1 the diagrams of Figure 7 show increase of c1(c2) to some positive (negative) values by decreasing |q|>1 in absence (presence) of NEM fields. r(s) behaves as increasing (decreasing) function by raising |q|>1. In Figure 6 we see increase (decrease) for s by raising |q|<1 in absence (presence) of NEM fields. r decreases (increase) by raising |q|<1 in absence (presence) of the NEM fields.Figure 5
Diagrams forc1, c2 are plotted versus |q|<1.Figure 6
Diagrams fors, r are plotted against |q|<1.Figure 7
Diagrams forc1, c2, s, r are plotted versus |q|>1.


## 5.3.1. Strong Lensing Magnifications

To calculatenth relativistic images magnification we should determine(93)μn≈βθndβdθn-1which by substituting (87) reads(94)μn≈ζnDθn0β.The above magnification is valid for a point source and for extended source there is obtained a different form of the magnification (see for instance (46) in [59]). Equation (94) shows that the first relativistic image is brightest one, and the magnifications decreases exponentially by raising “n”. Magnifications for retrolensing are obtained by (94) but by changing 2n→2n-1. Taking into account both sets of relativistic images, the total magnification is defined by μtotstrong=2∑n=1∞μn. Using the formula t/(1-t)=∑n=1∞tn and inserting (84), (85) and (94) we obtain(95)μtotstrong=1β2ups2ec2/c1Dc1dOL21+ec2/c1+e2π/c1e4π/c1-1for a point source where(96)upsxps=xpsΩxps,(97)c1xps=Γ0xpsΩ2xps,c2xps=-π+3Ups+Vps+2ln⁡2Γ0xpsΩ2xps-Γ0xpsΩ2xpsln⁡2ΩxpsΩ2xps.Equations (97) are obtained by substituting (59) into the relations (80).As an example we obtain the lensing observable defined by Bozza [43] as(98)s=θ1-θ∞,(99)r=μ1∑n=2∞μnwhich are useful when the outermost relativistic image can be resolved from the rest. “s” represents the angular separation between the first image and the limiting value of the images for n→∞. “r” is the ratio between the flux of the first image and sum of the fluxes of the other images. Applying (85) and (87) one can show exact form of the equation (98) which without the negligible term 0<ζ1/D≪1 becomes(100)s≈θ∞ec2-2π/c1in which we defined(101)θ∞=upsd~OLand numerical value of d~LO is used here for Sgr A∗ observed galactic black hole as follows.(102)d~OL=dOLRSCH=2.47×1017m1010m=2.47×107.One can infer that (99) can be rewritten as(103)r=2μ1μtotstrong-2μ1which by inserting (94) and (95) reads(104)r=e2π/c1+ec2/c1-e-2π/c1+ec2/c2e-4π/c11+e-2π/c1-ec2/c1e-4π/c1and its Taylor series expansion about e-2π/c1→0 becomes(105)r≈e2π/c1+ec2/c1-1.We plot diagrams of c1,2 against |q| in Figure 5 for |q|<1 and Figure 7 for |q|>1. They show increase of c1,2 to large negative values in absence of NEM fields. While in presence of NEM fields c1 takes on some positive values for |q|<0.6 and reaches to negative values for 0.6<|q|<1. c2 decreases by increasing |q|<1 in presence of NEM fields. For |q|>1 the diagrams of Figure 7 show increase of c1(c2) to some positive (negative) values by decreasing |q|>1 in absence (presence) of NEM fields. r(s) behaves as increasing (decreasing) function by raising |q|>1. In Figure 6 we see increase (decrease) for s by raising |q|<1 in absence (presence) of NEM fields. r decreases (increase) by raising |q|<1 in absence (presence) of the NEM fields.Figure 5
Diagrams forc1, c2 are plotted versus |q|<1.Figure 6
Diagrams fors, r are plotted against |q|<1.Figure 7
Diagrams forc1, c2, s, r are plotted versus |q|>1.


## 6. Concluding Remark

As a black hole solution of Born Infeld Einstein nonlinear Maxwell gravity we use ABG nonsingular charged black hole to study its gravitational lensing in WDL and SDL. We set our calculations to SgrA∗ observed black hole date and study NEM field effects on the gravitational lensing. In short we obtained negligibility of NEM effects for large values of the black hole charge but not for its small values. Nonlinearity causes the photon sphere radius to be larger. Sign of deflection angle of bending light rays is changed just with NEM field effects with respect to state where the NEM field is absent. Einstein rings become larger and their magnifications become greater. Angular separation between the innermost and the outermost relativistic images decreases by increasing the charge parameter while their magnifications increase. As a subsequent work one can use strategy and results of this work to compare with results obtained by studying gravitational lensing of other nonsingular galactic black holes described in the introduction section and setting other observed black holes data.



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*Source: 3067272-2018-03-12.xml*