#### Abstract

Nonsingular Ayon-Beato-Garcia (ABG) spherically symmetric static black hole (BH) with charge to mass ratio is metric solution of Born Infeld nonlinear Maxwell-Einstein theory. Central region of the BH behaves as (anti-)de Sitter for In the case where , 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 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 but decreases by raising 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 Image locations in weak deflection limit (WDL) decrease (increases) by raising 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 between the innermost and outermost relativistic images increases (decreases) by increasing in absence (presence) of NEM fields. Corresponding magnification decreases (increases) by raising in absence (presence) of NEM fields.

#### 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 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 (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 Section 2 and its asymptotically behavior for different values of . 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 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 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 . 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]withand associated electric field and are total mass and electric charge parameters of the ABG BH, respectively. The line element (1) is nonsingular static solution of NEM-Einstein metric equationwhich satisfies the action functional where is Ricci scalar and is a functional of This metric solution has only the coordinate singularity called horizon singularity without causal singularity called event horizon singularity because the Ricci and the Kretschmann scalars become regular at all points of the space-time . Defining mass and charge functions asone can show that the ABG metric (1) reduces apparently to a variable mass-charge RN type of BH as follows.where and are ADM mass and electric charge viewed from observer located at infinity. Its central region behaves as vacuum de Sitter asymptotically:forand anti-de Sitterforrespectively, where we defined effective cosmological constant asIn the particular casethe effective cosmological parameter vanishes, , and so, near the center , the ABG black hole metric reduces to a flat Minkowski background asymptotically. Setting , (5) read , 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

Assuming , (4) leads to the well known linear Einstein-Maxwell gravity where the photon propagates by the null equation where is corresponding four-momentum of the photon, but in general form where 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]:whereIn absence of nonlinear counterpart of EM fields we can inferfor which 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 which satisfy the metric solution (1). We should first obtain corresponding Lagrangian density Authors of the original paper [13] used the following ansatz to solve (4) and to obtain (1).whereOne can obtain the above equation by substituting the following dimensionless parameters into the EM Lagrangian density Equation (19) behaves asymptotically as for large distances Comparing the latter result and (18) we can infer for large distances . One can integrate it to obtain linear Maxwell Lagrangian The latter result tells us NEM action functionals are negligible for regions of far from the black hole event horizon Applying (18) and (19) we obtain parametric form of the Lagrangian density as follows.which has exact solution asOne can inferwhere prime ′ denotes differentiation with respect to If we need to obtain analytic form for we should substitute from (19) into (22) but that form will be complicated. Hence we plot numerical diagram for by substituting numerical values of Tables 1 and 2 into (19) and (22) and by computing their numerical values. Diagram for given by Figure 1 shows that effects of the NEM fields are negligible for but not for However we will need to exact form of the functions 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) readswhere we definedRadius of the event horizon can be obtained by solving the equation 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 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 is the greatest positive solution of [48]Setting (17), (25) readswhich describe the original background space-time (1) without NEM field effects for which (26) reduces to the following form.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 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.which rises by increasing We calculated numerical values of the above photon sphere radius for 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 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 Thus we collect numerical solutions of the both photon sphere equations (26) and (28) for 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 and , respectively, as follows. At first step we proceed to evaluate numerical values of the deflection angle of bending light rays.

#### 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 center 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].whereInsertingthe integral equations (31) becomewhere we definedAccording to method given in [59], we calculate Taylor series expansion of the functions and versus powers of as follows.where we definedin which prime ′ denotes differentiation with respect to its argument Inserting (35) and neglecting their higher order terms, the integral equation (33) becomeswhere its solution readsWDL and SDL of bending light rays are regimes where and , respectively. This restricts us to choose particular regimes of the ratio given by (39). Substituting (34) and (37) into the photon sphere equation (26) and setting one can get . The latter condition is valid for moving light rays near the photon sphere for which In other words one infers for WDL. Hence we can use asymptotic expansion of the solution (39) for and as follows.

##### 4.1. Weak Lensing Deflection Angles

One can obtain asymptotic expansion series of the functions and which up to terms in order of become, respectively,Substituting (40) into the integral solution (39) and using some simple calculations, one can inferDefiningand inserting (41) the deflection angle (30) becomesfor WDL where we definedApplying (17) and (34) we obtainwhich are applicable for WDL in absence of the NEM field effects. In the latter case asymptotic behavior of the functions given by (45) for readsSubstituting (45) and (46) asymptotic series expansion of the solution (39) is obtained as follows.where by substituting it into (30) one can obtain weak deflection angle (WDA) of bending light rays in absence of the NEM field such thatwhere we definedDiagrams of (43) and (48) are plotted in Figure 2 for ansatz by inserting numerical values given in Table 1. It is suitable to obtain the following averaged deflection angles.where we defined all mean coefficientsin which Substituting numerical values of Table 1 we obtainfor which (50) and (51) become, respectively,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.

##### 4.2. Strong Lensing Deflection Angles

In case of SDL we write Taylor series expansion of the solution (39) at neighborhood of or equivalently In the latter case we must be obtain Taylor series expansion of the functions and which up to terms in order of become, respectively,where we definedInserting (56) into the solution (39) one can obtain SDL of bending angle (30) as follows.where we definedDivergency of (58) for can be described by Bozza formalism as follows.where , means th circulation of light rays around center of the lens to make th relativistic image with deflection angle where differences between these angles should satisfy the condition Nonrelativistic images are determined by setting and relativistic images with positive (negative) parity are determined by setting where one can obtain . 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 given in the formula (60) must be replaced with In case of SDL in absence of NEM fields we should use (58) but by insertingNow 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 , the source position , and the deflection angle of bending light rays as follows (see [62] for more discussions).in which we definedIn the above equations is distance between the observer and the source, is distance between the observer and the lens, and is distance between the lens and the source. is formed when a line passing through the observer and the image crosses the optical axis (line passing through the observer and the lens). 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.where It has some real solutions forIn 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]where 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 [48, 55–57]. Its mass is estimated as and its distance from the earth is with corresponding Schwarzschild radius We consider a source to be a star located at distance 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 haveThe relations (65) and (67) lead us to choosefor WDL of gravitational lensing where we definedin which the subscript denotes the word “Maximum.” For critical source the lens equation (64) readswhich for WDL leads to the following approximation.Defining normalized angular locations for the images and the source aswe can obtain Taylor series expansion of the lens equation (64) at neighborhood of as follows.which in limits becomeThis is primary image location and by transforming as one can obtain secondary image location for which describes right (left) handed bending angles of light rays. Setting , , and 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 and against different values in Figure 3. Also we substitute (54) into the weak lens equation (74) by setting and plot mean weak image locations in Figure 3 versus in Figure 3. Now we are in a position to obtain location of relativistic image in the subsequent section.

##### 5.2. Strong Lensing Images

The Virbhadra-Ellis lens equation given by [63]is useful to study gravitational lensing in SDL. In the above lens equation we have for standard lensing in which the lens is located between the observer and the source, for situations where the source is located between the observer and the lens, and 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 , are small and is close to for the standard lensing and for the retrolensing with , . In the latter case we can use the approximations and for the lens equation (75) and substitute such that (see in [59])Defining coordinate independent impact parameterone can obtain its Taylor series expansion by applying (56)-(57) as follows (see in [59])in which for small reduces to Inserting the latter relation and (78), the strong deflection angle (58) becomeswhere we definedOne can obtain by inverting (79) asand by substituting (60) the above equation can be rewritten asBy making first order Taylor series expansion of the above equation around the angular position of th relativistic image is obtained aswhere we definedEliminating between (76) and (83) we obtainin which and so we can use approximation In the latter case the equation (86) readsThe second term in the above lens equation is more smaller than the first term which means all relativistic image locations lie very close to There are other sets of relativistic images by changing into the above lens equation. In case of perfect aligned the above lens equation reaches towhich describes angular location of th relativistic Einstein ring.

###### 5.2.1. Magnifications

The magnification of an image is defined as the ratio of flux of the image to flux of unlensed source. It has two components called tangential and radial and their multiplication makes the magnificationThe above equation denotes primary image magnifications with positive parity. Inserting the secondary image location into the magnification equation (89) one can obtain secondary image magnification 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 and magnification-weighted-centroid defined by, respectively, [64]and we calculate them for WDL and SDL in the next section.

##### 5.3. Weak Lensing Magnifications

In case of WDL we can use approximations and to evaluate (89) as follows.which by substituting (74) readsInserting numerical values given in Table 2 we calculate numerical values for , , and and plot their diagrams versus in Figure 4. Diagrams show that decreases by increasing in presence and absence of NEM fields. has minimum value for in presence (absence) of NEM fields, while corresponding take 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.

###### 5.3.1. Strong Lensing Magnifications

To calculate th relativistic images magnification we should determinewhich by substituting (87) readsThe above magnification is valid for a point source and for extended source there is obtained a different form of the magnification (see for instance in [59]). Equation (94) shows that the first relativistic image is brightest one, and the magnifications decreases exponentially by raising “”. Magnifications for retrolensing are obtained by (94) but by changing Taking into account both sets of relativistic images, the total magnification is defined by Using the formula and inserting (84), (85) and (94) we obtainfor a point source whereEquations (97) are obtained by substituting (59) into the relations (80).

As an example we obtain the lensing observable defined by Bozza [43] aswhich are useful when the outermost relativistic image can be resolved from the rest. “” represents the angular separation between the first image and the limiting value of the images for . “” 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 becomesin which we definedand numerical value of is used here for Sgr observed galactic black hole as follows.One can infer that (99) can be rewritten as