Advances in High Energy Physics

Volume 2018, Article ID 3067272, 18 pages

https://doi.org/10.1155/2018/3067272

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

Faculty of Physics, Semnan University, Semnan 35131-19111, Iran

Correspondence should be addressed to H. Ghaffarnejad; ri.ca.nanmes@dajenrafahgh

Received 6 July 2017; Accepted 22 January 2018; Published 12 March 2018

Academic Editor: Andrea Coccaro

Copyright © 2018 H. Ghaffarnejad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### 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.