(ABG) spherically symmetric sta
tic 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 case of the BH central region behaves as Minkowski flat metric. Nonlinear Electromagnetic (NEM) fields counterpart causes to deviate light geodesics and so light rays will forced to move on effective metric. In this paper we study weak and strong gravitational lensing of light rays by seeking affects of NEM fields counterpart on image locations and corresponding magnification. We set our calculations to experimentally observed Sgr A BH. In short we obtained: For large distances the NEM counterpart is negligible reaching to linear Maxwell fields. The NEM makes enlarge the BH photon sphere radius as linearly by raising but deceases 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 raises by increasing Weak image locations decreases (increases) by raising in presence (absence) of NEM fields. By raising the closest distance of the bending light rays weak image locations changes from left (right) to right (left) in absence (presence) of NEM fields. Einstein rings radius and corresponding magnification centroid become larger (smaller) in presence (absence) of NEM fields in case of weak lensing. Angular separation 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. raises (decreases) by increasing
Gravitational lensing of charged Ayon-Beato-Garcia black holes and non-linear effects of Maxwell fields
H. Ghaffarnejad111E-mail address: email@example.com , M. A. Mojahedi222E-mail address: firstname.lastname@example.org and H. Niad 333E-mail address: email@example.com Faculty of Physics, Semnan University, Zip Code 35131-19111, Iran
Since the advent of Einstein’s general relativity theory, black
holes and the singularity problem of curved space times become
challenging subjects in modern physics because of presence of
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 . Penrose cosmic censorship conjecture states
that the causal singularities must be covered by the event horizon
and so causes to disconnect interior and exterior regions of the
space time [2,3]. However nonsingular metric solutions are also
obtained from the Einstein field equation (see for instance
[4-23]). In the latter situations the Einstein field equation is
coupled to 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
are collected in ref. . Inspiring a physical central core idea,
Bardeen suggested the first spherically symmetric static regular
black hole in 1968 containing a horizon without singularity .
After his work, other regular black holes were designed based on
this model which we call here for instance ABG [11-14], Hayward
(HAY)  and Neves-Saa (NS) [16,17]. Non-singular property of
all of these solutions are 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 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  and reference therein). The solutions of rotating
regular black holes have been introduced in several articles
[18-24]. A very important source of strong gravity is the
Kerr-Newman-de Sitter (KNDS) and/or Kerr-Newman- anti-de Sitter
(KNADS) black hole. Kraniotis studied gravitational lensing of
KNDS and KNADS black hole in ref. , where closed form analytic
solutions of the null geodesics and the gravitational lens
equations have been obtained in terms of 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 . Ultra-high
energy particle collisions are studied on the regular black holes
 and backgrounds containing naked singularity . Motion of
test particles is studied in regular black hole spacetimes in
ref. . Circular geodesics are obtained for BAR and ABG regular
black-holes in ref. . The optical effects related to Keplerian
discs orbiting Kehagias-Sfetsos (KS) naked singularities was
investigated in ref. . Authors of the latter work are also
mentioned to be close similarity between circular geodesics in KS
and properties of the circular geodesics of the RN naked singular
space times. Schee et al studied also profiled spectral lines
generated by keplerian discs orbiting in the Bardeen and ABG space
times in ref. . Correspondence between the black holes and the
FRW geometries are studied for non-relativistic gravity models in
ref. . RN black hole gravitational lensing is studied in ref.
. Gravitational lensing from regular black holes is studied in
weak deflection limits of light rays [34-36] and in strong
deflection limits of light rays [37-41]. Strong deflection limits
of light rays can be distinguish gravitational lensing between
naked singularity and regular black holes background .
There is significant difference between optical
phenomena characters of the singular space-times such as SCH, RSN,
and non-singular space-times as HAY, BAR, ABG . It is related
to the fact that the regular space-times reach to a de Sitter
and/or anti-de Sitter like approximately at center (see
Eqs. (2.7) and (2.9)). Furthermore we should point that the
nonsingular charged black holes obtained from NEM models in
curved space times cause that the photons do not move along null
geodesics. As an applicable approach we must be obtain
corresponding effective metric for geodesics of moving photons
[42-45] and so 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 for instance ). 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
Briefly, we introduce in section 2 regular ABG black hole metric and its asymptotically behavior against 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 . We solve numerically the photon sphere equation of effective metric and obtain photon sphere radius against different charge values 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  to determine non-relativistic image locations against source positions for observed Sgr black hole [48-51]. 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 [37,38] to obtain logarithmic form of the deflection angle. We obtain relative distance between innermost and outermost relativistic images and corresponding magnification and then plot their diagrams. Section 5 denotes to concluding remark.
2 ABG space time
The ABG spherically symmetric black hole metric defined by Schwarzschild coordinates is 
and associated electric field
and are total mass and electric charge parameters of the BH respectively. The line element (2.1) is non-singular static solution of NEM-Einstein metric equation
which satisfies the action functional where is Ricci scalar and is a functional of This metric solution has only the coordinate singularity called as horizon singularity because the Ricci and the Kretschmann scalars become regular at all points of the space time . Defining mass and charge functions as
one can show that the ABG metric (2.1) reduces apparently to a variable mass-charge RN type of BH as
where and are ADM mass and electric charge viewed from observer located at infinity. Its central region behaves as vacuum de Sitter asymptotically:
and anti de Sitter
respectively where we defined effective cosmological constant as
In particular case
the effective cosmological parameter vanishes and so near the center , the ABG black hole metric reduces to a flat Minkowski background asymptotically. Setting the equations (2.5) read for which the metric solution (2.1) leads to singular charge-less Schwarzschild BH. Nonlinear counterpart of the Maxwell stress tensor causes to deviate the photon geodesics where the photons do not move along the null geodesics. Usually one use an effective metric to study gravitational lensing of the light rays moving on such a charged black holes metric [41-44]. In the following section we seek effective metric of the ABG black hole for photon trajectories.
3 Effective metric for photon trajectories
Assuming the equation (2.4) 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 (2.3), is self-interacting and so directly is reflected on the photon propagation. In the latter case the photons do not move along null geodesics (3.1) but instead, photons propagate along null geodesics of an effective geometry which depends on used nonlinear theories [43,44,52] as
In absence of nonlinear counterpart of EM fields we must be set
for which the effective metric reaches We are now in position to obtain effective metric of spherically symmetric static space time (2.1). To do so we must be obtain all quantities defined by which satisfy the metric solution (2.1). We should first obtain corresponding Lagrangian density We use result of the original paper  where its authors are used following ansatz to solve (2.4) and obtain (2.1).
comes from (2.3) by inserting dimensionless electric charge and radial coordinate
into One can obtain asymptotically behavior of the equation (3.7) for large distances as Comparing the latter result and (3.6) we infer which by integrating leads to linear Maxwell Lagrangian The latter result tells us NEM action functionals are negligible for regions of far from the black hole event horizon Applying (3.6) and (3.7) we obtain parametric form of the Lagrangian density as follows.
which has exact solution as
where over-prime denotes to differentiation with respect to If we need to obtain exact form of the functional we must be remove between (3.7) and (3.10) but it will take more complex form. Hence we plot numerical diagram of by inserting numerical values of tables 1 and 2 in figure 1. The diagram shows that negligibility of NEM fields 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. For metric solution (2.1) one can show that the effective metric (3.3) become
where we defined
The radius of the event horizon is given by the greatest positive root of the equation in absence (presence) of nonlinear counterpart of EM field. According to study of black hole gravitational lensing, photon sphere construction is one of important characters which must be considered here. It comes from energy condition  and is a particular hyper-surface () which does not evolve with time. In other words any null geodesic initially tangent to the photon sphere hyper-surface will remain tangent to it. It is made from circulating photons turn turning around the black hole center. Radius of the photon sphere is the greatest positive solution of the equation 
Setting (3.5) the equations (3.14), (3.15) and (3.16) read
describing original space time (2.1) in absence of the nonlinear EM fields effects for which (3.17) become
Diagrams of the equations (3.17) and (3.19) are plotted for larger solutions in figure 1. Linear branch of the right panel of 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 raises by increasing We calculated numerical values of the above photon sphere radius for and collected in the table 2. Corresponding diagram is given in figure 1. One can result from the figure 1 that we have small scale photon sphere for from both of the effective metric (3.13) and the original one (2.1). Hence obtained gravitational lensing results from (2.1) can be compared with ones which obtained from (3.13) only for Thus we collect numerical solutions of the both photon sphere equations (3.17) and (3.19) 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 regimes and as follows. We first apply to evaluate numerical values of the deflection angle of bending light rays.
4 Deflection angle
When light ray moves at neighborhood of the ABG black hole and deflects without turning around the black hole center then gravitational lensing takes ‘weak deflection limits‘ approach. 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 non-relativistic images are usually formed. They are called as primary and secondary images. In general, bending angle of light rays is obtained by solving null geodesics equation defined by (3.1) as follows .
the integral equations (4.2) become
where we defined
According to method given in ref. , we now expand and in powers of as follows.
where we defined
in which over-prime denotes to differentiation with respect to its argument Inserting (4.6) and (4.7) and neglecting their higher order terms, the integral equation (4.4) become
which has solution as follows.
Weak (strong) deflection limits of bending light rays are regimes where This restrict us to choose particular regimes of the ratio given by (4.11). Inserting (4.5) and (4.9) into the photon sphere equation (3.17) and setting one can result . The latter condition is valid for moving light rays near the photon sphere for which In other words one infers for weak deflection limits and so we can use asymptotic expansion form of the integral solution (4.11) for and as follows.
4.1 Weak lensing deflection angles
One can obtain asymptotic expansion series form of the functions and which up to terms in order of become respectively
Inserting (4.12), (4.13), (4.14), (4.15) and (4.16) into the integral solution (4.11) and using some simple calculations, one infers
and inserting (4.17) the deflection angle (4.1) become
for weak gravitational lensing where we defined
Applying (3.5) and (4.5) we obtain
which are applicable for weak deflection angle in absence of NEM field effects. In the latter case asymptotic behavior of the function (4.22) are obtained for as follows.
Inserting (4.22), (4.23) and (4.24) one can obtain asymptotic series expansion of the integral solution (4.11) as follows.
which by inserting into (4.1) one can obtain weak deflection angle of bending light rays in absence of NEM field such that
where we defined
Diagrams of the equations (4.19) and (4.26) are plotted in figure 2 for ansatz by inserting numerical values given in the table 1. It is suitable to obtain the following averaged deflection angles.
where we defined all mean coefficients
in which Inserting numerical values of the table 1 we obtain
for which (4.29) and (4.30) become respectively
Diagrams of the above mean weak deflection angles are given in the figure 2. They show that the sign of deflection angle is changed in presence of NEM fields with respect to sign of deflection angle in absence of it.
4.2 Strong lensing deflection angles
In case of strong deflection limits we write Taylor series expansion of the integral solution (4.11) at neighborhood of 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 defined
Inserting (4.35), (4.36), (4.37), (4.38) and (4.39) into the integral solution (4.11) one can obtain strong deflection limits of bending light ray angle (4.1) as follows.
where we defined
Divergency of the above equation in limits can be described by Bozza formalism as follows.
where means circulation of light rays around the lens center to make relativistic images by deflecting Non-relativistic images are determined by setting and relativistic images with positive (negative) parity are determined by setting where one can obtain . In case of retro-lensing where observer is located between source and lens, the light rays come back after than that turning around the lens (see figure 1 at ref. [40,41]). In the latter case the parameter given in the formula (4.44) must be replaced with In case of strong deflection limits in absence of NEM fields we should use (4.41) but by inserting
We now study image locations in weak and strong deflection limits.
5 Images locations
In order to calculate the weak deflection images we choose Ohanian lens equation  which has high accuracy and so lower errors with respect to other lens equations . It has the advantage of being the closest relative of the exact lens equation, since it only contains the asymptotic approximation without any additional assumptions. It can be rewritten against observational coordinates as image position , source position and deflection angle of bending light rays as follows (see  for more discussions).
in which we defined
In the above equations is distance between observer and source, is distance between the observer and the lens, is distance between the lens and the source. is formed when a line passing through the observer and the image is coincide optical axis (line passing through the observer and the lens). is formed when a line passing through the observer and the source is coincide the optical axis. One can obtain general solutions of the lens equation (5.1) as follows.
where It has some real solutions for
In the following we use (5.3) to obtain non-relativistic and relativistic image locations.
5.1 Weak lensing images
In weak deflection limits with large distances between lens, source and observer located in a straight line approximately, one can infer 
where must be inserted 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 as Sgr A [48-51]. 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 fall toward the black hole center. For the latter black hole we will have
The relations (5.4) and (5.6) leads us to choose
in weak deflection limits of gravitational lensing where we defined
in which the subscript denotes to the word ‘Maximum‘. For critical source the lens equation (5.3) reads
which for weak deflection limits leads to the following approximation.
we can obtain Taylor series expansion of the lens equation (5.3) at neighborhood of as follows.
which in limits become
This is primary image location and by transforming as one can obtain secondary image location. The parameter describes right-handed (+1)and/or left-handed (-1 ) bending angles. Setting and inserting numerical values of the deflection angles (4.19) and (4.26) via numerical values of the table 1 (see diagrams of the figure 2) we plot numerical values of and aga