Corrigendum Corrigendum to ( Null Geodesics and Strong Field Gravitational Lensing in a String Cloud Background )

In the paper titled “Null Geodesics and Strong Field Gravitational Lensing in a String Cloud Background,” recently, we have studied null geodesics and strong field gravitational lensing of a black hole with string cloud background [1]. References of some equations weremissing in the paper titled “Null Geodesics and Strong Field Gravitational Lensing in a String Cloud Background.” Here we add these references. Equations (10), (13), and (22) were originally derived in the following references:


Introduction
The study of null geodesics is interesting from both astrophysical and theoretical points of view.This phenomenon helps to understand geometrical structure of spacetime as well as explaining high energy phenomenon occurring near black hole (BH) such as accretion disks where particles move in circular orbits and formation of jets in which particles escape.The dynamics of test particle is useful to understand the observational effects such as deflection of light, time delays, and perihelion shift.Chandrasekhar [1] did the pioneer work in the geodesic study of Schwarzschild, Reissner-Nordstr ö m, and Kerr BHs.There has been growing interest to explore the behavior of geodesics around BHs [2][3][4][5] in the last few years.
Deflection of light ray by massive objects near a gravitational field is one of the important results in general relativity (GR) known as gravitational lensing.The deflection angle of light depends on the nature as well as distance of lens to an observer.Gravitational lensing is an effective tool to test predictions of GR as well as to study nonluminous objects such as extra solar planets, distant stars, detection of dark matter, estimation of cosmological parameter, detection of gravitational waves, and cosmic censorship conjuncture [6][7][8][9].
There are two types of gravitational lensing: weak and strong field limit.Initially, it was believed that gravitational lensing is based only on weak field and small deflection angles.Weak field approximation is very useful for investigating the properties of stars and galaxies [10,11].However, in the last decade, the gravitational lensing in strong field regime (testing the lensing properties near photon sphere) has gained much attention.The deflection of light in strong field provides a platform to test a theory of gravitation in its general form.Since, in weak field approximation, alternative gravitational theories must agree with GR, so it would be interesting to study strong field approximation to show deviations from GR [12].The gravitational field around the collapsed objects (BHs and neutron stars) is very strong.An astrophysical system involving such objects provides a way to investigate its properties in the context of strong field limit.The accretion matter of BHs and neutron stars emits radiations which originate from deep gravitational fields (at a distance of gravitational radius   = / 2 ).Thus the motion of accretion matter also motivates to study strong field gravity [13].
Virbhadra and Ellis [14] studied strong field gravitational lensing of Schwarzschild BH.They found a sequence of relativistic images on both sides of optical axis due to large deflection of light near the photon sphere.Frittelli et al. [15] proposed an exact lens equation without background spacetime and showed that the thin lens equation in strong field is remarkably accurate; even light rays take several Advances in High Energy Physics rounds around the lens before reaching the observer.Bozza [16] studied spherically symmetric BHs in strong field limit and expanded the deflection angle near photon sphere.He remarked that this method is valid for a generic spherically symmetric metric regardless of the field equations, assuming that the light follows geodesic equation.
There is no direct observational evidence of gravitational lensing by BH or other compact objects.The detection of images for small BH is difficult whereas supermassive BHs such as Sgr  * provide a good example to test the bending of light in strong gravity regimes [17].Many people [18][19][20] studied various aspects of gravitational lensing for rotating BHs.Eiroa and Sendra [21] explored Bardeen regular BH as gravitational lens and compared the results with Schwarzschild BH.Many astrophysical spacetimes such as fermion stars [22], naked singularities [23], magnetized [24] as well as alternative gravity BHs [25], and wormholes [26] are analyzed as gravitational lenses.Recently, Wei et al. [27] investigated gravitational lensing of Hayward BH and found that nonsingularity parameter has negligible effect in the weak field while it has a significant influence in the strong field limit.
Gravitational lensing for BHs in string and other higher dimensional theories has recently attained much attention.Bhadra [28] studied gravitational lensing of charged BH of heterotic string theory and found no significant string in strong field regime.Eiroa and Sendra [29] studied massless braneworld BH in weak as well as strong field and compared the results with Schwarzschild and Reissner-Nordstr ö m BHs.Tsukamoto et al. [30] examined lensing properties of Tangherlini spacetime and concluded that images have little effect on the total light curve in strong field.
In this paper, we study null geodesics as well as gravitational lensing of a spherically symmetric BH with string cloud background in strong field.The paper is organized as follows.
In Section 2, we introduce metric for the string cloud and study the behavior of null geodesics.Section 3 evaluates exact deflection angle using Bozza method.In Section 4, we explore positions, magnifications, and observables of the relativistic images for the supermassive galactic BH.In the last section, we conclude our results.

Basic Equations and Null Geodesics
Cosmic strings are considered as a generic outcome of symmetry breaking phase transitions of the early universe and play a vital role in the formation of large scale structure of the universe [31,32].It is believed that strings may exist in the early universe and are very important in creation of density inhomogeneities [33].String theory can describe many features of BHs.The association of strings with BHs is suggested by the relationship between entropy of BH horizon and string states [34].We consider static spherically symmetric spacetime with string cloud background [35] as where (2) is the string cloud parameter and  is the mass of BH (independent of ).This metric represents BH related to spherical mass  surrounded by a cloud of strings which can also be considered as a metric associated to a global monopole.The corresponding event horizon is In the limit  → 0, the Schwarzschild radius is recovered while it approaches infinity when  → 1.Thus, for a realistic model, 0 <  < 1.Also, for  = 0, it does not have any horizon and generates a naked singularity at  = 0.
The Lagrangian in the equatorial plane ( = /2) [1] for a photon traveling in string cloud is Using the Euler-Lagrange equations for null geodesics and an affine parameter , we have where  and  are the energy and angular momentum per unit mass.The Hamiltonian is given as where  is an integral of motion and  = −1, 0, 1 correspond to spacelike, null, and time like geodesics, respectively.For null geodesics, the radial equation of motion is where Figure 1 shows the behavior of effective potential for different values of  (0 <  < 1) and angular momentum .The dashed lines represent event horizons ( ℎ = 2.2, 2.5, 3.33, 5 for  = 0.1, 0.2, 0.4, 0.6).We see that maximum values exist outside the event horizons.Since there does not exist any minimum value of  eff , hence only unstable circular orbits exist.

Deflection Angle
In this section, following [16], we calculate deflection angle for (1) in the strong field limit.We consider photon sphere as an initial point which is the strong gravitational region in spacetime such that photons are forced to travel in their where The radius of photon sphere  ps is the largest positive root of the equation and yields The behavior of  ps is shown in Figure 2 which increases with the increasing values of  and is larger than that of the Schwarzschild radius.Also, for  = 0, radius of photon sphere of string cloud coincides with that of Schwarzschild.The deflection angle for a photon (from infinity) is calculated as a function of distance of the closest approach (minimum distance)  0 : where The deflection angle depends upon the relation between  0 and  ps which grows if  0 approaches  ps .Following [16], we define a new variable : Replacing  in (13), it follows that where All functions other than the subscript 0 are evaluated at The function (,  0 ) is regular for all values of  and  0 while ℎ(,  0 ) diverges for  → 0. Thus we expand this function inside the square root up to second order in  using Taylor's series expansion as where Using these values in (13), we have where When  is nonzero ( 0 ̸ =  ps ), the order of divergence in ℎ 0 is approximately 1/√ while, for  to be zero ( 0 =  ps ), the divergence is 1/.Thus   is the term which contains divergence whereas   is regular as the divergence is subtracted out [16].The impact parameter  is related to  0 by the conservation of angular momentum: Bozza [16] showed that the deflection angle has logarithmic divergence for the photons traveling near to photon sphere given as where  ps = √ 3 ps /( ps (1 − ) − 1) is the value of impact parameter at  0 =  ps and Advances in High Energy Physics Since   cannot be calculated directly, so we expand it using Taylor's series expansion up to second order in terms of  as where  ,0 corresponds to the uncharged (Schwarzschild) BH [16] and  ,2 = 3 ln(6(2 − √ 3)) ≈ −0.708333.The plots of  1 ,  2 , and  ps along with comparison of Schwarzschild BH are shown in Figure 3.We see that  1 increases positively with the increasing values of  while  2 is negative for small values of  but it grows positively with large values of .
Both coefficients are greater than the Schwarzschild limit.The impact parameter  ps is monotonically increasing function of  and also greater than the Schwarzschild limit.

Positions, Magnifications, and Observables
In this section, we study positions, magnifications, and observables of the relativistic images in the strong field limit, which are directly related by the deflection angle using the lens equation.The detailed discussion is available in [21,29]; here we give a review of the basic equations and derive our results numerically.The lens equation helps to explore the bending of light by relating the image to source position and has the following form [14]: The basic lens formalism is shown in Figure 4 which is helpful in understanding the lens equation.When objects are aligned, the lensing effects become more important; hence we consider the case in which  and  are small.The nonzero values of  give two weak deflection images (primary and secondary) and two infinite sets of point relativistic images.
The deflection angle from the first set (relativistic images) is  = Δ  + 2,  ∈  and 0 < Δ  ≪ 1.In this approximation, the lens equation can be written as [12] For the other set of images ( = −Δ  − 2), the quantity Δ  is replaced by −Δ  in the above equation.The deflection angle can be expressed in terms of angular position of the image  and the lens-observer distance   .According to lens geometry ( =   sin  ≈   ), (23) can be written as Figure 5 indicates that deflection angle increases for light propagating in the strong field limit near a BH with string cloud background.We see that deflection angle is much larger than the Schwarzschild BH.
Inverting the above equation and using a first order Taylor's series expansion around  = 2, one can obtain th image position as where Using ( 28) and (30),   can be written as Assuming 0 <   /  ≪ 1, the angular position of two sets of relativistic images are [21,29] For  = 0 in (33), an infinite sequence of Einstein rings is obtained as The magnification of th relativistic image is given by the quotient of solid angles subtended by the image and source: Keeping the angles small and using (33) and (36), the th relativistic image can be written as A first order approximation in     /  gives When the first image is the outer most image with angular position  1 , then the angular separation of other relativistic images is where  ∞ is the asymptotic position of the set of relativistic images obtained in the limit  → ∞.In the simple case, when only  1 is resolved as a single image, the angular separation between first and limiting value of the set of images is and ratio between the flux of first and sum of other images is Using the concept of perfect alignment and the strong field limit, the observables can be written as The existence of galactic supermassive BHs in our galaxy as well as in other galaxies is indicated by nuclear stellar dynamics [37,38].We study the Einstein rings, magnifications, and observables in strong field limit for the supermassive BH.For the numerical example, we consider the supermassive BH in the center of galaxy NGC4486B, which has mass 5.7×10 8 M ⊙ and distance from the Earth   = 15.3Mpc [39].For the sake of simplicity, we consider that the lens is placed midway between the source and observer; that is,   /  = 1/2 [14].Tables 1 and 2 indicate results for the first ( = 1) and second image ( = 2).All the quantities have been found in  and  is converted into magnitudes ( mag = 2.5 log ) [16].We see that not only the number of Einstein rings increases with  but also more Einstein rings (   ) are formed for the first image than the second one.The image magnification (  ) also shows increasing behavior for larger values of  and images for  = 1 are brighter than that of  = 2.The graphical behavior of  ∞ , , and  mag is shown in Figure 6 which indicates that, for the string cloud, magnification of the outer most image  mag is decreasing function of  while the angular position  ∞ and angular separation  grow with the increase of .It is also seen that  mag is smaller while  ∞ and  are larger than the Schwarzschild limit.Some numerical values of observables are presented in Table 3.

Final Remarks
In this paper, we have investigated null geodesics and gravitational lensing in the strong field limit for spherically symmetric BH with string cloud background.We have taken the example of supermassive BH in the center of galaxy NGC4486B for the numerical study and the results are compared with the Schwarzschild limit.We know that minimum

Figure 2 :
Figure 2: Radius of photon sphere ( ps ) for the string cloud versus string parameter .Red curve corresponds to  ps whereas the green line (dashed) indicates Schwarzschild limit.

Figure 3 :
Figure 3: Behavior of strong field deflection coefficients ( 1 (a) and  2 (b)) and parameter  ps (c) as functions of  are represented by yellow, purple, and green while red (dashed) line shows Schwarzschild limit.

Figure 4 :Figure 5 :
Figure4: Basic lens formalism: position of source, image, lens, and observer are represented by , , , and .The distance between lens-source, lens-observer, and source-observer are denoted by   ,   , and   , respectively.The line connecting  and  is called optical axis of the lens geometry.The angle  is the location of the source , while the position of image is denoted by angle  with respect to optical axis.The impact parameter is given by [36].
mag Figure 6: Plots of the observables corresponding to .Dashed lines show the value for the Schwarzschild BH.

Table 3 :
Observables for the supermassive BH at the galactic center.