Null Geodesics and Strong Field Gravitational Lensing of Black Hole with Global Monopole

We study two interesting features of a black hole with an ordinary as well as phantom global monopole. Firstly, we investigate null geodesics which imply unstable orbital motion of particles for both cases. Secondly, we evaluate deflection angle in strong field regime. We then find Einstein rings, magnifications, and observables of the relativistic images for supermassive black hole at the center of galaxy NGC4486B.We also examine time delays for different galaxies and present our results numerically. It is found that the deflection angle for ordinary/phantom global monopole is greater/smaller than that of Schwarzschild black hole. In strong field limit, the remaining properties of these black holes are quite different from the Schwarzschild black hole.


Introduction
Geodesics are associated with the motion of free particles traveling along their trajectories whose nature depends upon the spacetime. There are two types of geodesics followed by physical particles, that is, timelike and null (light-like), related to the propagation of massive and massless particles. The study of motion of massless particles such as photons is important from both astrophysical and theoretical points of view. It has been observed that light path is affected by gravity which means that path of a photon through spacetime may be bent by the gravitational field of a massive object such as a star or black hole (BH). The dynamics of test particle not only helps to understand geometrical structure of spacetime but also explains high energy phenomenon occurring near BH such as accretion disks where particles move in circular orbits and formation of jets in which particles escape.
Chandrasekhar [1] was the pioneer to investigate geodesic motion of a test particle around Schwarzschild, Reissner-Nordström (RN), and Kerr BHs. Fernando et al. [2] constructed geodesic structure of static charged BHs of dilaton gravity and studied orbital motion of test particles. Konoplya [3] analyzed motion of both massless and massive particles around magnetized BHs and concluded that tidal force has considerable effect on the motion of test particles. Leiva et al. [4] studied geodesics of the Schwarzschild BH in rainbow gravity and found that geodesics remain unchanged under the influence of semiclassical effects. Guha and Bhattacharya [5] determined that the null geodesics of five-dimensional RN anti-de Sitter BH have a unique fixed point and are terminating orbits. Pradhan [6] found conditions for the existence of ISCO (inner most stable circular orbit), marginally bound circular orbit, and null circular geodesics in equatorial plane for Kerr-Newman-Taub-NUT BH.
Deflection of light in gravitational field around a massive object is referred to as gravitational lensing and an object causing deflection is called gravitational lens. Gravitational lensing is a powerful tool in cosmology as well as in astrophysics to understand distribution of mass in the large scale structures of the universe as well as cluster of galaxies and halos. It provides a useful way to estimate Hubble parameter and detection of dark mater, dark energy, exoplanet, gravitational waves, and so forth. This phenomenon is divided into two regimes: weak and strong lensing. Weak gravitational lensing produces weakly distorted images of the source. In this case, the gravitational lens is not strong enough to form 2 Advances in High Energy Physics multiple images and high magnification. It helps in the measurement of distribution of luminous as well as dark matter in the universe. If the lens is massive enough and the source and lens are highly aligned, then multiple images are formed from the background source. This phenomenon is known as strong gravitational lensing. The distortion and position of such multiple images carry important information about distribution of mass in faraway galaxies and background sources at large distance.
The theory of gravitational lensing was initially developed in weak field approximation but this approach cannot describe the phenomena like high bending of light rays and formation of infinite series of images. This motivates studying the strong gravitational lensing, which not only helps to understand these phenomena but also explains the winding of light rays multiple times around a massive object before reaching to the observer. After the pioneer work of Darwin [7], much work has been done in the context of gravitational lensing in strong field [8][9][10][11]. Virbhadra and Ellis [12] studied strong field gravitational lensing of Schwarzschild BH and 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. [13] proposed an exact thin-lens equation whose accuracy was shown in the strong field. Bozza [14] developed a useful technique for spherically symmetric BHs in strong field by expanding the deflection angle near the photon sphere.
The image detection for low mass BHs is difficult but the supermassive BHs such as Sgr * are an interesting example of deflection of light in strong field [15,16]. Ding et al. [17] considered noncommutative BH as gravitational lens and found effect of noncommutative parameter similar to charge by comparing with RN BH. Deng [18] studied gravitational lensing of magnetically charged RN BH pierced by a cosmic string in strong field and found increase in the deflection angle. Sahu et al. [19] showed that strong gravitational lensing can be used to distinguish BHs from naked singularities. Wei et al. [20] explored strong lensing of Kerr-Taub-NUT BH and found significant effect of NUT charge. Different authors [21][22][23][24][25][26][27][28][29] studied gravitational lensing of many other astrophysical spacetimes in strong field limit. The fact that the universe is in the phase of accelerating expansion is a major turning point in cosmology which indicates the existence of dark energy supported by several observational evidences. Dark energy is an elusive force having large negative pressure. To understand its exact nature, several dynamical models have been proposed out of which phantom field is a strange kind of dark energy with equation of state parameter < −1 violating the null energy condition. Exact BH solutions including phantom fields are called phantom BHs. Babichev et al. [30] investigated that phantom energy accretion onto a BH causes a continuous decrease in BH mass. Bronnikov and Fabris [31] found an interesting regular phantom BH solution which is asymptotically flat, de Sitter, and anti-de Sitter. Eiroa and Sendra [32] studied regular phantom BHs as gravitational lens and compared their results with Schwarzschild and vacuum Brans-Dicke BHs. Some people [33][34][35] have discussed light paths and gravitational lensing of phantom BHs.
Global monopoles are topological defects of vacuum manifold that arise from the phase transition in the early universe. Their formation depends upon the gauge symmetry breaking with a choice of suitable scalar field. It can be shown that their energy is concentrated near the monopole core into a small region. Barriola and Vilenkin [36] found static spherically symmetric BH with a global monopole. Many authors [37][38][39][40][41][42] studied physical properties of BHs with global monopole.
In this paper, we study null geodesics as well as strong gravitational lensing of spherically symmetric BHs (with ordinary and phantom global monopoles). The format of the paper is as follows. In the next section, we introduce the metric having both ordinary and global monopoles and study the behavior of null geodesics. In Section 3, we evaluate exact deflection angle using Bozza's method. Section 4 explores Einstein rings, magnifications, and observables of the relativistic images. In Section 5, we numerically study the observables for the central supermassive BH. Section 6 is devoted to the study of time delays between the relativistic images in different galaxies. In the last section, we summarize the results.

Null Geodesics
We consider static spherically symmetric BH with a global monopole [43]. This was obtained by global SO(3) symmetry breaking of a triplet scalar field in the Schwarzschild BH background. This is topologically different from Schwarzschild BH due to the existence of global monopole. The metric of this BH is described as where is the mass of BH, is the energy scale of symmetry breaking, and is the term describing kinetic energy of the BH. If = 1, it represents an ordinary global monopole originating from positive kinetic energy of scalar field [36]. If = −1, the phantom global monopole is formed originating from negative kinetic energy of scalar field. The corresponding event horizon is Notice that the Schwarzschild radius is recovered for = 0, it does not have any horizon for = 0, and it generates a naked singularity at = 0. There are several phantom BHs having negative kinetic energy and pressure; it would be interesting to study the behavior of energy density and pressure of the BH with phantom global monopole. It can be seen in Figure 1 that energy density and pressure admit the properties of a phantom model which is almost similar to [35]. The expressions of and are given in the Appendix. The Lagrangian in the equatorial plane ( = /2) [1] for a photon traveling around a BH with global monopole is Using the Euler-Lagrange equations for null geodesics and an affine parameter , we havė where and are the energy and angular momenta per unit mass. The Hamiltonian is given as where is an integral of motion and = −1, 0, 1 correspond to spacelike, null, and timelike geodesics, respectively. For null geodesics, the radial equation of motion iṡ where eff = ( / 2 )(1 − 8 2 − 2 / ). Figure 2 represents the behavior of effective potential for different values angular momentum . We observe that, for = −1, eff has only maximum values and hence only unstable circular orbits exist. For = 1, maximum value exists at ≈ 0.002 which corresponds to unstable circular orbits; however, eff continuously decreases after this point leading to stable circular orbits.

Deflection Angle
Here, we study gravitational lensing in strong field limit and the effect of with ordinary and phantom global monopoles. We obtain deflection angle for (1) in the strong field using Bozza's technique. We can write (1) in terms of Schwarzschild radius by defining the transformations = /2 and = /2 as where The radius of photon sphere ( ps ) is the largest positive root of the equation [44]: Figure 3 describes the behavior of ps for ordinary and phantom global monopoles. In the first case, the radius of photon sphere is an increasing function of and greater than the Schwarzschild limit while, for the second case, it shows a decreasing behavior as increases and is less than the Schwarzschild limit. The deflection angle for a photon (from infinity) is calculated as a function of distance of the closest approach 0 [45,46]: where The deflection angle depends upon the relation between 0 and ps which grows if 0 approaches ps . Following [14], we introduce a new variable : Replacing in (13), it follows that where and all functions other than the subscript 0 are evaluated at The function ( , 0 ) is regular for all values of and 0 while ( , 0 ) diverges when tends to zero. We expand the function inside the square root (by applying Taylor's series approximation) up to second order in [14]: where Advances in High Energy Physics 5 Using the above expressions, integral (13) can be separated as where When is nonzero ( 0 ̸ = ps ), the order of divergence in 0 is approximately 1/√ , while, for to be zero ( 0 = ps ), the divergence is 1/ . Thus, includes divergence whereas R is regular as the divergent term is subtracted out [14]. By the conservation of angular momentum, the impact parameter can be defined as a function of 0 [46,47]: The deflection angle for the photons propagating near the photon sphere is given as where = 2 ( ps ) Since R cannot be calculated directly, using Taylor's series approximation we expand it up to second order in terms of as where R,0 corresponds to the uncharged (Schwarzschild) BH and R,2 ≈ −2.26271 (8 ). The graphical analysis of the deflection coefficients 1 and 2 and impact parameter ps is shown in Figure 4 (ordinary as well as phantom global monopoles). We observe that, for = 1, 1 and ps are increasing functions of and greater than the Schwarzschild limit, while 2 diverges for large values of and is less than Schwarzschild limit. The situation is totally different for the phantom global monopole ( = −1). We see that 1 and ps decrease with the increase of and are smaller than the Schwarzschild limit, whereas 2 has an increasing behavior for all values of and is larger than the Schwarzschild limit.

Magnifications, Einstein Rings, and Observables
In this section, we study the effect of global monopole (ordinary as well as phantom) on magnifications, Einstein rings, and observables for the relativistic images in the strong field limit. These quantities are directly related to the deflection angle by the lens equation and have been discussed in detail [14,32]. We briefly discuss basic equations and the results are given numerically. In the context of gravitational lensing, the deflection angle is associated with angles and describing the source and the image position. The lens equation for lens l situated between source s and observer o is given as [12] tan = tan − ls os {tan ( − ) + tan } .
The distances between observer-source, observer-lens, and lens-source are denoted by os , ol , and ls , respectively. These distances are measured from the optical axis (line joining observer o and lens l) and are considered much greater than the horizon. We consider the case when objects are highly aligned; that is, and are small. The deflection angle for a photon traveling near the photon sphere is = Δ +2 , ∈ , where 0 < Δ ≪ 1. In this approximation, the lens equation takes the form [48] = − ls os Δ .
This equation describes only images on the same side of the source by taking positive . To obtain images on the opposite side, the same equation can be solved with the source placed in − . The deflection angle can be expressed in terms of angular position of the image and the observer-lens distance ol . According to lens geometry ( = ol sin ≈ ol ), we have The behavior of deflection angle for both cases (ordinary and phantom global monopoles) can be seen in Figure 5. For the first case, is a monotonically increasing function of while it decreases in the second case. Also, for ordinary global monopole, is larger than Schwarzschild limit (after = 0.1), while, for phantom global monopole, the deflection angle is very much less than Schwarzschild limit as well as for ordinary global monopole. We find that a light ray passing near a BH with global monopole makes a large deflection angle as compared to the light ray traveling near a BH with phantom global monopole. An infinite sequence of Einstein rings is given as where = ps 1 ol The first image is the outer most image having angular position 1 and the remaining images are given as where ∞ is the asymptotic position of the set of relativistic images obtained in the limit → ∞. For high alignment in the strong field limit, the observable (the angular separation ( ) between the first and remaining images and ratio ( ) between flux of the first and sum of other images) can be written as Analysis of these parameters is useful to understand the nature of BH lens.

Numerical Analysis of the Observables
There is a strong evidence of the existence of supermassive BHs (with masses 10 6 -10 9 ) in the nuclei of galaxies of local universe [49][50][51]. It is believed that supermassive BHs at the galactic center have an intense effect on the evolution of host galaxies. Such BHs release a great amount of energy into surroundings which is helpful in star formation and growth of BH [52][53][54]. We examine the Einstein rings, magnifications, and observables in strong field by taking the example of supermassive BH in the center of galaxy NGC4486B, which has mass 5.7 × 10 8 ⊙ and distance from the Earth ol = 15.3 Mpc [55]. For the sake of simplicity, we assume that the lens is placed midway between the source and observer; that is, ls / os = 1/2 [12].
The graphical analysis of observables is shown in Figure 6.   position ∞ ; also both of these quantities are greater than those of Schwarzschild. The magnification of the outer most image mag = 2.5 log appears as a decreasing function of and much less than Schwarzschild limit. For phantom global monopole, shows increasing behavior in the beginning but decreases gradually after ≈ 0.075. Thus, for low values of , angular separation is greater while becoming smaller than Schwarzschild limit for high values of (after = 0.19). The magnification of the outer most image continuously increases and is greater than Schwarzschild limit while angular position is less than the Schwarzschild limit. We observe that for = 1 images become more separated for large values of . However, for = −1, images are well separated in the region  = 0.05 − 0.09 and after = 0.09, it is difficult to distinguish between the first and the remaining images. The relative magnitude mag for ordinary global monopole is smaller than the BH with phantom global monopole. The angular position ∞ for the case = 1 is larger than the case of = −1.
Tables 1-4 indicate the numerical results for Einstein rings and image magnifications corresponding to the first ( = 1) and second images ( = 2). In the first case (ordinary global monopole), numbers of Einstein rings and magnifications gradually increase for both images ( = 1,2) but the image is brighter than the second one. Also, there exist more Einstein rings for the first image than the second. In the second case (phantom global monopole), the behavior 8 Advances in High Energy Physics of Einstein rings and magnifications is entirely different; both quantities decrease with increasing ; that is, strong lensing of BHs with phantom global monopole produces less Einstein rings and the images are fainter than the BH with global monopole.

Time Delays between Relativistic Images
In the strong field regime, the deflection angle (around a BH) could be more than 2 and multiple images might be formed. In this case, the time for the light propagation along each path corresponding to different images is not the same. Thus, intrinsic variations from a distant source may appear at different times in the multiple images. This time lag is called the time delay between two images of a lensed source. Refsdal [55] firstly proposed that Hubble parameter can be described by time delay. Following [56], we can calculate time delays between relativistic images.
The time delay between th and th relativistic images from different paths of photons formed on the same side of the lens is given by [32] Δ , = ps [2 ( − ) where positive (negative) sign indicates if both images are on the same (opposite) side of the source. For images on the opposite side of the lens, we have [44] Δ , = ps [2 ( − ) The winding number denotes image on the same side of the source and represents image on the opposite side of the source. Some results for ordinary and phantom global monopoles are presented in Tables 5-10 for the specific masses and distances taken from [56].
In Tables 5-10, the subscripts + and − represent the case of both images which are on the same or opposite side of the source. We observe that, with increasing values of , time delays for the BH with global monopole increase. The situation is totally opposite for the BH with phantom global monopole where the time delays have smaller values for larger values of .

Final Remarks
In this paper, we have examined the effects of energy scale of symmetry breaking on the null geodesics as well as strong gravitational lensing for BHs having ordinary and phantom global monopoles. The results are shown graphically and are compared with the Schwarzschild BH. The analysis of eff represents that only unstable orbits of particles exist for = −1, whereas eff for = 1 leads to stable circular orbits. We have adopted the technique of Bozza to calculate the deflection angle and all the related quantities for both cases ( = 1, −1). The deflection angle for the BH with ordinary global monopole is monotonically increasing function of and larger than the Schwarzschild limit. For the phantom global monopole, the deflection angle continuously decreases and is less than the deflection angle of Schwarzschild as well as BH with ordinary global monopole. The behavior of deflection angle for phantom global monopole is entirely different from [35], where ( ) is an increasing function of phantom constant .
We have considered the example of supermassive BH at the center of galaxy NGC4486B for Einstein rings, magnifications, and observables. For the BH with ordinary global monopole, the images are well separated for high values of . The inner most angular positions are greater while relative magnitude is smaller than the Schwarzschild limit. The analysis of phantom global monopole shows that the separation among images is high for initial values of but it gradually reduces when increases. The inner most angular positions are less while relative magnitude is larger than Schwarzschild BH. When = 1, images are more prominent and formation of Einstein rings increases as increases. For the BH with phantom global monopole ( = −1), the images become fade for large , and also numbers of Einstein rings gradually decrease. It is worth mentioning that the results of observables for the case of phantom global monopole are contrary to [35].
Finally, we have studied the time delays between relativistic images by taking the example of different galaxies having supermassive BHs in their center. We have considered two cases for both ordinary and global monopoles, when the images (first and second) are on the same side of the lens and when the images are on the opposite side of the lens. The time delays for an ordinary global monopole become larger with large values of , whereas for phantom global monopole they have smaller values as increases. We conclude that the parameter (energy scale of symmetry breaking) highly affects the propagation of light in the strong field limit. It is worthwhile to mention here that the strong field lensing properties in both cases (ordinary and phantom global monopoles) are different not only from each other but also from the Schwarzschild BH.