Advances in High Energy Physics

Volume 2015 (2015), Article ID 854264, 11 pages

http://dx.doi.org/10.1155/2015/854264

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

^{1}Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan^{2}Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan

Received 16 July 2015; Revised 6 August 2015; Accepted 17 August 2015

Academic Editor: Rong-Gen Cai

Copyright © 2015 M. Sharif and Sehrish Iftikhar. 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

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.

#### 1. 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 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 [8–11]. 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 [21–29] 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 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 [33–35] 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 [37–42] 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.

#### 2. 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 aswhere is the mass of BH, is the energy scale of symmetry breaking, and is the term describing kinetic energy of the BH. If , it represents an ordinary global monopole originating from positive kinetic energy of scalar field [36]. If , the phantom global monopole is formed originating from negative kinetic energy of scalar field. The corresponding event horizon isNotice that the Schwarzschild radius is recovered for , it does not have any horizon for , and it generates a naked singularity at . 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.