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Advances in High Energy Physics

Volume 2015, Article ID 635625, 9 pages

http://dx.doi.org/10.1155/2015/635625

## Null Geodesics and Strong Field Gravitational Lensing in a String Cloud Background

^{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 2 March 2015; Accepted 1 May 2015

Academic Editor: Juan José Sanz-Cillero

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

This paper is devoted to studying two interesting issues of a black hole with string cloud background. Firstly, we investigate null geodesics and find unstable orbital motion of particles. Secondly, we calculate deflection angle in strong field limit. We then find positions, magnifications, and observables of relativistic images for supermassive black hole at the galactic center. We conclude that string parameter highly affects the lensing process and results turn out to be quite different from the Schwarzschild black hole.

#### 1. 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-Nordström, and Kerr BHs. There has been growing interest to explore the behavior of geodesics around BHs [2–5] 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 [6–9].

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 ). 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 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 [18–20] 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-Nordström 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.

#### 2. 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] aswhere 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 isIn the limit , the Schwarzschild radius is recovered while it approaches infinity when . Thus, for a realistic model, . Also, for , it does not have any horizon and generates a naked singularity at .

The Lagrangian in the equatorial plane () [1] for a photon traveling in string cloud isUsing the Euler-Lagrange equations for null geodesics and an affine parameter , we havewhere and are the energy and angular momentum per unit mass. The Hamiltonian is given aswhere is an integral of motion and correspond to spacelike, null, and time like geodesics, respectively. For null geodesics, the radial equation of motion iswhere . Figure 1 shows the behavior of effective potential for different values of ) and angular momentum . The dashed lines represent event horizons ( for ). We see that maximum values exist outside the event horizons. Since there does not exist any minimum value of , hence only unstable circular orbits exist.