Advances in High Energy Physics

Volume 2015 (2015), Article ID 454217, 11 pages

http://dx.doi.org/10.1155/2015/454217

## Null Geodesics and Gravitational Lensing in a Nonsingular Spacetime

^{1}Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China^{2}School of Science, Xi’an Jiaotong University, Xi’an 710049, China

Received 25 August 2014; Revised 9 December 2014; Accepted 22 December 2014

Academic Editor: Elias C. Vagenas

Copyright © 2015 Shao-Wen Wei et al. 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

The null geodesics and gravitational lensing in a nonsingular spacetime are investigated. According to the nature of the null geodesics, the spacetime is divided into several cases. In the weak deflection limit, we find the influence of the nonsingularity parameter on the positions and magnifications of the images is negligible. In the strong deflection limit, the coefficients and observables for the gravitational lensing in a nonsingular black hole background and a weakly nonsingular spacetime are obtained. Comparing these results, we find that, in a weakly nonsingular spacetime, the relativistic images have smaller angular position and relative magnification but larger angular separation than those of a nonsingular black hole. These results might offer a way to probe the spacetime nonsingularity parameter and put a bound on it by the astronomical instruments in the near future.

#### 1. Introduction

The cosmic censorship hypothesis [1, 2] says that singularities that arise in the solutions of Einstein’s equations are typically hidden within event horizons and therefore cannot be seen from the rest of spacetime. However, in a semiclassical approximation [3], black holes tend to shrink until the central singularities are reached, which will lead to the breakdown of the theory.

Motivated by the idea of the free singularities, there are several ways to obtain black hole spacetime with no singularities at the center. The one presented in [4, 5] was inspired by the noncommutative geometry. The points on the classical commutative manifold are replaced by states on a noncommutative algebra, and the point-like objects are replaced by smeared objects. Thus the singularity problem is cured at the terminal stage of black hole evaporation.

Another way is to introduce a de Sitter core to replace the central singularity. The first one constructed in this way is the Bardeen regular black hole [6–9], which was found to have both an event horizon and a Cauchy horizon. Recently, Hayward proposed a nonsingular black hole solution [10] (Poisson and Israel also derived an equivalent solution based on a simple relation between vacuum energy density and curvature [11, 12]), which is a minimal model satisfying the asymptotically flat and flatness conditions at the center. Its static region is Bardeen-like. In this nonsingular spacetime, a black hole could be generated from an initial vacuum region and then subsequently evaporate to a vacuum region without singularity [10]. This case was extended to the dimensional spacetime and some interesting results were obtained [13]. The quasinormal frequency of this nonsingular spacetime has been recently analyzed in [14] with a significant difference from the singular spacetime. In fact, according to the nature of this spacetime, we can divide it into several cases, that is, the nonsingular black hole, the extremal nonsingular black hole, the weakly nonsingular spacetime, the marginally nonsingular spacetime, and the strongly nonsingular spacetime. Other regular black hole solutions [15–22] can be constructed with the introduction of some external form of matter, such as nonlinear magnetic monopole, electrodynamics, or Gaussian sources, which leads to the fact that they are not vacuum solutions of Einstein’s equations.

The subject of gravitational lensing by black holes and compact stars has received great attention in the last ten years, basically due to the strong evidence of the presence of supermassive black holes at the center of galaxies. The study can be traced back to [23], where the author examined the gravitational lensing when the light passes near the photon sphere of Schwarzschild spacetime. In [24], the authors showed that, in the case of large values of the scalar charge, the lensing characteristics were significantly different. And the result provides preliminary knowledge on the naked singularity lens. This resurrects the study of the gravitational lensing. After modelling the massive dark object at the galactic center as a Schwarzschild black hole lens, it was found that [25–27], similar to Darwin’s paper, apart from a primary image and a secondary image resulting by small bending of light in a weak gravitational field, there is theoretically an infinite sequence of very demagnified images on both sides of the optical axis. A similar result was also found in [28, 29]. These images were named as the “relativistic images” by Virbhadra and Ellis [25] and that term was extensively used in later work. Based on lens equation [25, 30], Bozza et al. [31–35] developed a semianalytical method to deal with it. This method has been applied to other black holes [36–61]. These results suggest that, through measuring the relativistic images, gravitational lensing could act as a probe to these black holes, as well as a profound verification of alternative theories of gravity in the strong field regime [37–39]. Furthermore, it can also guide us to detect the gravitational waves at proper frequency [62, 63]. It is also worthwhile to mention that, in [27], the author pointed out that Bozza’s semianalytical method gives small percentage difference of the deflection angle, angular position, and angular separation compared to their accurate values, while it gives large percentage difference of magnification and differential time delays among the relativistic images. Thus one must pay great attention to studying the differential time delays among the relativistic images, and we will not consider that case in this paper.

In [54, 55], the authors studied the strong gravitational lensing by a regular black hole with noncommutative corrected parameter. The result showed that gravitational lensing in the strong deflection limit could provide a probe to the noncommutative parameter. In this paper, we mainly focus on the exploration of the lensing features in a nonsingular spacetime with the central singularity replaced by a de Sitter core. At first, we study the nature of the spacetime in different range of the nonsingularity parameter . And, according to it, the spacetime is classified into the nonsingular black hole , the extremal nonsingular black hole , the weakly nonsingular spacetime , the marginally nonsingular spacetime , and the strongly nonsingular spacetime . Then, under this classification, we study the lensing features in a nonsingular spacetime in both weak and strong deflection limits. The result shows that, in the weak deflection limit, the influence of the nonsingularity parameter on the lensing is negligible. Compared with it, has a significant effect in the strong deflection limit, which is very helpful for detecting the nonsingularity of our universe in the future astronomical observations.

The paper is structured as follows. In Section 2, we study the null geodesics and photon sphere for this nonsingular spacetime. In Section 3, the influence of the nonsingularity parameter on the lensing in the weak and strong deflection limits is investigated, respectively. In Section 4, supposing that the gravitational field of the supermassive black hole at the center of our Milky Way can be described by the nonsingular metric, we estimate the numerical values of the coefficients and observables for gravitational lensing in the strong deflection limit. A brief discussion is given in Section 5.

#### 2. Null Geodesics and Photon Sphere

In [10], Hayward suggested that a nonsingular spacetime, as a minimal model, can be described by the metric where the metric function reads and its behavior is It is quite clear that the spacetime described by the above metric is similar to a Schwarzschild spacetime at large distance, while, at small distance, there is an effective cosmological constant, which leads to regularity at . The parameter is a new fundamental constant on the same ground as and . In order to keep some degree of generality, we consider as a free, model-dependent parameter. On the other hand, it is clear that when , there will be a spacetime singularity at . So, we can name as a nonsingularity parameter measuring the nonsingularity of a spacetime.

The outer and inner horizons are determined by . And, for the spacetime with large mass, we approximately have and . In order to compute the null geodesics in this nonsingular spacetime, we follow [64]. Here, we only restrict our attention to the equatorial orbits with . The Lagrangian is The generalized momentum can be defined from this Lagrangian as with its components given by Substituting (5) and (6) into (4), we find that the Lagrangian is independent of both and . Thus, we immediately get two integrals of the motion: and . Solving (5) and (6), we easily obtain the motion and motion: The Hamiltonian is given by Here is another integral of the motion. And are for spacelike, null, and timelike geodesics, respectively. Since we consider the null geodesics, we choose here. Then the radial motion can be expressed as with the effective potential . Then the circular geodesics satisfy Moreover, a stable (unstable) circular orbit requires (0), which admits a minimum (maximum) of the effective potential. Solving (12), we have And, for the metric (2), this equation reduces to where, for simplicity, we measure all quantities with the Schwarzschild radius, which is equivalent to putting in all equations. Solving (14), we can obtain the stable and unstable circular orbits for this nonsingular spacetime. For a spherically symmetric and static spacetime, the photon sphere is known as an unstable circular orbit of photon (other definitions can be found in [25, 65]). So, we can obtain the photon sphere for this nonsingular spacetime from (14) with the unstable condition . It is obvious that this relation (14) is quite different from that in the Schwarzschild black hole spacetime, which implies that, in the strong field limit, there exist some distinct effects of on the gravitational lensing. The stable circular orbit can also be got by imposing . The event horizons, photon sphere, and stable circular orbit are plotted in Figure 1 as a function of . It is clear that there are several distinct ranges of the parameter emerging where the structures of the horizons and circular geodesics will be qualitatively different; namely, , , , , and . In the following we will discuss these cases, respectively.