Null Geodesics and Gravitational Lensing in a Nonsingular Spacetime

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 q 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.

Shao-Wen Wei 1 and Yu-Xiao Liu 1 and Chun-E Fu 2

Academic Editor:Elias C. Vagenas

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; 13 January 2015

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 SCOAP³ .

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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . And, according to it, the spacetime is classified into the nonsingular black hole [figure omitted; refer to PDF] , the extremal nonsingular black hole [figure omitted; refer to PDF] , the weakly nonsingular spacetime [figure omitted; refer to PDF] , the marginally nonsingular spacetime [figure omitted; refer to PDF] , and the strongly nonsingular spacetime [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] on the lensing is negligible. Compared with it, [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] where the metric function [figure omitted; refer to PDF] reads [figure omitted; refer to PDF] and its behavior is [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . The parameter [figure omitted; refer to PDF] is a new fundamental constant on the same ground as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In order to keep some degree of generality, we consider [figure omitted; refer to PDF] as a free, model-dependent parameter. On the other hand, it is clear that when [figure omitted; refer to PDF] , there will be a spacetime singularity at [figure omitted; refer to PDF] . So, we can name [figure omitted; refer to PDF] as a nonsingularity parameter measuring the nonsingularity of a spacetime.

The outer and inner horizons [figure omitted; refer to PDF] are determined by [figure omitted; refer to PDF] . And, for the spacetime with large mass, we approximately have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] . The Lagrangian is [figure omitted; refer to PDF] The generalized momentum can be defined from this Lagrangian as [figure omitted; refer to PDF] with its components given by [figure omitted; refer to PDF] Substituting (5) and (6) into (4), we find that the Lagrangian [figure omitted; refer to PDF] is independent of both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus, we immediately get two integrals of the motion: [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Solving (5) and (6), we easily obtain the [figure omitted; refer to PDF] motion and [figure omitted; refer to PDF] motion: [figure omitted; refer to PDF] The Hamiltonian is given by [figure omitted; refer to PDF] Here [figure omitted; refer to PDF] is another integral of the motion. And [figure omitted; refer to PDF] are for spacelike, null, and timelike geodesics, respectively. Since we consider the null geodesics, we choose [figure omitted; refer to PDF] here. Then the radial motion can be expressed as [figure omitted; refer to PDF] with the effective potential [figure omitted; refer to PDF] . Then the circular geodesics satisfy [figure omitted; refer to PDF] Moreover, a stable (unstable) circular orbit requires [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] 0), which admits a minimum (maximum) of the effective potential. Solving (12), we have [figure omitted; refer to PDF] And, for the metric (2), this equation reduces to [figure omitted; refer to PDF] where, for simplicity, we measure all quantities with the Schwarzschild radius, which is equivalent to putting [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] on the gravitational lensing. The stable circular orbit can also be got by imposing [figure omitted; refer to PDF] . The event horizons, photon sphere, and stable circular orbit are plotted in Figure 1 as a function of [figure omitted; refer to PDF] . 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, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . In the following we will discuss these cases, respectively.

Figure 1: Horizons and circular orbits for the nonsingular spacetime. The thin solid and dashed lines are for the outer and inner horizons, respectively. And the thick solid and dashed lines are for the photon sphere and stable circular orbit, respectively. The parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Case 1 (nonsingular black hole [figure omitted; refer to PDF] ).

Since [figure omitted; refer to PDF] , the horizon locates at [figure omitted; refer to PDF] . And note that the local minimum and maximum of [figure omitted; refer to PDF] correspond to the stable and unstable circular orbits, respectively. Then we can easily read the property of different cases from Figure 2.

For the case of a nonsingular black hole with [figure omitted; refer to PDF] , there are two horizons [figure omitted; refer to PDF] , one photon sphere [figure omitted; refer to PDF] , and one stable circular orbit [figure omitted; refer to PDF] . And they satisfy the following relation: [figure omitted; refer to PDF] This implies that the outer horizon is always covered by a photon sphere, and the stable circular orbit locates between these two horizons.

In Figure 2(a), the general behavior of the effective potential [figure omitted; refer to PDF] is shown as a function of [figure omitted; refer to PDF] for different values of the angular momentum [figure omitted; refer to PDF] . We find that the effective potential admits two zeros corresponding to the outer and inner horizons, as well as one maximum and one minimum corresponding to the photon sphere and stable circular orbit. We can also see that the local minimum point always lies between these two zeros indicating the stable circular orbit lies in the region between the two horizons.

Behavior of the effective potential [figure omitted; refer to PDF] as a function of [figure omitted; refer to PDF] with different values of [figure omitted; refer to PDF] .

(a) The nonsingular black hole

[figure omitted; refer to PDF]

(b) The extremal nonsingular black hole

[figure omitted; refer to PDF]

(c) The weakly nonsingular spacetime

[figure omitted; refer to PDF]

(d) The marginally nonsingular spacetime

[figure omitted; refer to PDF]

(e) The strongly nonsingular spacetime

[figure omitted; refer to PDF]

Case 2 (extremal nonsingular black hole [figure omitted; refer to PDF] ).

For this case, we have the following relation: [figure omitted; refer to PDF] The first " [figure omitted; refer to PDF] " means that the two horizons coincide with each other. This case corresponds to an extremal nonsingular black hole. The second " [figure omitted; refer to PDF] " implies that the degenerate horizon is also a stable circular orbit against small perturbation.

The detailed behavior of the effective potential [figure omitted; refer to PDF] is presented in Figure 2(b). It shows that the effective potential has one zero and one minimum located at the same point. It also admits a maximum corresponding to the photon sphere.

Case 3 (weakly nonsingular spacetime [figure omitted; refer to PDF] ).

For the case of [figure omitted; refer to PDF] , we clearly see from Figure 1 that the horizon disappears. So this case describes a nonsingular spacetime without a black hole. Obviously, the radius of the photon sphere and stable circular orbit satisfy [figure omitted; refer to PDF] This tells that the stable circular orbit is always covered by a photon sphere. The effective potential [figure omitted; refer to PDF] is plotted in Figure 2(c), from which we find that [figure omitted; refer to PDF] is positive for all [figure omitted; refer to PDF] 's. Thus, the horizons do not exist in this case, which is consistent with the result from Figure 1. On the other hand, there are one minimum and one maximum corresponding to the stable circular orbit and photon sphere, respectively.

Case 4 (marginally nonsingular spacetime [figure omitted; refer to PDF] ).

For this case, the horizon also disappears. And the stable and unstable circular orbits coincide with each other; that is, [figure omitted; refer to PDF] . For this circular orbit, we have [figure omitted; refer to PDF] . This result can be seen from Figure 2(d). One the other hand, we will see in the next section that some strong deflection limit coefficients will diverge in this case caused by [figure omitted; refer to PDF] .

Case 5 (strongly nonsingular spacetime [figure omitted; refer to PDF] ).

For the last case, all the horizons, photon sphere, and the stable circular orbit disappear. The effective potential [figure omitted; refer to PDF] is plotted in Figure 2(e). It monotonically decreases from infinity to zero as [figure omitted; refer to PDF] goes from 0 to [figure omitted; refer to PDF] .

From the above discussion, we find that the photon sphere of a nonsingular black hole is always larger than that of a weakly nonsingular spacetime. Therefore, we come to the conclusion that a photon is more easily captured by a nonsingular black hole. Since the photon sphere disappears for a strongly nonsingular spacetime, we will not focus on the lensing for a strongly nonsingular spacetime.

3. Lensing in Nonsingular Spacetime

In this section, we will study the lensing in nonsingular spacetime. The influence of nonsingularity parameter [figure omitted; refer to PDF] on the position and magnification of the nonrelativistic and relativistic images will be analyzed.

3.1. The Deflection Angle and Lens Geometry

Taking into account the spherical symmetry of this spacetime, we just consider the case that both the observer and the source lie in the equatorial plane ( [figure omitted; refer to PDF] ), resulting in the fact that the whole trajectory of the photon is also restricted in the same plane. Then the deflection angle for the photon coming from infinity and returning to infinity is [figure omitted; refer to PDF] with the total azimuthal angle given by [figure omitted; refer to PDF] where (9) and (11) are used and [figure omitted; refer to PDF] is set to one. In a black hole spacetime, the deflection angle [figure omitted; refer to PDF] is a monotonically decreasing function with [figure omitted; refer to PDF] . So it is easy to imagine that the light ray can make a complete loop or more than one loop before reaching the observer. As other authors have pointed out, the value of the deflection angle will be unboundedly large when the photon sphere is reached.

Defining [figure omitted; refer to PDF] , the total azimuthal angle can be expressed as [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

Here we would like to give a brief study on the lens geometry. The lens configuration is supposed in which the black hole is situated between the light source and observer; both of them are far from the black hole lens, so that the gravitational fields are very weak and the spacetime there can be described by a flat metric. An important element of the lens geometry is the optical axis, which is defined as the line joining the observer and the lens. The angular positions of the source (S) and the images (I), seen from the observer, are denoted by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then the lens equation reads [25] [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the observer-lens, lens-source, and observer-source distances, respectively. It was pointed out by Bozza [66] that these distances are not the true distances between different positions; however, if the source is very close to the optical axis, they are a reasonably good approximation.

3.2. Nonrelativistic Images

Let us first consider the lensing in the weak deflection limit, where photon has a large impact parameter; that is, [figure omitted; refer to PDF] . Then the function [figure omitted; refer to PDF] in (21) can perform a Taylor expansion around [figure omitted; refer to PDF] , which is given by [figure omitted; refer to PDF] Thus, the deflection angle is calculated as [figure omitted; refer to PDF] with the coefficients [figure omitted; refer to PDF] Note that the nonsingularity parameter [figure omitted; refer to PDF] only affects the deflection angle in the fourth order. Since [figure omitted; refer to PDF] has a large value, the influence of [figure omitted; refer to PDF] on the nonrelativistic images is very weak. Restoring the dimension, the first three coefficients are exactly consistent with that of [67].

For high alignment and using lens equation (22), the image positions and magnifications in the weak deflection limit can be written as a series expansion of the form [67] [figure omitted; refer to PDF] where the expansion parameter [figure omitted; refer to PDF] denotes the angle subtended by the gravitational radius normalized by the angular Einstein radius. It is easy to check that the coefficients [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are independent of [figure omitted; refer to PDF] , and their forms can be found in [67]. Thus, the nonsingularity parameter [figure omitted; refer to PDF] only affects the position and magnification of images of more than the third order in [figure omitted; refer to PDF] . As a result, we come to the conclusion that the influence of the nonsingularity parameter can be ignored in the weak deflection limit.

3.3. Relativistic Images

In this case, the spacetime always has a photon sphere. And a photon before reaching the observer can do many loops around the black hole; therefore the photon should pass very near the photon sphere. Adopting the method developed by Bozza et al., we define a variable [31] [figure omitted; refer to PDF] For the photon at infinity [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . And when [figure omitted; refer to PDF] , one easily gets [figure omitted; refer to PDF] . Then the total azimuthal angle (20) can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . Note that the function [figure omitted; refer to PDF] is regular for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , while [figure omitted; refer to PDF] diverges at [figure omitted; refer to PDF] . So, we split the integral (28) into two parts: [figure omitted; refer to PDF] where the regular and divergent parts [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are, respectively, given by [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . In order to find the divergence of the integrand, we do a Taylor expansion of the function inside the square root of [figure omitted; refer to PDF] and obtain the function [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where the coefficients [figure omitted; refer to PDF] and [figure omitted; refer to PDF] read [figure omitted; refer to PDF] We find that when [figure omitted; refer to PDF] approaches [figure omitted; refer to PDF] , the coefficient [figure omitted; refer to PDF] vanishes, and the leading term of the divergence in [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , which would lead to the logarithmic divergence of the integrand. Thus, near [figure omitted; refer to PDF] , the deflection angle can be assumed in the form [figure omitted; refer to PDF] Under this assumption, the minimum impact parameter [figure omitted; refer to PDF] and the strong deflection limit coefficients [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] where [figure omitted; refer to PDF] For this nonsingular spacetime, the coefficient [figure omitted; refer to PDF] cannot be calculated analytically. In order to obtain it, we expand [figure omitted; refer to PDF] around [figure omitted; refer to PDF] : [figure omitted; refer to PDF] Therefore, ignoring the higher-order terms, we get [figure omitted; refer to PDF] With this equation, we can numerically calculate [figure omitted; refer to PDF] . It is worth pointing out that this result is accurate for the case [figure omitted; refer to PDF] , while it is invalid for [figure omitted; refer to PDF] . From these strong deflection limit coefficients, one easily sees that there is a significant effect of the nonsingularity parameter [figure omitted; refer to PDF] on the strong gravitational lensing. When [figure omitted; refer to PDF] , these parameters will reduce to the case of the Schwarzschild black hole [31, 36].

The behaviors of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are presented in Figure 3. The result shows that the minimum impact parameter has a similar behavior as the radius of the photon sphere. We can also find that the strong deflection limit coefficient [figure omitted; refer to PDF] grows with [figure omitted; refer to PDF] , while [figure omitted; refer to PDF] decreases. For a nonsingular black hole, both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a finite value. Compared with the result of nonsingular black hole, in the weakly nonsingular spacetime, it has small values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , while it has large value of [figure omitted; refer to PDF] . In particular, [figure omitted; refer to PDF] goes to positive infinity and [figure omitted; refer to PDF] goes to negative infinity when the nonsingularity parameter [figure omitted; refer to PDF] approaches [figure omitted; refer to PDF] , where the marginally nonsingular spacetime arrives. With the values of these coefficients, one can obtain the behavior of the deflection angle. For fixed impact parameter [figure omitted; refer to PDF] , the deflection angle increases with [figure omitted; refer to PDF] shown in Figure 4(a). And, in Figure 4(b), it is shown that [figure omitted; refer to PDF] decreases with [figure omitted; refer to PDF] for fixed [figure omitted; refer to PDF] . However, the angle will be unboundedly large when [figure omitted; refer to PDF] , which corresponds to the case that [figure omitted; refer to PDF] approaches [figure omitted; refer to PDF] .

Behaviors of the minimal impact parameter [figure omitted; refer to PDF] and the strong deflection limit coefficients [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(a) The deflection angle [figure omitted; refer to PDF] as a function of [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] from the bottom to top. (b) The deflection angle [figure omitted; refer to PDF] as a function of [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , 0.3, and 0.42 from the bottom to top.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Here, we consider that the source, lens, and observer are highly aligned; that is, [figure omitted; refer to PDF] ; lens equation (22) is reduced to [31, 36] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denotes the number of the loops that the photon did around the lens. Using the lens geometry, we can obtain the angular position and magnification of the images [31, 36]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Since the magnification [figure omitted; refer to PDF] , the first image is the brightest one. We can also see that [figure omitted; refer to PDF] is proportional to the small quantity [figure omitted; refer to PDF] , which leads to faint images. Thus, we have the conclusion that, for nonzero [figure omitted; refer to PDF] , the first image is the brightest one among these relativistic images and its brightness decreases quickly with the distance [figure omitted; refer to PDF] . On the other hand, it is worth noting that when [figure omitted; refer to PDF] , the magnification (41) will be no longer valid.

4. Numerical Estimation of the Observables for the Supermassive Galactic Black Hole Lensing

In this section, we first introduce three observables and then estimate the numerical values for the observables of gravitational lensing in the strong field limit.

Consider the case that the outermost image with angular position [figure omitted; refer to PDF] is a single image and the others are packed together at [figure omitted; refer to PDF] . Then we have the observables [36]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] measures the angular separation between the first image and other ones and [figure omitted; refer to PDF] denotes the flux of the first image and the sum of the others.

Next, we would like to estimate the numerical values for the observables of gravitational lensing in the strong field limit by supposing that the gravitational field of the supermassive black hole at the center of our Milky Way can be described by the nonsingular metric (1), and the gravitational field near the light source and the observer is very weak so that it can be described by a flat metric. The mass of the supermassive black hole and the distance between the observer and the black hole are estimated to be [figure omitted; refer to PDF] and [figure omitted; refer to PDF] kpc with [figure omitted; refer to PDF] being the mass of the sun. Under this assumption, the angular position can be calculated with the relation [figure omitted; refer to PDF] From this equation, we get that the angular position [figure omitted; refer to PDF] depends on the parameter [figure omitted; refer to PDF] and mass [figure omitted; refer to PDF] determined by the nature of the gravitational lens and also on the observer-lens distance [figure omitted; refer to PDF] determined by the lens geometry. In order to obtain a large value of [figure omitted; refer to PDF] for a lens of the same type, it should have large mass and small [figure omitted; refer to PDF] . For different values of the nonsingularity parameter [figure omitted; refer to PDF] , the numerical values for the strong deflection limit coefficients and observables are listed in Table 1. It is clear that these results reduce to the Schwarzschild black hole spacetime when [figure omitted; refer to PDF] . Moreover, we found that as [figure omitted; refer to PDF] increases, the angular position of the relativistic images [figure omitted; refer to PDF] and the relative magnitudes [figure omitted; refer to PDF] decrease, while the angular separation [figure omitted; refer to PDF] increases.

Table 1: Numerical estimation for main observables and the strong field limit coefficients for a Schwarzschild black hole, a nonsingular black hole, and a weakly nonsingular spacetime, which is supposed to describe the object at the center of our galaxy; [figure omitted; refer to PDF] .

	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	1.0000	1.0402	1.1094	1.2625	1.3771	1.6277
[figure omitted; refer to PDF]	-0.4002	-0.4446	-0.5369	-0.8112	-1.0782	-1.8353
[figure omitted; refer to PDF]	2.5981	2.5659	2.5209	2.4537	2.4217	2.3806
[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] arcsec)	0.0211	0.0259	0.0350	0.0578	0.0750	0.1055
[figure omitted; refer to PDF] (magnitude)	6.8219	6.5583	6.1490	5.4036	4.9539	4.1911
[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] arcsec)	16.8699	16.6610	16.3688	15.9321	15.7243	15.4580

The behaviors of the observables can also be found in Figure 5. We can find that, in the weakly nonsingular spacetime, the angular position [figure omitted; refer to PDF] of the relativistic images [figure omitted; refer to PDF] and the relative magnitudes [figure omitted; refer to PDF] decrease more quickly, and the angular separation [figure omitted; refer to PDF] increases more rapidly than the case of the nonsingular black hole. It is also clear that the angular position has a maximum value [figure omitted; refer to PDF] [figure omitted; refer to PDF] at [figure omitted; refer to PDF] . Compared with a Schwarzschild black hole or a nonsingular black hole, a weakly nonsingular spacetime has a smaller angular position of the relativistic images and relative magnification of the outermost relativistic image with the other relativistic images. However, it has a larger angular separation for these relativistic images. From Figure 5(a), we find that the numerical value for the angular position of the innermost relativistic images [figure omitted; refer to PDF] of a nonsingular spacetime is of about 15.3 [figure omitted; refer to PDF] 16.8 [figure omitted; refer to PDF] . In principle, such a resolution is reachable by very long baseline interferometry (VLBI) projects and advanced radio interferometry between space and Earth (ARISE), which have the angular resolution of 10 [figure omitted; refer to PDF] 100 [figure omitted; refer to PDF] in the near infrared [68, 69]. Therefore we are hopeful to observe these relativistic images within a not so far future.

Gravitational lensing by the galactic center black hole. Variation of the values of the angular position [figure omitted; refer to PDF] , angular separation [figure omitted; refer to PDF] , and relative magnitudes [figure omitted; refer to PDF] with the parameter [figure omitted; refer to PDF] in the nonsingular spacetime; [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

5. Summary

In this paper, we have shown that, in the weak deflection limit, the influence of the nonsingularity parameter [figure omitted; refer to PDF] on the gravitational lensing is negligible. However, in the strong deflection limit, [figure omitted; refer to PDF] has a significant effect, which may offer a potentially powerful tool to probe the nonsingularity of spacetime.

First, we investigated the null geodesics of a nonsingular spacetime. According to the nature of the nonsingular spacetime, it is classified into several cases, such as the nonsingular black hole, the extremal nonsingular black hole, the weakly nonsingular spacetime, the marginally nonsingular spacetime, and the strongly nonsingular spacetime. We found that the photon sphere can exist not only in a nonsingular black hole background but also in a spacetime without a black hole. The result also shows that the photon is more easily captured by a nonsingular black hole rather than by a weakly nonsingular spacetime.

Second, based on the null geodesics, lensing in the weak and strong deflection limits was studied. For the first case, we found that the influence of the nonsingularity parameter [figure omitted; refer to PDF] on the positions and magnifications of the images is negligible. Thus, we cannot distinguish a nonsingular black hole from a Schwarzschild one. In the strong deflection limit, these strong deflection limit coefficients were also obtained. For a nonsingular black hole, these coefficients are always finite for any value of the nonsingularity parameter [figure omitted; refer to PDF] . Compared with the result of nonsingular black hole, in the weakly nonsingular spacetime, it has small values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , while it has large value of [figure omitted; refer to PDF] . And when the nonsingular spacetime approaches the marginally nonsingular one with [figure omitted; refer to PDF] , we found that [figure omitted; refer to PDF] goes to positive infinity and [figure omitted; refer to PDF] goes to negative infinity. These results tell that the gravitational lensing by a weakly nonsingular spacetime is more obvious than a nonsingular black hole.

The model was also applied to the supermassive black hole hosted in the center of our Milky Way. It was shown that, with the increase of the nonsingularity parameter [figure omitted; refer to PDF] , the angular position of the relativistic images [figure omitted; refer to PDF] and the relative magnitudes [figure omitted; refer to PDF] decrease, while the angular separation [figure omitted; refer to PDF] increases. We found that, in a weakly nonsingular spacetime, the angular position of the relativistic images [figure omitted; refer to PDF] and relative magnitudes [figure omitted; refer to PDF] decrease more quickly, and the angular separation [figure omitted; refer to PDF] increases more rapidly than a nonsingular black hole.

As pointed out in [27], the relative magnitude [figure omitted; refer to PDF] obtained in this paper may have a large difference from its accurate value. However, the angular position [figure omitted; refer to PDF] and the angular separation [figure omitted; refer to PDF] are accurate enough. Thus, combined with the expected data from VLBI and ARISE, the strong gravitational lensing may offer a possible way to distinguish a weakly nonsingular spacetime from a nonsingular black hole. Furthermore, we are also allowed to probe the spacetime nonsingularity parameter [figure omitted; refer to PDF] by the astronomical instruments in the near future astronomical observations.

Acknowledgments

The authors would like to thank the anonymous referees whose comments largely helped them in improving the original paper. Shao-Wen Wei wishes to thank Dr. Changqing Liu for useful discussions and valuable comments. This work was supported by the National Natural Science Foundation of China (Grants nos. 11205074, 11075065, and 11375075) and the Fundamental Research Funds for the Central Universities (Grants nos. lzujbky-2013-21 and lzujbky-2013-18).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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