Strong Gravitational Lensing in a Charged Squashed Kaluza- Klein G\"{o}del Black hole

In this paper we investigate the strong gravitational lansing in a charged squashed Kaluza-Klein G\"{o}del black hole. The deflection angle is considered by the logarithmic term proposed by Bozza et al. Then we study the variation of deflection angle and its parameters $\bar{a}$ and $\bar{b}$ . We suppose that the supermassive black hole in the galaxy center can be considered by a charged squashed Kaluza-Klein black hole in a G\"{o}del background and by relation between lensing parameters and observables, we estimate the observables for different values of charge, extra dimension and G\"{o}del parameters.


Introduction
As we know the light rays or photons would be deviated from their straight way when they pass close to the massive object such as black holes. This deflection of light rays is known as gravitational lensing. This gravitational lensing is one of the applications and results of general relativity [1] and is used as an instrument in astrophysics, because it can help us to extract the information about stars. In 1924 Chwolson pointed out that when a star(source), a deflector(lens) and an observer are perfectly aligned, a ring-shape image of the star appears which is called 'Einstein ring'. Other studies have been led by Klimov, Liebes, refsdal and Bourassa and Kantowski [2]. Klimov investigated the lensing of galaxies by galaxies [3], but Liebes studied the lensing of stars by stars and also stars by clusters in our galaxy [4]. Refsdal showed that the geometrical optics can be used for investigating the gravitational lenses properties and time delay resulting from it [5,6]. The gravitational lensing has been presented in details in [7] and reviewed by some papers (see for examples [8]- [11]). At this stage, the gravitational lensing is developed for weak field limit and could not describe some phenomena such as looping of light rays near the massive objects. Hence, scientists started to study these phenomena from another point of view and they proposed gravitational lensing in a strong field limit. When the source is highly aligned with lens and the observer, one set of infinitive relativistic "ghost" images would be produce on each side of black hole. These images are produced when the light rays that pass very close to black hole, wind one or several times around the black hole before reaching to observer. At first, this phenomenon was proposed by Darwin [12] and revived in Refs. [13]- [15]. Darwin proposed a surprisingly easy formula for the positions of the relativistic images generated by a Schwarzschild black hole. Afterward several studies of null geodesics in strong gravitational fields have been led in literatures: a semi-analytical investigation about geodesics in Kerr geometry has been made in [16], also the appearance of a black hole in front of a uniform background was studied in Refs. [17,18]. Recently, Virbhadra and Ellis formulated lensing in the "strong field limit" and obtain the position and magnification of these images for the Schwarzschild black hole [19,20]. In Ref [21], by an alternative formulation, Frittelli, Kiling and Newman attained an exact lens equation, integral expressions for its solutions, and compared their result with Virbhadra and Ellis. Afterwards, the new method was proposed by Bozza et al. in which they revisited the schwarzschild black hole lensing by retaining the first two leading order terms [22]. This technic was used by Eiroa, Romero and Torres to study a Reissner-Nordstrom black hole [23] and Petters to calculate relativistic effects on microlensing events [24]. Finally, the generalization of Bozza's method for spherically symetric metric was developed in [25]. Bozza compared the image patterns for several interesting backgrounds and showed that by the separation of the first two relativistic images we can distinguish two different collapsed objects. Further development for other black holes can be found in [26]- [33]. Several interesting studeies are devoted to lensing by naked singularities [34,35], Janis-Newman-Winicour metric [25] and role of scaler field in gravitational lensing [36]. In Ref [36], Virbhadra et al. have considered a static and circularly symmetric lens characterized by mass and scalar charge parameter and investigated the lensing for different values of charge parameter. The gravitational lenses are important tools for probing the universe. Narasimha and Chitre predicted that the gravitational lening of dark matter can give the useful data about of position of dark matter in the universe [37,39]. Also in some papers gravitational lens is used to detect exotic objects in the universe, such as cosmic strings [40]- [42]. Recently, the idea of large extra dimensions has attracted much attention to construct theories in which gravity is unified with other forces [43]. One of the most interesting problems is the verification of extra dimensions by physical phenomena. For this purpose higherdimensional black holes in accelerators [44,45] and in cosmic rays [46]- [49] and gravitational waves from higher-dimensional black holes [50] are studied. The five-dimensional Einstein-Maxwell theory with a Chern-Simons term predicted five-dimensional charged black holes [51]. Such a higher-dimensional black holes would reside in a spacetime that is approximately isotropic in the vicinity of the black holes, but effectively four-dimensional far from the black holes [52]. These higher dimensional black holes are called Kaluza-Klein black holes. The presence of extra dimension is tested by quasinormal modes from the perturbation around the higher dimensional black hole [53]- [59] and the spectrum of Hawking radiation [60]- [63]. The gravitational lensing is another method to investigate the extra dimension. Thus, the study of strong gravitational lensing by higher dimensional black hole can help us to extract information about the extra dimension in astronomical observations in the future. The Kaluza-Klein black holes with squashed horizon [64] is one of the extra dimensional black holes and it's Hawking radiation and quasinormal modes have been investigated in some papers [65]- [68]. Also, the gravitational lensing of these black holes is studied in several papers. Liu [71]. One the other hands we know that our universe is rotational and it is reasonable to consider Gödel background for our universe. An exact solution for rotative universe was obtained by Gödel. He solved Einstein equation with pressureless matter and negative cosmological constant [72]. The solutions representing the generalization of the Gödel universe in the minimal five dimensional gauged supergravity are considered in many studies [74]- [80]. The properties of various black holes in the Gödel background are investigated in many works [?]. The strong gravitational lensing in a Squashed Kaluza-Klein Black hole in a Gödel universe is investigated in Ref. [70]. In this paper, we tudy the strong gravitational lensing in a charged squashed Kaluza-Klein Gödel black hole. In that case, we see the effects of the scale of the extra dimension, charge of black hole and Gödel parameter on the coefficients and observables of strong gravitational lensing. So, this paper is organized as follows: Section 2 is briefly devoted to charged squashed Kaluza-Klein Gödel black hole background. In section 3 we use the Bozza's method [26,27] to obtain the deflection angle and other parameters of strong gravitational lensing as well as variation of them with extra dimension, Gödel parameter and charge of black hole . In section 4, we suppose that the supermassive object at the center of our galaxy can be considered by the metric of charged squashed Kaluza-Klein Gödel black hole. Then, we evaluate the numerical results for the coefficients and observables in the strong gravitational lensing . In the last Section, we present a summary of our work.

The charged squashed Kaluza-Klein Gödel black hole metric
The charged squashed Kaluza-Klein Gödel black hole spacetime is given by [74], where σ 1 = cos ψ dθ + sin ψ sin θ dφ, σ 2 = − sin ψ dθ + cos ψ sin θ dφ, σ 3 = dψ + cos θ dφ. (2) (3) and 0 ≤ θ < π, 0 ≤ φ < 2π, 0 ≤ ψ < 4π and 0 < r < r ∞ . Here M and q are the mass and charge of the black hole respectively and j is the parameter of Gödel background. The killing horizon of the black hole is given by equation V (r) = 0 , where We see that the black hole has two horizons. As q −→ 0 the horizon of the squashed Kaluza-Klein Gödel black hole is recovered [70] and when q and j tend to zero, we have r 2 h = 2M, which is the horizon of five-dimensional Schwarzschild black hole. Here we note that the argument of square root constraints the mass, charge and Gödel parameter values. When r ∞ −→ ∞, we have k(r) −→ 1, which means that the squashing effect disappears and the five-dimensional charged black hole is recovered.
By using the transformations, ρ = ρ 0 ρ 0 (ρ 0 +ρq) 2 , the metric (1) can be written in the following form, with Where ρ h+ and ρ h− denote the outer and inner horizons of the black hole in the new coordinate and ρ 0 is a scale of transition from five-dimensional spacetime to an effective four-dimensional one. Here The Komar mass of black hole is related to ρ M with ρ M = 2G 4 M, where G 4 is the four dimensional gravitational constant. By using relations (4) and (7) we can obtain ρ h± in the following coupled equations, where when ρ q −→ 0, the horizon of black hole becomes [70].
In case of ρ q −→ 0 and j −→ 0, we have ρ h = ρ M which is consistent with neutral squashed Kaluza-Klein black hole [69]. You note that the square root in relation (9) constrains the values of ρ 0 , ρ q and j. In the case j = 0, the permissive regime is shown in Ref. [71]. For any value of ρ q there is allowed rang for ρ 0 . Hence these parameters can not select any value and when j increases from zero, permissive regime for ρ q and ρ 0 becomes more confined. We solved the above coupled equations numerically and results are shown in figure 1. We see that the outer horizon of black hole increases with the size of extra dimension, ρ 0 and decreases with j and ρ q . Note that two horizons of black hole coincide in especial values of parameters, which in that case we have an extremal black hole.

Geodesic equations, Deflection angle
In this section, we are going to investigate the deflection angle of light rays when they pass close to a charged squashed Kaluza-Klein Gödel black hole. We also study the effect of the charge parameter ρ q , the scale of extra dimension ρ 0 and Gödel parameter on the deflection angle and it's coefficients in the equatorial plane (θ = π/2). In this plane, the squashed Kaluza-Klein Gödel metric reduces to where The null geodesic equations are,ẍ where whereẋ is the tangent vector to the null geodesics and the dote denotes derivative with respect to affine parameter. We use equation (12) and obtain the following equations, where E, L φ and L ψ are constants of motion. Also, the θ-component of equation (12) in equatorial plane θ = π/2, is given by, Ifφ = 0, then deflection angle will be zero and this is illegal, So we set L ψ = D(ρ)ψ−H(ρ)ṫ = 0. By using equation (15) one can obtain following expression for the impact parameter, and the minimum of impact parameter takes place in photon sphere radius r ps , that is given by the root of following equation [81], Here ρ s is the closet approach for light ray and the prime is derivative with respect to ρ s . The analytical solution for the above equation is very complicated, so we calculated the equation (18) numerically. Variations of r photon sphere radius are plotted with respect to the charge ρ q , the scale of extra dimension ρ 0 and Gödel parameter in the figure 2. Also figure 3 shows variations of impact parameter in it's minimum value (at radius of photon sphere). These figures show that by adding the charge to the black hole, the behavior of the photon sphere radius and minimum of impact parameter is different compare with the neutral black hole [69]. As ρ 0 approaches to it's minimum values the radius of photon sphere and impact parameter become divergent. By using the chain derivative and equation (15), the deflection angle in the charged squashed Kaluza-Klein Gödel black hole can be written as, with When we decrease the ρ s (and consequently u) the deflection angle increases. At some points, the deflection angle exceeds from 2π so that the light ray will make a complete loop around the compact object before reaching at the observer. By decreasing ρ s further, the photon will wind several times around the black hole before emerging. Finally, for ρ s = ρ sp the deflection angle diverges and the photon is captured by the black hole. Moreover, from equations (20) and (21), we can find that in the Charged Squashed Kaluza-Klein Gödel black hole, both of the deflection angles depend on the parameters j, ρ 0 and ρ q , which implies that we could detect the rotation of universe, the extra dimension and charge of black hole in theory by gravitational lens. Note that the I φ (ρ s ) depend on j 2 , not j. It shows that the deflection angle is independent of the direction of rotation of universe. But, from equation (21) we find that the integral I ψ (ρ) contains the factor j, then the deflection angle α ψ (ρ s ) for the photon traveling around the lens in two opposite directions is different .The main reason is that the equatorial plan is parallel with Gödel rotation plan [70]. When j vanishes, the deflection angle of ψ tends zero [69,71]. We focus on the deflection angle in the φ direction, So we can rewrite the equation (20) as, with and where z = 1 − ρs ρ . The function R(z, ρ s ) is regular for all values of z and ρ s , while f (z, ρ s ) diverges as z approaches to zero. Therefore, we can split the integral (23) in two parts, the divergent part I D (ρ s ) and the regular one I R (ρ s ), which are given by, Here, we expand the argument of the square root in f (z, ρ s ) up to the second order in z [70], where (30) For ρ s > ρ ps , p(ρ s ) is nonzero and the leading order of the divergence in f 0 is z −1/2 , which have a finite result. As ρ s −→ ρ ps , p(ρ s ) approaches zero and divergence is of order z −1 , that makes the integral divergent. Therefor, the deflection angle can be approximated in the following form [25], By using (31) and (32), we can investigate the properties of strong gravitational lensing in the charged squashed Kaluza-Klein Gödel black hole. In this case, variations of the coefficientsā andb, and the deflection angle α have been plotted with respect to the extra dimension ρ 0 , charge of the black hole ρ q , and Gödel parameter j in figures 4-6.
As j tends to zero, these quantities reduce to charged squashed Kaluza-klein black hole [71] and with ρ q = 0 the squashed Kaluza-klein black hole recovers [69]. One can see that the deflection angle increases with extra dimension and decreases with ρ q . By comparing these parameters with those in four-dimensional Schwarzschild and Reissner-Nordström black holes , we could extract information about the size of extra dimension as well as the charge of the black hole by using strong field gravitational lensing.

Observables estimation
In the previous section, we investigated the strong gravitational lensing by using a simple and reliable logarithmic formula for deflection angle, which was obtain by Bozza et al. Now, by using relations between the parameters of the strong gravitational lensing and  observables, estimat the position and magnification of the relativistic images. By comparing these observables with the data from the astronomical observation, we could detect properties of an massive object. We suppose that the spacetime of the supermassive object at the galaxy center of Milky Way can be considered as a charged squashed Kaluza-Klein Gödel black hole, then we can estimate the numerical values for observables. We can write the lens equation in strong gravitational lensing, as the source, lens, and observer are highly aligned as follows [22], where D LS is the distance between the lens and source. D OS is the distance between the observer and the source so that, D OS = D LS + D OL . β and θ are the angular position of the source and the image with respect to lens, respectively. ∆α n = α − 2nπ is the offset of deflection angle with integer n which indicates the n-th image. The n-th image position θ n and the n-th image magnification µ n can be approximated as follows [22,25], θ 0 n is the angular position of α = 2nπ. In the limit n −→ ∞, the relation between the minimum of impact parameter u ps and asymptotic position of a set of images θ ∞ can be expressed by u ps = D OL θ ∞ . In order to obtain the coefficientsā andb, in the simplest case, we separate the outermost image θ 1 and all the remaining ones which are packed together at θ ∞ , as done in Refs [22,25]. Thus s = θ 1 − θ ∞ is considered as the angular separation between the first image and other ones and the ratio of the flux of them is given by, We can simplify the observables and rewrite them in the following form [22,25], Thus, by measuring the s, R and θ ∞ , one can obtain the values of the coefficientsā,b and u sp . If we compare these values by those obtained in the previous section, we could detect the size of the extra dimension, charge of black hole and rotation of universe. Another observable for gravitational lensing is relative magnification of the outermost relativistic image with the other ones. This observable is shown by r m which is related to R by,   Figure 9: The variation of angular position θ ∞ with respect to j, ρ 0 and ρ q that is given in µarcseconds.
Using θ ∞ = usp D OL and equations (32), (37) and (38) we can estimate the values of the observable in the strong field gravitational lensing. The variation of the observables θ ∞ , s and r m are plotted in figures 7-9. Note that the mass of the central object of our galaxy is estimated to be 4.31 × 10 6 M ⊙ and the distance between the sun and the center of galaxy is D OL = 8.5 kpc [82]. For different ρ 0 , ρ q and j, the numerical values for the observables are listed in Table 1. One can see that our results reduce to those in the four-dimensional Schwarzschild black hole as ρ 0 −→ 0. Also our results are in agreement with the results of Ref. [70] in the limit ρ q −→ 0 and in the limit j −→ 0, the results of Ref. [71] are recovered.

Summary
The light rays can be deviated from the straight way in the gravitational field as predicted by General Relativity. This deflection of light rays is known as gravitational lensing. In the strong field limit, the deflection anglethe of the light rays which pass very close to the black hole, becomes so large that, it winds several times around the black hole before appearing at the observer. Therefore the observer would detect two infinite set of faint relativistic images produced on each side of the black hole. On the other hand the extra dimension is one of the important predictions in the string theory which is believed to be a promising candidate for the unified theory. Also it is reasonable to consider a rotative universe with global rotation. Hence the five-dimensional Einstein-Maxwell theory with a Chern-Simons term in string theory predicted five-dimensional charged black holes in the Gödel background. In our study, we considered the charged squashed Kaluza-Klein Gödel black hole spacetime and investigated the strong gravitational lensing by this metric. We obtained theoretically the deflection angle and other parameters of strong gravitational lensing . Finally, we suppose that the supermassive black hole at the galaxy center of Milky Way can be considered by this spacetime and we estimated numerically the values of observables that are realated to the lensing parameters. Theses observable parameters are θ ∞ , s and R, where θ ∞ is the position of relativistic images, s angular separation between the first image θ 1 and other ones θ ∞ and R is the ratio of the flux from the first image and those from all the other images. Our results are presented in figures 1-9 and Table 1. By comparatione observable parameters with observational data measured by the astronomical instruments in the future, we can discuss the properties of the massive object in the center of our galaxy.