# Strong Gravitational Lensing in a Charged Squashed Kaluza- Klein Gödel Black
hole

J. Sadeghi a and H. Vaez a
a _Physics Department, Mazandaran University_ ,
_P.O.Box 47416-95447, Babolsar, Iran_
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we investigate the strong gravitational lansing in a charged
squashed Kaluza-Klein Gödel 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 Gödel background and by relation
between lensing parameters and observables, we estimate the observables for
different values of charge, extra dimension and Gödel parameters.

PACS numbers: 95.30.sf, 04.70.-s, 98.62.sb
Keywords: Gravitational lensing; Charged Squashed Kaluza-Klein Gödel Black
hole

##  1 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 et al. have
studied the gravitational lensing by squashed Kaluza-Klein black holes in Refs
[69, 70] and Sadeghi et al. investigated the charged type of this black hole
[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 [gghghghhg]. 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.

##  2 The charged squashed Kaluza- Klein Gödel black hole metric

The charged squashed Kaluza- Klein Gödel black hole spacetime is given by
[74],

$ds^{2}=-f(r)dt^{2}+\frac{k^{2}(r)}{V(r)}dr^{2}-2g(r)\sigma_{3}dt+h(r)\sigma^{2}_{3}+\frac{r^{2}}{4}[k(r)(\sigma^{2}_{1}+\sigma^{2}_{2})+\sigma^{2}_{3}],$
(1)

where

$\displaystyle\sigma_{1}=\cos\psi\,d\theta+\sin\psi\,\sin\theta\,d\phi,$
$\displaystyle\sigma_{2}=-\sin\psi\,d\theta+\cos\psi\,\sin\theta\,d\phi,$
$\displaystyle\sigma_{3}=d\psi+\cos\theta\,d\phi.$ (2) $\displaystyle
f(r)=1-\frac{2M}{r^{2}}+\frac{q^{2}}{r^{4}},\,\,\,\,\,\,\,g(r)=j(r^{2}+3q),\,\,\,\,\,h(r)=-j^{2}r^{2}(r^{2}+2M+6q),\,\,\,\,\,\,$
$\displaystyle
V(r)=1-\frac{2M}{r^{2}}+\frac{16j^{2}(M+q)(M+2q)}{r^{2}}+\frac{q^{2}(1-8j^{2}(M+3q))}{r^{4}},\,\,\,\,\,\,k(r)=\frac{V(r_{\infty})r_{\infty}^{4}}{(r^{2}-r_{\infty}^{2})^{2}}.$
(3)

and $0\leq\theta<\pi$, $0\leq\phi<2\pi$, $0\leq\psi<4\pi$ and
$0<r<r_{\infty}$. 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

$\displaystyle
r_{h}^{2}=M-8j^{2}(M+q)(M+2q)\pm\sqrt{[M+q-8j^{2}(M+2q)^{2}][M-q-8j^{2}(M+q)^{2}]}.$
(4)

We see that the black hole has two horizons. As $q\longrightarrow 0$ the
horizon of the squashed Kaluza- Klein Gödel black hole is recovered [70] and
when $q$ and $j$ tend to zero, we have $r_{h}^{2}=2M$, 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
$r_{\infty}\longrightarrow\infty$, we have $k(r)\longrightarrow 1$, which
means that the squashing effect disappears and the five-dimensional charged
black hole is recovered.
By using the transformations,
$\rho=\rho_{0}\frac{r^{2}}{r^{2}_{\infty}-r^{2}}$,
$\tau=\sqrt{\frac{\rho_{0}(1+\alpha)}{\rho_{0}+\rho_{M}}}t$ and
$\alpha=\frac{\rho_{q}^{2}(\rho_{0}+\rho_{M})}{\rho_{0}(\rho_{0}+\rho_{q})^{2}}$,
the metric (1) can be written in the following form,

$ds^{2}=-\mathcal{F}(\rho)d\tau^{2}+\frac{K(\rho)}{\mathcal{G}(\rho)}d\rho^{2}+\mathcal{C}(\rho)(d\theta^{2}+sin^{2}\theta\,d\phi^{2})-2H(\rho)\sigma_{3}d\tau+\mathcal{D}(\rho)\sigma_{3}^{2},$
(5)
$\displaystyle\mathcal{F}(\rho)=1-\frac{\rho_{M}-2\alpha\rho_{0}}{(1+\alpha)\rho_{M}}(\frac{\rho_{M}}{\rho})+\frac{\alpha}{1+\alpha}(\frac{\rho_{0}}{\rho_{M}})^{2}(\frac{\rho_{M}}{\rho})^{2},$
$\displaystyle
K(\rho)=1+\frac{\rho_{0}}{\rho}\,,\,\,\,\,\,\mathcal{G}=(1-\frac{\rho_{h+}}{\rho})(1-\frac{\rho_{h-}}{\rho}),$
$\displaystyle\mathcal{C}(\rho)=\rho^{2}K(\rho),\,\,\,\,H(\rho)=jr_{\infty}^{2}\left(\frac{1}{K(\rho)}+\frac{3\rho_{q}}{\rho_{0}+\rho_{q}}\right)\sqrt{\frac{\rho_{0}+\rho_{M}}{\rho_{0}(1+\alpha)}},$
$\displaystyle\mathcal{D}(\rho)=\frac{r^{2}_{\infty}}{4K(\rho)}-\frac{j^{2}\rho
r_{\infty}^{2}}{(\rho+\rho_{0})^{2}(\rho_{M}+\rho_{0})(\rho_{q}+\rho_{0})}\times$
$\displaystyle\left\\{\rho[\rho_{0}(\rho_{0}+2\rho_{M})+7\rho_{0}\rho_{q}+8\rho_{M}\rho_{q}]+\rho_{0}[\rho_{0}(\rho_{M}+6\rho_{q})+7\rho_{M}\rho_{q}]\right\\},$
(6)

with

$\displaystyle\rho_{M}=\rho_{0}\frac{2M}{r^{2}_{\infty}-2M}\,\,,\,\,\,\,\,\,\,\rho_{q}=\rho_{0}\frac{q}{r^{2}_{\infty}-q},\,\,\,\,\,\,\rho_{h\pm}=\rho_{0}\frac{r_{h\pm}^{2}}{r^{2}_{\infty}-r_{h\pm}^{2}}\,.$
(7)

Figure 1: The plots show the variation of horizon radiuses with respect to
$j$, $\rho_{0}$ and $\rho_{q}$ (Note that in each figure, for $\rho_{q}\neq
0$, two horizons merge at a point. This point has been shown for one of
figures. )

Where $\rho_{h+}$ and $\rho_{h-}$ denote the outer and inner horizons of the
black hole in the new coordinate and $\rho_{0}$ is a scale of transition from
five-dimensional spacetime to an effective four-dimensional one. Here
$\rho_{0}^{2}=\frac{r^{2}_{\infty}}{4}V(r_{\infty})$, so that
$r^{2}_{\infty}=4(\rho_{0}+\rho_{h+})(\rho_{0}+\rho_{h-})$. The Komar mass of
black hole is related to $\rho_{M}$ with $\rho_{M}=2G_{4}M$, where $G_{4}$ is
the four dimensional gravitational constant. By using relations (4) and (7) we
can obtain $\rho_{h\pm}$ in the following coupled equations,

$\displaystyle
2\left[\rho_{0}(\rho_{h+}+\rho_{h-})+2\rho_{h+}\rho_{h-}\right]={a}(\rho_{h+},\rho_{h-}),$
$\displaystyle
2\left[\rho_{0}(\rho_{h+}-\rho_{h-})\right]={b}(\rho_{h+},\rho_{h-}),$ (8)

where

$\displaystyle
a=\frac{\rho_{M}r_{\infty}^{2}}{2(\rho_{0}+\rho_{M})}-2j^{2}r_{\infty}^{4}\frac{\left(\rho_{M}\rho_{0}+3\rho_{M}\rho_{0}+2\rho_{q}\rho_{0}\right)\left(\rho_{M}\rho_{0}+5\rho_{M}\rho_{q}+4\rho_{q}\rho_{0}\right)}{(\rho_{0}+\rho_{M})^{2}(\rho_{0}+\rho_{q})^{2}},$
$\displaystyle
b=\left\\{a^{2}-4\frac{\rho_{q}^{2}r_{\infty}^{4}}{(\rho_{0}+\rho_{q})^{2}}\left(1-4j^{2}r_{\infty}^{2}\frac{(\rho_{0}\rho_{M}+7\rho_{M}\rho_{q}+6\rho_{q}\rho_{0})}{(\rho_{0}+\rho_{M})(\rho_{0}+\rho_{q})}\right)\right\\}^{\frac{1}{2}}.$
(9)

when $\rho_{q}\longrightarrow 0$, the horizon of black hole becomes
$\rho_{h}=\frac{2(\rho_{0}+\rho_{M})}{\sqrt{1+64j^{2}\rho_{M}^{2}}+}-\rho_{0}$,
as obtained in [70]. In case of $\rho_{q}\longrightarrow 0$ and
$j\longrightarrow 0$, we have $\rho_{h}=\rho_{M}$ which is consistent with
neutral squashed Kaluza-Klein black hole [69]. You note that the square root
in relation (2) constrains the values of $\rho_{0}$, $\rho_{q}$ and $j$. In
the case $j=0$, the permissive regime is shown in Ref.[71]. For any value of
$\rho_{q}$ there is allowed rang for $\rho_{0}$. Hence these parameters can
not select any value and when $j$ increases from zero, permissive regime for
$\rho_{q}$ and $\rho_{0}$ 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, $\rho_{0}$
and decreases with $j$ and $\rho_{q}$. Note that two horizons of black hole
coincide in especial values of parameters, which in that case we have an
extremal black hole.

##  3 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 $\rho_{q}$, the scale of
extra dimension $\rho_{0}$ and Gödel parameter on the deflection angle and
it’s coefficients in the equatorial plane ($\theta=\pi/2$).
In this plane, the squashed Kaluza-Klein Gödel metric reduces to

$ds^{2}=-\mathcal{F}(\rho)d\tau^{2}+\mathcal{B}(\rho)d\rho^{2}+\mathcal{C}(\rho)\,d\phi^{2}+\mathcal{D}(\rho)d\psi^{2}-2H(\rho)dtd\psi,$
(10)

where

$\mathcal{B}(\rho)=\frac{K(\rho)}{\mathcal{G}(\rho)}.$ (11)

The null geodesic equations are,

${\ddot{x}_{i}}+\Gamma_{jk}^{i}\,\dot{x}^{j}\,\dot{x}^{k}=0,$ (12)

where

$g_{ij}\dot{x}^{i}\,\dot{x}^{j}=0,$ (13)

where $\dot{x}$ 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,

$\displaystyle\dot{t}=\frac{\mathcal{D}(\rho)E-H(\rho)L_{\psi}}{H^{2}(\rho)+\mathcal{F}(\rho)\mathcal{D}(\rho)},$
$\displaystyle\dot{\phi}=\frac{L_{\phi}}{\mathcal{C}(\rho)},$
$\displaystyle\dot{\psi}=\frac{H(\rho)E+\mathcal{F}(\rho)L_{\psi}}{H^{2}(\rho)+\mathcal{F}(\rho)\mathcal{D}(\rho)},$
(14)
$(\dot{\rho})^{2}=\frac{1}{\mathcal{B}(\rho)}\left[\frac{\mathcal{D}(\rho)E-2H(\rho)EL_{\psi}-\mathcal{F}(\rho)L^{2}_{\psi}}{H^{2}(\rho)+\mathcal{F}(\rho)\mathcal{D}(\rho)}-\frac{L^{2}_{\phi}}{\mathcal{C}(\rho)}\right].$
(15)

where $E$, $L_{\phi}$ and $L_{\psi}$ are constants of motion. Also, the
$\theta$-component of equation (12) in equatorial plane $\theta=\pi/2$, is
given by,

$\displaystyle\dot{\phi}\left[\mathcal{D}(\rho)\dot{\psi}-H(\rho)\dot{t}\right]=0.$
(16)

If $\dot{\phi}=0$, then deflection angle will be zero and this is illegal, So
we set $L_{\psi}=\mathcal{D}(\rho)\dot{\psi}-H(\rho)\dot{t}=0$. By using
equation (15) one can obtain following expression for the impact parameter,

$L_{\phi}=u=\sqrt{\frac{\mathcal{C}(\rho_{s})\mathcal{D}(\rho_{s})}{H^{2}(\rho_{s})+\mathcal{F}(\rho_{s})\mathcal{D}(\rho_{s})}},$
(17)

and the minimum of impact parameter takes place in photon sphere radius
$r_{ps}$, that is given by the root of following equation [81],

$\mathcal{D}(\rho_{s})\,\left[H(\rho_{s})^{2}+\mathcal{F}(\rho_{s})\mathcal{D}(\rho_{s})\right]\mathcal{C}^{\prime}(\rho_{s})-\mathcal{C}(\rho_{s})\,\left[\mathcal{D}(\rho_{s})^{2}\mathcal{F}^{\prime}(\rho_{s})+2\mathcal{D}(\rho_{s})H(\rho_{s})H^{\prime}(\rho_{s})-H(\rho_{s})^{2}\mathcal{D}^{\prime}(\rho_{s})\right]=0.$
(18)

Figure 2: The variation of photon sphere radius with respect to $j$,
$\rho_{0}$ and $\rho_{q}$.

Figure 3: The variation of impact parameter as a function of $j$, $\rho_{0}$
and $\rho_{q}$.

Figure 4: The variation of $\bar{a}$ with respect to $j$, $\rho_{0}$ and
$\rho_{q}$.

Here $\rho_{s}$ is the closet approach for light ray and the prime is
derivative with respect to $\rho_{s}$. 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
$\rho_{q}$, the scale of extra dimension $\rho_{0}$ 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
$\rho_{0}$ 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,

$\displaystyle\alpha_{\varphi}(\rho_{s})=I_{\varphi}(\rho_{s})-\pi,$
$\displaystyle\alpha_{\psi}(\rho_{s})=I_{\psi}(\rho_{s})-\pi,$ (19)
$\displaystyle
I_{\varphi}(\rho_{s})=2\int^{\infty}_{\rho_{s}}\frac{\sqrt{\mathcal{B}(\rho)\mathcal{A}(\rho)\mathcal{C}(\rho_{s})}}{\mathcal{C}(\rho)}\frac{1}{\sqrt{\mathcal{F}(\rho_{s})-\mathcal{F}(\rho)\frac{\mathcal{C}(\rho_{s})}{\mathcal{C}(\rho)}}}\,\,\,d\rho,$
(20) $\displaystyle
I_{\psi}(\rho_{s})=2\int^{\infty}_{\rho_{s}}\frac{H(\rho)}{\mathcal{D}(\rho)}\sqrt{\frac{\mathcal{B}(\rho)\mathcal{A}(\rho_{s})}{\mathcal{A}(\rho)}}\frac{1}{\sqrt{\mathcal{F}(\rho_{s})-\mathcal{F}(\rho)\frac{\mathcal{C}(\rho_{s})}{\mathcal{C}(\rho)}}}\,\,\,d\rho,$
(21)

with

$\mathcal{A}(\rho)=\frac{H^{2}(\rho)+\mathcal{F}(\rho)\mathcal{D}(\rho)}{\mathcal{D}(\rho)}.$
(22)

When we decrease the $\rho_{s}$ (and consequently $u$) the deflection angle
increases. At some points, the deflection angle exceeds from $2\pi$ so that
the light ray will make a complete loop around the compact object before
reaching at the observer. By decreasing $\rho_{s}$ further, the photon will
wind several times around the black hole before emerging. Finally, for
$\rho_{s}=\rho_{sp}$ 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 $j$, $\rho_{0}$ and $\rho_{q}$, 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
$I_{\phi}(\rho_{s})$ 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 $I_{\psi}(\rho)$ contains the factor
$j$, then the deflection angle $\alpha_{\psi}(\rho_{s})$ 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 $\psi$ tends zero [69, 71]. We
focus on the deflection angle in the $\phi$ direction,
So we can rewrite the equation (20) as,

$I(\rho_{s})=\int^{1}_{0}R(z,\rho_{s})f(z,\rho_{s})\,dz,$ (23)

with

$R(z,\rho_{s})=2\frac{\rho}{\rho_{s}\mathcal{C}(\rho)}\sqrt{\mathcal{B}(\rho_{s})\mathcal{A}(\rho)\mathcal{C}(\rho_{s})},$
(24)

and

$f(z,\rho_{s})=\frac{1}{\sqrt{\mathcal{A}(\rho_{s})-\mathcal{A}(\rho)\mathcal{C}(\rho_{s})/\mathcal{C}(\rho)}},$
(25)

where $z=1-\frac{\rho_{s}}{\rho}$. The function $R(z,\rho_{s})$ is regular for
all values of $z$ and $\rho_{s}$, while $f(z,\rho_{s})$ diverges as $z$
approaches to zero. Therefore, we can split the integral (23) in two parts,
the divergent part $I_{D}(\rho_{s})$ and the regular one $I_{R}(\rho_{s})$,
which are given by,

$I_{D}(\rho_{s})=\int^{1}_{0}R(0,\rho_{ps})f_{0}(z,\rho_{s})\,dz,$ (26)
$I_{R}(\rho_{s})=\int^{1}_{0}\left[R(z,\rho_{s})f(z,\rho_{s})-R(0,\rho_{ps})f_{0}(z,\rho_{s})\right]\,dz.$
(27)

Here, we expand the argument of the square root in $f(z,\rho_{s})$ up to the
second order in $z$ [70],

$f_{0}(z,\rho_{s})=\frac{1}{\sqrt{p(\rho_{s})z+q(\rho_{s})z^{2}}},$ (28)

where

$p(\rho_{s})=\frac{\rho_{s}}{\mathcal{C}(\rho_{s})}\left[\mathcal{C}^{\prime}(\rho_{s})\mathcal{A}(\rho_{s})-\mathcal{C}(\rho_{s})\mathcal{A}^{\prime}(\rho_{s})\right],$
(29)
${q}(\rho_{s})=\frac{\rho_{s}^{2}}{2\mathcal{C}(\rho_{s})}\left[2\mathcal{C}^{\prime}(\rho_{s})\mathcal{C}(\rho_{s})\mathcal{A}^{\prime}(\rho_{s})-2\mathcal{C}^{\prime}(\rho_{s})^{2}\mathcal{A}(\rho_{s})+\mathcal{A}(\rho_{s})\mathcal{C}(\rho_{s})\mathcal{C}^{\prime\prime}(\rho_{s})-\mathcal{C}^{2}(\rho_{s})\mathcal{A}^{\prime\prime}(\rho_{s})\right].$
(30)

For $\rho_{s}>\rho_{ps}$, $p(\rho_{s})$ is nonzero and the leading order of
the divergence in $f_{0}$ is $z^{-1/2}$, which have a finite result. As
$\rho_{s}\longrightarrow\rho_{ps}$, $p(\rho_{s})$ 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],

$\alpha=-\bar{a}\,log\left(\frac{u}{u_{sp}}-1\right)+\bar{b}+O(u-u_{sp}),$
(31)

where

$\displaystyle\bar{a}=\frac{R(0,\rho_{ps})}{2\sqrt{q(\rho_{ps})}}\,,$
$\displaystyle\bar{b}=-\pi+b_{R}+\bar{a}\,log\frac{\rho_{ps}^{2}\left[\mathcal{C}^{\prime\prime}(\rho_{ps})\mathcal{A}(\rho_{ps})-\mathcal{C}(\rho_{ps})\mathcal{A}^{\prime\prime}(\rho_{ps})\right]}{u_{ps}\sqrt{\mathcal{A}^{3}(\rho_{ps})\mathcal{C}(\rho_{ps})}}\,,$
$\displaystyle
b_{R}=I_{R}(\rho_{ps}),\,\,\,\,\,\,u_{ps}=\sqrt{\frac{\mathcal{C}(\rho_{ps})}{\mathcal{A}(\rho_{ps})}}\,.$
(32)

By using (31) and (3), we can investigate the properties of strong
gravitational lensing in the charged squashed Kaluza- Klein Gödel black hole.
In this case, variations of the coefficients $\bar{a}$ and $\bar{b}$, and the
deflection angle $\alpha$ have been plotted with respect to the extra
dimension $\rho_{0}$, charge of the black hole $\rho_{q}$, 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 $\rho_{q}=0$ the squashed Kaluza-klein black hole
recovers [69]. One can see that the deflection angle increases with extra
dimension and decreases with $\rho_{q}$. 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.

Figure 5: The variation of $\bar{b}$ with respect to $j$, $\rho_{0}$ and
$\rho_{q}$.

Figure 6: Deflection angle as a function of $j$, $\rho_{0}$ and $\rho_{q}$ at
$x_{s}=x_{ps}+0.05$. Note that $\alpha$ is given in Radian.

Figure 7: The variation of $s$ with respect to $j$, $\rho_{0}$ and $\rho_{q}$ . The angular separation is expressed in $\mu$arcseconds.  $\rho_{q}$ | $\rho_{0}$ |  | $\theta_{\infty}$ |  |  | $s$ |  |  | $r_{m}$ |
---|---|---|---|---|---|---|---|---|---|---
|  | $j=0$ | $j=0.03$ | $j=0.06$ | $j=0$ | $j=0.03$ | $j=0.06$ | $j=0$ | $j=0.03$ | $j=0.06$
| $0$ | 26.007 | 25.473 | 24.013 | 0.0325 | 0.0319 | 0.0300 | 6.8219 | 6.8219 | 6.8219
| $0.2$ | 27.669 | 26.962 | 25.080 | 0.0339 | 0.0390 | 0.0365 | 6.6838 | 6.8212 | 6.6736
| $0.4$ | 29.214 | 28.327 | 25.987 | 0.0476 | 0.0465 | 0.0434 | 6.5678 | 6.5617 | 6.5440
0 | $0.6$ | 30.662 | 29.583 | 26.735 | 0.0556 | 0.0543 | 0.0507 | 6.4681 | 6.4583 | 6.4269
| $0.8$ | 32.032 | 30.749 | 27.370 | 0.0639 | 0.0625 | 0.0587 | 6.3830 | 6.3660 | 6.3188
| $1$ | 33.335 | 31.836 | 27.899 | 0.0726 | 0.0710 | 0.0672 | 6.3046 | 6.2835 | 6.2167
| $0.2$ | 29.389 | 28.121 | 25.005 | 0.0171 | 0.0164 | 0.0150 | 7.7688 | 7.7606 | 7.7378
| $0.4$ | 29.260 | 27.914 | 24/574 | 0.0411 | 0.0397 | 0.0361 | 6.7556 | 6.7450 | 6.7142
0.03 | $0.6$ | 30.594 | 29.093 | 25.339 | 0.0531 | 0.0512 | 0.0477 | 6.5281 | 6.5136 | 6.4706
| $0.8$ | 31.953 | 30.258 | 26.000 | 0.0627 | 0.0607 | 0.0557 | 6.4063 | 6.3872 | 6.3270
| $1$ | 33.261 | 31.344 | 26.525 | 0.0719 | 0.0697 | 0.0647 | 6.3163 | 6.2907 | 6.2107
| $0.2$ | 34.882 | 33.270 | 29.337 | 0.0048 | 0.0047 | 0.0042 | 9.3012 | 9.2919 | 9.2658
| $0.4$ | 29.395 | 27.677 | 23.623 | 0.0292 | 0.0281 | 0.0251 | 7.1808 | 7.1680 | 7.1155
0.06 | $0.6$ | 30.412 | 28.527 | 24.031 | 0.0469 | 0.0450 | 0.0404 | 6.6806 | 6.6598 | 6.5995
| $0.8$ | 31.738 | 29.648 | 24.632 | 0.0593 | 0.0575 | 0.0515 | 6.4727 | 6.4470 | 6.3704
| $1$ | 33.052 | 30.736 | 25.132 | 0.0699 | 0.0674 | 0.0616 | 6.3480 | 6.3159 | 6.2190
| $0.2$ | 48.924 | 46.807 | 41.546 | 0.0021 | 0.0020 | 0.0018 | 10.476 | 10.470 | 10.452
| $0.4$ | 29.635 | 27.645 | 23.073 | 0.0191 | 0.0183 | 0.0163 | 7.6851 | 7.6624 | 7.6058
0.09 | $0.6$ | 30.144 | 27.932 | 22.858 | 0.0403 | 0.0338 | 0.0337 | 6.8883 | 6.8601 | 6.7795
| $0.8$ | 31.408 | 28.961 | 23.318 | 0.0548 | 0.0523 | 0.0468 | 6.5684 | 6.5351 | 6.4372
| $1$ | 32.732 | 30.027 | 23.761 | 0.0672 | 0.0643 | 0.0582 | 6.3948 | 6.3555 | 6.2362
| $0.2$ | 111.910 | 107.525 | 96.386 | 0.0019 | 0.0018 | 0.0017 | 11.362 | 11.360 | 11.353
| $0.4$ | 30.003 | 27.821 | 22.893 | 0.0124 | 0.0118 | 0.0105 | 8.1843 | 8.1580 | 8.0848
0.12 | $0.6$ | 29.809 | 27.331 | 21.824 | 0.0324 | 0.0308 | 0.0274 | 7.1226 | 7.0867 | 6.9846
| $0.8$ | 30.984 | 28.221 | 22.075 | 0.0497 | 0.0473 | 0.0422 | 6.6823 | 6.6418 | 6.5175
| $1$ | 32.311 | 29.255 | 22.433 | 0.0639 | 0.0610 | 0.0550 | 6.4516 | 6.4031 | 6.2563
Table 1: Numerical estimations for the coefficients and observables of strong
gravitational lensing by considering the supermmasive object of galactic
center be a charged squashed Kaluza-Klein Gödel black hole. (Not that the
numerical values for $\theta_{\infty}$ and $s$ are of order microarcsec)

##  4 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],

$\beta=\theta-\frac{D_{LS}}{D_{OS}}\Delta\alpha_{n},$ (33)

where $D_{LS}$ is the distance between the lens and source. $D_{OS}$ is the
distance between the observer and the source so that, $D_{OS}=D_{LS}+D_{OL}$.
$\beta$ and $\theta$ are the angular position of the source and the image with
respect to lens, respectively. $\Delta\alpha_{n}=\alpha-2n\pi$ is the offset
of deflection angle with integer $n$ which indicates the $n$-th image.
The $n$-th image position $\theta_{n}$ and the $n$-th image magnification
$\mu_{n}$ can be approximated as follows [22, 25],

Figure 8: The variation of $r_{m}$ with $j$, $\rho_{0}$ and $\rho_{q}$.

Figure 9: The variation of angular position $\theta_{\infty}$ with respect to
$j$, $\rho_{0}$ and $\rho_{q}$ that is given in $\mu$arcseconds.

$\theta_{n}=\theta^{0}_{n}+\frac{u_{ps}(\beta-\theta_{n}^{0})e^{\frac{\bar{b}-2n\pi}{\bar{a}}}D_{OS}}{\bar{a}D_{LS}D_{OL}},$
(34)
$\mu_{n}=\frac{u_{ps}^{2}(1+e^{\frac{\bar{b}-2n\pi}{\bar{a}}})e^{\frac{\bar{b}-2n\pi}{\bar{a}}}D_{OS}}{\bar{a}\beta
D_{LS}D_{OL}^{2}}.$ (35)

$\theta_{n}^{0}$ is the angular position of $\alpha=2n\pi$. In the limit
$n\longrightarrow\infty$, the relation between the minimum of impact parameter
$u_{ps}$ and asymptotic position of a set of images $\theta_{\infty}$ can be
expressed by $u_{ps}=D_{OL}\theta_{\infty}$. In order to obtain the
coefficients $\bar{a}$ and $\bar{b}$, in the simplest case, we separate the
outermost image $\theta_{1}$ and all the remaining ones which are packed
together at $\theta_{\infty}$, as done in Refs [22, 25]. Thus
$s=\theta_{1}-\theta_{\infty}$ is considered as the angular separation between
the first image and other ones and the ratio of the flux of them is given by,

$\mathcal{R}=\frac{\mu_{1}}{\sum_{n=2}^{\infty}\mu_{n}}.$ (36)

We can simplify the observables and rewrite them in the following form [22,
25],

$\displaystyle
s=\theta_{\infty}e^{\frac{\bar{b}}{\bar{a}}-\frac{2\pi}{\bar{a}}},$ (37)
$\displaystyle\mathcal{R}=e^{\frac{2\pi}{\bar{a}}}.$

Thus, by measuring the $s$, $\mathcal{R}$ and $\theta_{\infty}$, one can
obtain the values of the coefficients $\bar{a}$, $\bar{b}$ 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 $\mathcal{R}$ by,

$\displaystyle r_{m}=2.5\,\log\mathcal{R}.$ (38)

Using $\theta_{\infty}=\frac{u_{sp}}{D_{OL}}$ and equations (3), (37) and (38)
we can estimate the values of the observable in the strong field gravitational
lensing. The variation of the observables $\theta_{\infty}$, $s$ and $r_{m}$
are plotted in figures 7-9. Note that the mass of the central object of our
galaxy is estimated to be $4.31\times 10^{6}M_{\odot}$ and the distance
between the sun and the center of galaxy is $D_{OL}=8.5\,kpc$ [82].
For different $\rho_{0}$, $\rho_{q}$ 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
$\rho_{0}\longrightarrow 0$. Also our results are in agreement with the
results of Ref. [70] in the limit $\rho_{q}\longrightarrow 0$ and in the limit
$j\longrightarrow 0$, the results of Ref. [71] are recovered.

##  5 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 $\theta_{\infty}$, $s$
and $R$, where $\theta_{\infty}$ is the position of relativistic images, $s$
angular separation between the first image $\theta_{1}$ and other ones
$\theta_{\infty}$ 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.

### Acknowledgments

H. Vaez would like to thank Afshin. Ghari for helpful comments.

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