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Strong Gravitational Lensing in a Charged Squashed

Kaluza- Klein Godel Black hole

J. Sadeghi a∗ and H. Vaez a†a Physics Department, Mazandaran University,

P.O.Box 47416-95447, Babolsar, Iran

July 16, 2018

Abstract

In this paper we investigate the strong gravitational lansing in a charged squashed

Kaluza-Klein Godel 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 a and 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 Godel

background and by relation between lensing parameters and observables, we estimate

the observables for different values of charge, extra dimension and Godel parameters.

PACS numbers: 95.30.sf, 04.70.-s, 98.62.sb

Keywords: Gravitational lensing; Charged Squashed Kaluza-Klein Godel Black hole

1 Introduction

As we know the light rays or photons would be deviated from their straight way whenthey pass close to the massive object such as black holes. This deflection of light rays isknown as gravitational lensing. This gravitational lensing is one of the applications andresults of general relativity [1] and is used as an instrument in astrophysics, because it canhelp us to extract the information about stars. In 1924 Chwolson pointed out that whena star(source), a deflector(lens) and an observer are perfectly aligned, a ring-shape imageof 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 galaxiesby galaxies [3], but Liebes studied the lensing of stars by stars and also stars by clustersin our galaxy [4]. Refsdal showed that the geometrical optics can be used for investigating

∗Email: pouriya@ipm.ir†Email: h.vaez@umz.ac.ir

1

the gravitational lenses properties and time delay resulting from it [5, 6]. The gravitationallensing 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 couldnot 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 proposedgravitational lensing in a strong field limit. When the source is highly aligned with lens andthe observer, one set of infinitive relativistic ”ghost” images would be produce on each sideof black hole. These images are produced when the light rays that pass very close to blackhole, 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 pro-posed a surprisingly easy formula for the positions of the relativistic images generated by aSchwarzschild black hole. Afterward several studies of null geodesics in strong gravitationalfields have been led in literatures: a semi-analytical investigation about geodesics in Kerrgeometry has been made in [16], also the appearance of a black hole in front of a uniformbackground was studied in Refs. [17, 18]. Recently, Virbhadra and Ellis formulated lensingin the ”strong field limit” and obtain the position and magnification of these images for theSchwarzschild black hole [19, 20]. In Ref [21], by an alternative formulation, Frittelli, Kil-ing and Newman attained an exact lens equation, integral expressions for its solutions, andcompared their result with Virbhadra and Ellis. Afterwards, the new method was proposedby Bozza et al. in which they revisited the schwarzschild black hole lensing by retaining thefirst two leading order terms [22]. This technic was used by Eiroa, Romero and Torres tostudy a Reissner-Nordstrom black hole [23] and Petters to calculate relativistic effects onmicrolensing events [24]. Finally, the generalization of Bozza’s method for spherically symet-ric metric was developed in [25]. Bozza compared the image patterns for several interestingbackgrounds and showed that by the separation of the first two relativistic images we candistinguish two different collapsed objects. Further development for other black holes can befound 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 lenscharacterized by mass and scalar charge parameter and investigated the lensing for differentvalues of charge parameter.The gravitational lenses are important tools for probing the universe. Narasimha and Chitrepredicted that the gravitational lening of dark matter can give the useful data about ofposition of dark matter in the universe [37, 39]. Also in some papers gravitational lens isused 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 theo-ries in which gravity is unified with other forces [43]. One of the most interesting problems isthe verification of extra dimensions by physical phenomena. For this purpose higherdimen-sional black holes in accelerators [44, 45] and in cosmic rays [46]-[49] and gravitational wavesfrom higher-dimensional black holes [50] are studied. The five-dimensional Einstein-Maxwelltheory with a Chern- Simons term predicted five-dimensional charged black holes [51]. Sucha higher-dimensional black holes would reside in a spacetime that is approximately isotropicin the vicinity of the black holes, but effectively four-dimensional far from the black holes

2

[52]. These higher dimensional black holes are called Kaluza-Klein black holes. The presenceof extra dimension is tested by quasinormal modes from the perturbation around the higherdimensional black hole [53]-[59] and the spectrum of Hawking radiation [60]-[63]. The grav-itational lensing is another method to investigate the extra dimension. Thus, the study ofstrong gravitational lensing by higher dimensional black hole can help us to extract informa-tion about the extra dimension in astronomical observations in the future. The Kaluza-Kleinblack holes with squashed horizon [64] is one of the extra dimensional black holes and it’sHawking 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 considerGodel background for our universe. An exact solution for rotative universe was obtainedby Godel. He solved Einstein equation with pressureless matter and negative cosmologicalconstant [72]. The solutions representing the generalization of the Godel universe in theminimal five dimensional gauged supergravity are considered in many studies [74]-[80]. Theproperties of various black holes in the Godel background are investigated in many works [?].The strong gravitational lensing in a Squashed Kaluza-Klein Black hole in a Godel universeis investigated in Ref. [70].In this paper, we tudy the strong gravitational lensing in a charged squashed Kaluza-KleinGodel black hole. In that case, we see the effects of the scale of the extra dimension, chargeof black hole and Godel parameter on the coefficients and observables of strong gravitationallensing.So, this paper is organized as follows: Section 2 is briefly devoted to charged squashedKaluza-Klein Godel 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 wellas variation of them with extra dimension, Godel parameter and charge of black hole . Insection 4, we suppose that the supermassive object at the center of our galaxy can be consid-ered by the metric of charged squashed Kaluza-Klein Godel black hole. Then, we evaluatethe 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 Godel black

hole metric

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

ds2 = −f(r)dt2 + k2(r)

V (r)dr2 − 2g(r)σ3dt+ h(r)σ2

3 +r2

4[k(r)(σ2

1 + σ22) + σ2

3], (1)

3

where

σ1 = cosψ dθ + sinψ sin θ dφ,

σ2 = − sinψ dθ + cosψ sin θ dφ,

σ3 = dψ + cos θ dφ. (2)

f(r) = 1− 2M

r2+q2

r4, g(r) = j(r2 + 3q), h(r) = −j2r2(r2 + 2M + 6q),

V (r) = 1− 2M

r2+

16j2(M + q)(M + 2q)

r2+q2(1− 8j2(M + 3q))

r4, k(r) =

V (r∞)r4∞(r2 − r2∞)2

. (3)

and 0 ≤ θ < π, 0 ≤ φ < 2π, 0 ≤ ψ < 4π and 0 < r < r∞. Here M and q are the massand charge of the black hole respectively and j is the parameter of Godel background. Thekilling horizon of the black hole is given by equation V (r) = 0 , where

r2h =M − 8j2(M + q)(M + 2q)±√

[M + q − 8j2(M + 2q)2][M − q − 8j2(M + q)2]. (4)

We see that the black hole has two horizons. As q −→ 0 the horizon of the squashed Kaluza-Klein Godel black hole is recovered [70] and when q and j tend to zero, we have r2h = 2M ,which is the horizon of five-dimensional Schwarzschild black hole. Here we note that theargument of square root constraints the mass, charge and Godel parameter values. Whenr∞ −→ ∞, we have k(r) −→ 1, which means that the squashing effect disappears and thefive-dimensional charged black hole is recovered.

By using the transformations, ρ = ρ0r2

r2∞−r2

, τ =√

ρ0(1+α)ρ0+ρM

t and α =ρ2q(ρ0+ρM )

ρ0(ρ0+ρq)2, the metric

(1) can be written in the following form,

ds2 = −F(ρ)dτ 2 +K(ρ)

G(ρ) dρ2 + C(ρ)(dθ2 + sin2θ dφ2)− 2H(ρ)σ3dτ +D(ρ)σ2

3, (5)

F(ρ) = 1− ρM − 2αρ0(1 + α)ρM

(ρMρ

) +α

1 + α(ρ0ρM

)2(ρMρ

)2,

K(ρ) = 1 +ρ0ρ, G = (1− ρh+

ρ)(1− ρh−

ρ),

C(ρ) = ρ2K(ρ), H(ρ) = jr2∞

(

1

K(ρ)+

3ρqρ0 + ρq

)√

ρ0 + ρMρ0(1 + α)

,

D(ρ) =r2∞

4K(ρ)− j2ρr2∞

(ρ+ ρ0)2(ρM + ρ0)(ρq + ρ0)×

{ρ[ρ0(ρ0 + 2ρM) + 7ρ0ρq + 8ρMρq] + ρ0[ρ0(ρM + 6ρq) + 7ρMρq]} , (6)

with

ρM = ρ02M

r2∞ − 2M, ρq = ρ0

q

r2∞ − q, ρh± = ρ0

r2h±r2∞ − r2h±

. (7)

4

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

h-

h+

h/M

j

q= 0

q= 0.03

q= 0.06

q= 0.09

q= 0.12

0 = 0.6

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

h-

h+

h/M

j

0 = 0.2

0 = 0.4

0 = 0.6

0 = 0.8

0 = 1

q = 0.06

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0

0.2

0.4

0.6

0.8

1.0

0/ M

h-

h/M

h+

j = 0.6 q= 0

q= 0.03

q= 0.06

q= 0.09

q= 0.12

0.05 0.10 0.15 0.20 0.25 0.300.0

0.2

0.4

0.6

0.8

h-

h+

q = 0.6

0/ M

h/M

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

merging pointof two horizons

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0

0.2

0.4

0.6

0.8

1.0

h+

h-

j = 0.06

q/

M

h/M

0 = 0.2

0 = 0.4

0 = 0.6

0 = 0.8

0 = 1

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0

0.2

0.4

0.6

0.8

1.00 = 0.6

h+

h-

h/

M

q/ M

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

Figure 1: The plots show the variation of horizon radiuses with respect to j, ρ0 and ρq (Notethat in each figure, for ρq 6= 0, two horizons merge at a point. This point has been shownfor one of figures. )

5

Where ρh+ and ρh− denote the outer and inner horizons of the black hole in the newcoordinate and ρ0 is a scale of transition from five-dimensional spacetime to an effective

four-dimensional one. Here ρ20 =r2∞

4V (r∞), so that r2∞ = 4(ρ0 + ρh+)(ρ0 + ρh−). The Komar

mass of black hole is related to ρM with ρM = 2G4M , where G4 is the four dimensionalgravitational constant. By using relations (4) and (7) we can obtain ρh± in the followingcoupled equations,

2 [ρ0(ρh+ + ρh−) + 2ρh+ρh−] = a(ρh+, ρh−),

2 [ρ0(ρh+ − ρh−)] = b(ρh+, ρh−), (8)

where

a =ρMr

2∞

2(ρ0 + ρM)− 2j2r4∞

(ρMρ0 + 3ρMρ0 + 2ρqρ0) (ρMρ0 + 5ρMρq + 4ρqρ0)

(ρ0 + ρM )2(ρ0 + ρq)2,

b =

{

a2 − 4ρ2qr

4∞

(ρ0 + ρq)2

(

1− 4j2r2∞(ρ0ρM + 7ρMρq + 6ρqρ0)

(ρ0 + ρM)(ρ0 + ρq)

)}

1

2

. (9)

when ρq −→ 0, the horizon of black hole becomes ρh =2(ρ0+ρM )√1+64j2ρ2

M+− ρ0, as obtained in [70].

In case of ρq −→ 0 and j −→ 0, we have ρh = ρM which is consistent with neutral squashedKaluza-Klein black hole [69]. You note that the square root in relation (9) constrains thevalues of ρ0, ρq and j. In the case j = 0, the permissive regime is shown in Ref.[71]. Forany value of ρq there is allowed rang for ρ0. Hence these parameters can not select any valueand when j increases from zero, permissive regime for ρq and ρ0 becomes more confined.We solved the above coupled equations numerically and results are shown in figure 1. Wesee that the outer horizon of black hole increases with the size of extra dimension, ρ0 anddecreases with j and ρq. Note that two horizons of black hole coincide in especial values ofparameters, 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 passclose to a charged squashed Kaluza-Klein Godel black hole. We also study the effect of thecharge parameter ρq, the scale of extra dimension ρ0 and Godel parameter on the deflectionangle and it’s coefficients in the equatorial plane (θ = π/2).In this plane, the squashed Kaluza-Klein Godel metric reduces to

ds2 = −F(ρ)dτ 2 + B(ρ)dρ2 + C(ρ) dφ2 +D(ρ)dψ2 − 2H(ρ)dtdψ, (10)

where

B(ρ) = K(ρ)

G(ρ) . (11)

The null geodesic equations are,

xi + Γijk xj xk = 0, (12)

6

wheregijx

i xj = 0, (13)

where x is the tangent vector to the null geodesics and the dote denotes derivative withrespect to affine parameter. We use equation (12) and obtain the following equations,

t =D(ρ)E −H(ρ)LψH2(ρ) + F(ρ)D(ρ)

,

φ =LφC(ρ) ,

ψ =H(ρ)E + F(ρ)LψH2(ρ) + F(ρ)D(ρ)

, (14)

(ρ)2 =1

B(ρ)

[D(ρ)E − 2H(ρ)ELψ − F(ρ)L2ψ

H2(ρ) + F(ρ)D(ρ)−

L2φ

C(ρ)

]

. (15)

where E, Lφ and Lψ are constants of motion. Also, the θ-component of equation (12) inequatorial plane θ = π/2, is given by,

φ[

D(ρ)ψ −H(ρ)t]

= 0. (16)

If φ = 0, then deflection angle will be zero and this is illegal, So we set Lψ = D(ρ)ψ−H(ρ)t =0. By using equation (15) one can obtain following expression for the impact parameter,

Lφ = u =

√

C(ρs)D(ρs)

H2(ρs) + F(ρs)D(ρs), (17)

and the minimum of impact parameter takes place in photon sphere radius rps, that is givenby the root of following equation [81],

D(ρs)[

H(ρs)2 + F(ρs)D(ρs)

]

C′(ρs)−C(ρs)[

D(ρs)2F ′(ρs) + 2D(ρs)H(ρs)H

′(ρs)−H(ρs)2D′(ρs)

]

= 0.(18)

Here ρs is the closet approach for light ray and the prime is derivative with respect toρs. The analytical solution for the above equation is very complicated, so we calculated theequation (18) numerically. Variations of r photon sphere radius are plotted with respect tothe charge ρq, the scale of extra dimension ρ0 and Godel parameter in the figure 2. Alsofigure 3 shows variations of impact parameter in it’s minimum value (at radius of photonsphere). These figures show that by adding the charge to the black hole, the behavior ofthe photon sphere radius and minimum of impact parameter is different compare with theneutral black hole [69]. As ρ0 approaches to it’s minimum values the radius of photon sphereand impact parameter become divergent.By using the chain derivative and equation (15), the deflection angle in the charged squashedKaluza-Klein Godel black hole can be written as,

αϕ(ρs) = Iϕ(ρs)− π,

αψ(ρs) = Iψ(ρs)− π, (19)

7

0.00 0.05 0.10 0.15 0.20 0.25

0.6

0.8

1.0

1.2

1.4

1.6

0 = 0.6

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

q/ M

ps/

M

0.00 0.05 0.10 0.15 0.20 0.250.8

1.0

1.2

1.4

1.6

1.8

2.0

j = 0.60

0 =1

0 =0.8

0 =0.6

0 =0.4

0 =0.2

ps/

M

q/ M

0.2 0.4 0.6 0.8 1.0 1.2 1.40.8

1.0

1.2

1.4

1.6

1.8

2.0

ps

/M

0/ M

q = 0.06

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

0 1 2 31.1

1.2

1.3

1.4

1.5

j = 0.06

q= 0.12

q= 0.09

q= 0.06

q= 0.03

ps/

M

0/ M

q= 0

0.00 0.05 0.10 0.15 0.20 0.25

0.5

1.0

1.5

2.0

q = 0.06

0 = 0.2

0 = 0.4

0 = 0.6

0 = 0.8

0 = 1

ps/

Mj

0.00 0.05 0.10 0.15 0.20 0.250.0

0.5

1.0

1.5

ps/

M

j

q = 0

q = 0.03

q = 0.06

q = 0.09

q = 0.12

0= 0.6

Figure 2: The variation of photon sphere radius with respect to j, ρ0 and ρq.

8

0.00 0.05 0.10 0.15 0.20 0.25

1.0

1.5

2.0

2.5

3.0

ρ0 = 0.6

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

u ps/ρ

M

ρq/ρM

0.00 0.05 0.10 0.15 0.20 0.251.5

2.0

2.5

3.0

3.5

j = 0.60

ρ0 =1

ρ0 =0.8

ρ0 =0.6

ρ0 =0.4

ρ0 =0.2

u ps/ρ

M

ρq/ρM

0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

u ps

/ρM

ρ0/ρM

ρq = 0.06

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

0 1 2 3

1.8

2.0

2.2

2.4

2.6

2.8

3.0

u ps/ρ

M

ρ0/ρM

j = 0.06

ρq= 0

ρq= 0.03

ρq= 0.06

ρq= 0.09

ρq= 0.12

0.00 0.05 0.10 0.15 0.20 0.25

0

1

2

3

ρq = 0.06

ρ0 = 0.2

ρ0 = 0.4

ρ0 = 0.6

ρ0 = 0.8

ρ0 = 1

u ps

/ρM

j0.00 0.05 0.10 0.15 0.20 0.25

0

1

2

3

u ps/ρ

M

j

ρ0= 0.6

ρq = 0

ρq = 0.03

ρq = 0.06

ρq = 0.09

ρq = 0.12

Figure 3: The variation of impact parameter as a function of j, ρ0 and ρq.

9

0.00 0.05 0.10 0.15 0.20 0.25

1.0

1.1

1.2

1.3

1.4

0 = 0.6

q=0 q=0.03 q=0.06 q=0.09 q=0.12

a

j0.0 0.1 0.2 0.3 0.4 0.5

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

q = 0.06

a

j

0 = 0.2 0 = 0.4 0 = 0.6 0 = 0.8 0 = 1

0 1 2 3 4

0.8

1.0

1.2

1.4

1.6

1.8

2.0

a

0/ M

q = 0

q = 0.03

q = 0.06

q = 0.09

q = 0.12

j= 0.06

1 2 3 40.8

1.0

1.2

1.4

a

0/ M

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

q = 0.06

0.00 0.05 0.10 0.15 0.20 0.250.5

0.6

0.7

0.8

0.9

1.0

1.1

0 = 0.2

0 = 0.4

0 = 0.6

0 = 0.8

0 = 1

aq/ M

j = 0.06

0.00 0.05 0.10 0.15 0.200.85

0.90

0.95

1.00

1.05

1.10

0 = 0.6

q/ M

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

a

Figure 4: The variation of a with respect to j, ρ0 and ρq.

10

Iϕ(ρs) = 2

∫ ∞

ρs

√

B(ρ)A(ρ)C(ρs)C(ρ)

1√

F(ρs)− F(ρ)C(ρs)C(ρ)

dρ, (20)

Iψ(ρs) = 2

∫

∞

ρs

H(ρ)

D(ρ)

√

B(ρ)A(ρs)

A(ρ)

1√

F(ρs)−F(ρ)C(ρs)C(ρ)

dρ, (21)

with

A(ρ) =H2(ρ) + F(ρ)D(ρ)

D(ρ). (22)

When we decrease the ρs (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 aroundthe compact object before reaching at the observer. By decreasing ρs further, the photon willwind several times around the black hole before emerging. Finally, for ρs = ρsp the deflectionangle 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 Godel black hole, bothof the deflection angles depend on the parameters j, ρ0 and ρq, which implies that we coulddetect the rotation of universe, the extra dimension and charge of black hole in theory bygravitational lens. Note that the Iφ(ρs) depend on j2, not j. It shows that the deflectionangle is independent of the direction of rotation of universe. But, from equation (21) we findthat the integral Iψ(ρ) contains the factor j, then the deflection angle αψ(ρs) for the photontraveling around the lens in two opposite directions is different .The main reason is that theequatorial plan is parallel with Godel rotation plan [70]. When j vanishes, the deflectionangle of ψ tends zero [69, 71]. We focus on the deflection angle in the φ direction,So we can rewrite the equation (20) as,

I(ρs) =

∫ 1

0

R(z, ρs)f(z, ρs) dz, (23)

withR(z, ρs) = 2

ρ

ρsC(ρ)√

B(ρs)A(ρ)C(ρs), (24)

and

f(z, ρs) =1

√

A(ρs)−A(ρ)C(ρs)/C(ρ), (25)

where z = 1 − ρsρ. The function R(z, ρs) is regular for all values of z and ρs, while f(z, ρs)

diverges as z approaches to zero. Therefore, we can split the integral (23) in two parts, thedivergent part ID(ρs) and the regular one IR(ρs), which are given by,

ID(ρs) =

∫ 1

0

R(0, ρps)f0(z, ρs) dz, (26)

IR(ρs) =

∫ 1

0

[R(z, ρs)f(z, ρs)− R(0, ρps)f0(z, ρs)] dz. (27)

11

Here, we expand the argument of the square root in f(z, ρs) up to the second order in z [70],

f0(z, ρs) =1

√

p(ρs)z + q(ρs)z2, (28)

wherep(ρs) =

ρsC(ρs)

[C′(ρs)A(ρs)− C(ρs)A′(ρs)] , (29)

q(ρs) =ρ2s

2C(ρs)[

2C′(ρs)C(ρs)A′(ρs)− 2C′(ρs)2A(ρs) +A(ρs)C(ρs)C′′(ρs)− C2(ρs)A′′(ρs)

]

.

(30)For ρs > ρps, p(ρs) is nonzero and the leading order of the divergence in f0 is z−1/2, whichhave a finite result. As ρs −→ ρps, p(ρs) approaches zero and divergence is of order z−1,that makes the integral divergent. Therefor, the deflection angle can be approximated in thefollowing form [25],

α = −a log(

u

usp− 1

)

+ b+O(u− usp), (31)

where

a =R(0, ρps)

2√

q(ρps),

b = −π + bR + a logρ2ps [C′′(ρps)A(ρps)− C(ρps)A′′(ρps)]

ups√

A3(ρps)C(ρps),

bR = IR(ρps), ups =

√

C(ρps)A(ρps)

. (32)

By using (31) and (32), we can investigate the properties of strong gravitational lensingin the charged squashed Kaluza- Klein Godel black hole. In this case, variations of thecoefficients a and b, and the deflection angle α have been plotted with respect to the extradimension ρ0, charge of the black hole ρq, and Godel parameter j in figures 4-6.As j tends to zero, these quantities reduce to charged squashed Kaluza-klein black hole [71]and with ρq = 0 the squashed Kaluza-klein black hole recovers [69]. One can see that thedeflection angle increases with extra dimension and decreases with ρq. By comparing theseparameters with those in four-dimensional Schwarzschild and Reissner-Nordstrom black holes, we could extract information about the size of extra dimension as well as the charge of theblack hole by using strong field gravitational lensing.

4 Observables estimation

In the previous section, we investigated the strong gravitational lensing by using a simpleand 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

12

0.00 0.05 0.10 0.15-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

0 = 0.6

q=0

q=0.03

q=0.06

q=0.09

q=0.12

b

j0.00 0.05 0.10 0.15 0.20

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

b

j

0 = 0.2

0 = 0.4

0 = 0.6

0 = 0.8

0 = 1

q = 0.06

0.5 1.0 1.5 2.0 2.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

b

0/ M

q = 0

q = 0.03

q = 0.06

q = 0.09

q = 0.12

j = 0.06

1 2 3 4 5-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

b

0/ M

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

q = 0.06

0.00 0.05 0.10 0.15 0.20 0.25-0.40

-0.35

-0.30

-0.25

-0.20

-0.15j = 0.06

0 = 0.2 0 = 0.4 0 = 0.6 0 = 0.8 0 = 1

q/ M

b0.00 0.05 0.10 0.15 0.20 0.25

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

b

q/ M

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

0 = 0.6

Figure 5: The variation of b with respect to j, ρ0 and ρq.

13

0.00 0.05 0.10 0.15 0.20

5.8

6.0

6.2

6.4

6.6

6.8

7.0

q/ M

0 = 0.6

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

0.00 0.05 0.10 0.15 0.20

4.5

5.0

5.5

6.0

6.5

7.0

7.5

q = 0.06

0 = 0.2 0 = 0.4 0 = 0.8 0 = 0.6 0 = 1

j

0.0 0.5 1.0 1.5 2.0 2.5 3.04

5

6

7

8

9

0/

M

q = 0

q = 0.03

q = 0.06

q = 0.09

q = 0.12

j = 0.06

1 2 3 46.0

6.5

7.0

7.5

8.0

8.5

9.0

0/ M

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

q = 0.06

0.00 0.05 0.10 0.15 0.20 0.254.0

4.5

5.0

5.5

6.0

6.5

7.0

j = 0.06

q/ M

0 = 0.2 0 = 0.4 0 = 0.6 0 = 0.8 0 = 1

0.00 0.05 0.10 0.15 0.20

5.8

6.0

6.2

6.4

6.6

6.8

7.0

q/ M

0 = 0.6

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

Figure 6: Deflection angle as a function of j, ρ0 and ρq at xs = xps + 0.05. Note that α isgiven in Radian.

14

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.05

0.10

0.15

0.20

q = 0.06

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

0/ M

s

0.00 0.05 0.10 0.150.02

0.03

0.04

0.05

0.06

j

0 = 0.6

q=0 q=0.03 q=0.06 q=0.09 q=0.12

s

0.00 0.05 0.10 0.150.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 = 0.2 0 = 0.4 0 = 0.8 0 = 0.6 0 = 1

q = 0.06

j

s

0.5 1.0 1.5 2.0 2.5 3.00.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

s

0/ M

q = 0

q = 0.03

q = 0.06

q = 0.09

q = 0.12

j = 0.06

0.00 0.05 0.10 0.15 0.20 0.250.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

sq/ M

0 = 0.2 0 = 0.6 0 = 0.4 0 = 0.8

0 = 1

j = 0.06

0.00 0.05 0.10 0.15 0.20 0.25

0.01

0.02

0.03

0.04

0.05

0.06

0 = 0.6

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

q/ M

s

Figure 7: The variation of s with respect to j, ρ0 and ρq . The angular separation is expressedin µarcseconds.

15

observables, estimat the position and magnification of the relativistic images. By comparingthese observables with the data from the astronomical observation, we could detect propertiesof an massive object. We suppose that the spacetime of the supermassive object at the galaxycenter of Milky Way can be considered as a charged squashed Kaluza-Klein Godel 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, andobserver are highly aligned as follows [22],

β = θ − DLS

DOS∆αn, (33)

where DLS is the distance between the lens and source. DOS is the distance between theobserver and the source so that, DOS = DLS + DOL. β and θ are the angular position ofthe source and the image with respect to lens, respectively. ∆αn = α− 2nπ is the offset ofdeflection angle with integer n which indicates the n-th image.The n-th image position θn and the n-th image magnification µn can be approximated asfollows [22, 25],

θn = θ0n +ups(β − θ0n)e

b−2nπa DOS

aDLSDOL, (34)

µn =u2ps(1 + e

b−2nπa )e

b−2nπa DOS

aβDLSD2OL

. (35)

θ0n is the angular position of α = 2nπ. In the limit n −→ ∞, the relation between theminimum of impact parameter ups and asymptotic position of a set of images θ∞ can beexpressed by ups = DOLθ∞. In order to obtain the coefficients a and b, in the simplest case,we separate the outermost image θ1 and all the remaining ones which are packed togetherat θ∞, as done in Refs [22, 25]. Thus s = θ1 − θ∞ is considered as the angular separationbetween the first image and other ones and the ratio of the flux of them is given by,

R =µ1

∑∞

n=2 µn. (36)

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

s = θ∞eba−

2πa ,

R = e2πa . (37)

Thus, by measuring the s, R and θ∞, one can obtain the values of the coefficients a, b andusp. If we compare these values by those obtained in the previous section, we could detect thesize of the extra dimension, charge of black hole and rotation of universe. Another observablefor gravitational lensing is relative magnification of the outermost relativistic image with theother ones. This observable is shown by rm which is related to R by,

rm = 2.5 logR. (38)

16

0.00 0.05 0.10 0.15 0.20 0.25 0.30

4.5

5.0

5.5

6.0

6.5

7.0

0 = 0.6

q=0 q=0.03 q=0.06 q=0.09 q=0.12

r m

j0.0 0.1 0.2 0.3 0.4 0.53

4

5

6

7

8

9

j

q = 0.06 0 = 0.2 0 = 0.4 0 = 0.6 0 = 0.8 0 = 1

r m

0 1 2 3 44

5

6

7

j= 0.06

0/ M

q = 0

q = 0.03

q = 0.06

q = 0.09

q = 0.12

r m

0 1 2 3 44.0

4.5

5.0

5.5

6.0

6.5

7.0

0/ M

q = 0.06

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

rm

0.00 0.05 0.10 0.15 0.20 0.25

6

7

8

9

10

11

q/ M

j = 0.06

0 = 0.2

0 = 0.4

0 = 0.6

0 = 0.8

0 = 1

r m

0.00 0.05 0.10 0.15 0.206.2

6.4

6.6

6.8

7.0

7.2

7.4

7.6

7.8

q/ M

0 = 0.6

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

r m

Figure 8: The variation of rm with j, ρ0 and ρq.

17

0.00 0.05 0.10 0.15 0.20

5

10

15

20

25

30

j

q=0 q=0.03 q=0.06 q=0.09 q=0.12

0=0.6

0.0 0.1 0.2 0.3 0.4

5

10

15

20

25

30

35

q = 0.06

j

0 = 0.2 0 = 0.4 0 = 0.6 0 = 0.8 0 = 1

0 1 2 3 4

20

22

24

26

28

30

0/ M

j= 0.06

q = 0

q = 0.03

q = 0.06

q = 0.09

q = 0.12

0.0 0.5 1.0 1.5 2.0

10

20

30

40

50

0/ M

q = 0.06

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

0.00 0.05 0.10 0.15 0.20

20

25

30

35

40

q/ M

j = 0.06 0 = 0.2

0 = 0.4

0 = 0.6

0 = 0.8

0 = 1

0.00 0.05 0.10 0.15 0.20

10

15

20

25

30

q/ M

0 = 0.6

j = 0 j = 0.03 j = 0.06 j = 0.09 j = 0.12

Figure 9: The variation of angular position θ∞ with respect to j, ρ0 and ρq that is given inµarcseconds.

18

Using θ∞ = uspDOL

and equations (32), (37) and (38) we can estimate the values of the ob-servable in the strong field gravitational lensing. The variation of the observables θ∞, sand rm are plotted in figures 7-9. Note that the mass of the central object of our galaxy isestimated to be 4.31× 106M⊙ and the distance between the sun and the center of galaxy isDOL = 8.5 kpc [82].For different ρ0, ρq and j, the numerical values for the observables are listed in Table 1. Onecan see that our results reduce to those in the four-dimensional Schwarzschild black hole asρ0 −→ 0. Also our results are in agreement with the results of Ref. [70] in the limit ρq −→ 0and in the limit j −→ 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 predictedby General Relativity. This deflection of light rays is known as gravitational lensing. In thestrong field limit, the deflection anglethe of the light rays which pass very close to the blackhole, becomes so large that, it winds several times around the black hole before appearing atthe observer. Therefore the observer would detect two infinite set of faint relativistic imagesproduced on each side of the black hole. On the other hand the extra dimension is one ofthe important predictions in the string theory which is believed to be a promising candidatefor the unified theory. Also it is reasonable to consider a rotative universe with globalrotation. Hence the five-dimensional Einstein-Maxwell theory with a Chern- Simons termin string theory predicted five-dimensional charged black holes in the Godel background. Inour study, we considered the charged squashed Kaluza-Klein Godel black hole spacetime andinvestigated the strong gravitational lensing by this metric. We obtained theoretically thedeflection angle and other parameters of strong gravitational lensing . Finally, we supposethat the supermassive black hole at the galaxy center of Milky Way can be considered bythis spacetime and we estimated numerically the values of observables that are realated tothe lensing parameters. Theses observable parameters are θ∞, s and R, where θ∞ is theposition of relativistic images, s angular separation between the first image θ1 and otherones θ∞ and R is the ratio of the flux from the first image and those from all the otherimages. Our results are presented in figures 1-9 and Table 1. By comparatione observableparameters 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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23

ρq ρ0 θ∞ s rmj = 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 6.82190.2 27.669 26.962 25.080 0.0339 0.0390 0.0365 6.6838 6.8212 6.67360.4 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 6.42690.8 32.032 30.749 27.370 0.0639 0.0625 0.0587 6.3830 6.3660 6.31881 33.335 31.836 27.899 0.0726 0.0710 0.0672 6.3046 6.2835 6.21670.2 29.389 28.121 25.005 0.0171 0.0164 0.0150 7.7688 7.7606 7.73780.4 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 6.47060.8 31.953 30.258 26.000 0.0627 0.0607 0.0557 6.4063 6.3872 6.32701 33.261 31.344 26.525 0.0719 0.0697 0.0647 6.3163 6.2907 6.21070.2 34.882 33.270 29.337 0.0048 0.0047 0.0042 9.3012 9.2919 9.26580.4 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 6.59950.8 31.738 29.648 24.632 0.0593 0.0575 0.0515 6.4727 6.4470 6.37041 33.052 30.736 25.132 0.0699 0.0674 0.0616 6.3480 6.3159 6.21900.2 48.924 46.807 41.546 0.0021 0.0020 0.0018 10.476 10.470 10.4520.4 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 6.77950.8 31.408 28.961 23.318 0.0548 0.0523 0.0468 6.5684 6.5351 6.43721 32.732 30.027 23.761 0.0672 0.0643 0.0582 6.3948 6.3555 6.23620.2 111.910 107.525 96.386 0.0019 0.0018 0.0017 11.362 11.360 11.3530.4 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 6.98460.8 30.984 28.221 22.075 0.0497 0.0473 0.0422 6.6823 6.6418 6.51751 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 gravitationallensing by considering the supermmasive object of galactic center be a charged squashedKaluza-Klein Godel black hole. (Not that the numerical values for θ∞ and s are of ordermicroarcsec)

24