Strong gravitational lensing in a rotating Kaluza-Klein black hole with squashed horizons

We have investigated the strong gravitational lensing in a rotating squashed Kaluza-Klein (KK) black hole spacetime. Our result show that the strong gravitational lensings in the rotating squashed KK black hole spacetime have some distinct behaviors from those in the backgrounds of the four-dimensional Kerr black hole and of the squashed KK G\"{o}del black hole. In the rotating squashed KK black hole spacetime, the marginally circular photon radius $\rho_{ps}$, the coefficient $\bar{a}$, $\bar{b}$, the deflection angle $\alpha(\theta)$ in the $\phi$ direction and the corresponding observational variables are independent of whether the photon goes with or against the rotation of the background, which is different with those in the usual four-dimensional Kerr black hole spacetime. Moreover, we also find that with the increase of the scale of extra dimension $\rho_0$, the marginally circular photon radius $\rho_{ps}$ and the angular position of the relativistic images $\theta_\infty$ first decreases and then increases in the rotating squashed KK black hole for fixed rotation parameter $b$, but in the squashed KK G\"{o}del black hole they increase for the smaller global rotation parameter $j$ and decrease for the larger one. In the extremely squashed case $\rho_0=0$, the coefficient $\bar{a}$ in the rotating squashed KK black hole increases monotonously with the rotation parameter, but in the squashed KK G\"{o}del black hole it is a constant and independent of the global rotation of the G\"{o}del Universe.


I. INTRODUCTION
The Kaluza-Klein (KK) black holes with squashed horizons are a kind of interesting Kaluza-Klein type metrics with the special topology and asymptotical structure [1][2][3][4][5][6][7][8]. This family of black holes behave as fully five-dimensional black holes in the vicinity of horizon, while behave as four-dimensional black holes with a constant twisted S 1 fiber in the far region. In these black holes, the size of compactified extra dimension can be adjustable by a parameter r ∞ . Recent investigations show that the spectrum of Hawking radiation [9,10], the quasinormal frequencies [11,12] and the precession of a gyroscope in a circular orbit [13]  From the general theory of relativity, we know that photons would be deviated from their straight path when they pass close to a compact and massive body. The effects originating from the deflection of light rays in a gravitational field are known as gravitational lensing [14][15][16]. Like a natural and large telescope, gravitational lensing can help us to capture the information about the very dim stars which are far away from our Galaxy.
The strong gravitational lensing is caused by a compact object with a photon sphere. As photons pass close to the photon sphere, the deflection angles of of the light rays become very large, which yields that there exist two infinite sets of faint relativistic images on each side of the black hole. With these relativistic images, we could extract the information about black holes in the Universe and verify profoundly alternative theories of gravity in their strong field regime [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. Recently, we have studied the strong gravitational lensing in the background of a Schwarzschild squashed KK black hole [33] and a squashed KK Gödel black hole [34] and find that the size of the extra dimension and the the global rotation parameter j of the Gödel Universe affects the photon sphere radius, the deflection angle and the corresponding observational variables in the strong gravitational lensing. Sadeghi et al have considered the effect of the charge on the strong gravitational lensing in the squashed KK black holes [35,36]. These investigations could help us to understand further the effects of the scale of extra dimension on the strong gravitational lensings. However, to my knowledge, the strong gravitational lensing is still open in the background of a rotating squashed KK black hole. Since the rotation is a universal phenomenon for the celestial bodies in our Universe, it is necessary to study the strong gravitational lensing in the rotating squashed KK black hole spacetime. With the squashing transformation, Wang [6] obtained a rotating squashed KK black hole spacetime with two equal angular momenta in Einstein theory. This black hole solution with squashed horizons is geodesic complete and free of naked singularities.
It has the similar topology and asymptotical structure to that of the static squashed KK black hole, but with richer physical properties. In this paper, we are going to study the strong gravitational lensing in this rotating squashed KK black hole and probe the effects of the rotation parameter of black hole and the scale of extra dimension on the deflection angle and the coefficients in the strong field limit.
The plan of our paper is organized as follows. In Sec.II we introduce briefly the rotating squashed KK black hole [6]. In Sec.III we adopt to Bozza's method [22][23][24] and obtain the deflection angles for light rays propagating in the rotating squashed KK black hole. In Sec.IV we suppose that the gravitational field of the supermassive black hole at the center of our Galaxy can be described by this metric and then obtain the numerical results for the main observables in the strong gravitational lensing. Moreover, we also make a comparison among the strong gravitational lensings in the rotating squashed KK, the squashed KK Gödel and four-dimensional Kerr black hole spacetimes. At last, we present a summary.

II. THE ROTATING KALUZA-KLEIN BLACK HOLE SPACETIME WITH SQUASHED HORIZON
Let us now review briefly a rotating squashed KK black hole without charge, which can be obtained by applying the squashing transformation techniques to a five-dimensional Kerr black hole with two equal angular momenta [6]. In terms of Meurer-Cartan 1-forms, the metric of a rotating KK black bole has a form with σ 1 = − sinψdθ + cosψ sin θdφ, σ 2 = cosψdθ + sinψ sin θdφ, where 0 < θ < π, 0 < φ < 2π and 0 <ψ < 4π. The parameters are given by Σ 0 = r 2 (r 2 + a 2 ), The quantities M and a are related to the mass and angular momenta of black hole, respectively. r ∞ corresponds to the spatial infinity. The polar coordinate r is limited in the range 0 < r < r ∞ . The outer and inner horizons are located at r = r + and r = r − , which are relate to the parameters M , a by a 4 = (r + r − ) 2 and M − 2a 2 = r 2 + + r 2 − . The shape of black hole horizon is deformed by the parameter k(r + ).
In this black hole spacetime (1), the intrinsic singularity is the just one at r = 0, while r ± and r ∞ are coordinate singularities. As in [6], one can introduce a new radial coordinate and then rewrite the metric (1) as where The parameterρ 0 is a scale of transition from five-dimensional spacetime to an effective four-dimensional one.
As the rotation parameter a tends to zero, one can find that the metric (6) reduces to that of a five-dimensional Schwarzschild black hole with squashed horizon. In the limit ρ → ∞, i.e, r → r ∞ , it is easy to find that there is a cross-term between dt and σ 3 in the asymptotic form of the metric (6). However, this cross-term can be vanished by changing the coordinates as [6] t = h t,ψ = ψ − j t. where This means that the asymptotic topology of the spacetime (6) is the same as that of the Schwarzschild squashed KK black hole spacetime. The Komar mass M k of the rotating squashed KK black hole (6) can be given by [37,38] where G 5 and G 4 are the five-dimensional and four-dimensional gravitational constants, respectively. Therefore, in the rotating squashed KK black hole spacetime, the relationship between G 5 and G 4 can be expressed as with The expression of r ′ ∞ is more complicated than that of r ∞ . However, in the rotating squashed KK black hole spacetime (6), one can find [6] that the parameter r ′ ∞ for the compactified dimension is better than r ∞ because the geometric interpretation is clearer for r ′ ∞ than for r ∞ . As a disappears, we find that r ′ ∞ reduces to r ∞ and then the relationship (11) tend to the usual form ( i.e., G 5 = 2πr ∞ G 4 ) in the Schwarzschild squashed KK black hole spacetime. As in Ref. [13], we can rewritten Komar mass as which implies that ρ M can be expressed as In order to simplify the calculation, we here introduce a new radial coordinate change a new scale of extra dimension and a new rotation parameter we find that the radial coordinate (4) and the quantity ρ M can be rewritten as and respectively. With help of these operation, one can find that the metric of the rotating squashed KK black hole spacetime (6) can be expressed as with where Among the parameters ρ 0 , M , a, r ∞ , r ′ ∞ , ρ M and b, there are only three independent parameters. For simplicity, we chose the parameters ρ 0 , ρ M and b as the independent parameters. The others are related to the selected three parameters by With these quantities, we can find that all of coefficients in the metric (19) can be expressed as the functions of the parameters ρ 0 , ρ M and b, which means that we can study the strong gravitational lensing in the rotating squashed KK black hole spacetime (6) through the standard form used in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].

III. DEFLECTION ANGLE IN A ROTATING SQUASHED KALUZA-KLEIN BLACK HOLE SPACETIME
In this section, we will study deflection angles of the light rays when they pass close to a rotating squashed KK black hole, and then probe the effects of the rotation parameter b and the scale of extra dimension ρ 0 on the deflection angle and the coefficients in the strong field limit. For simplicity, we here just consider that both the observer and the source lie in the equatorial plane in the rotating squashed KK black hole spacetime (19) and the whole trajectory of the photon is limited on the same plane With this condition θ = π 2 , we get the reduced metric in the form From the null geodesics, it is easy to obtain three constants of motion where a dot represents a derivative with respect to affine parameter λ along the geodesics. E is the energy of the phone, L φ and L ψ correspond to its angular momentum in the φ and ψ directions, respectively. Making use of these three constants, one can find that the equations of motion of the photon can be expressed further Considering the θ-component of the null geodesics in the equatorial plane (θ = π 2 ), we have which implies that either dφ dλ = 0 or L ψ = D(ρ) dψ dλ − H(ρ) dt dλ = 0. As done in the usual squashed KK black hole spacetimes [33,34], here we set L ψ = 0, which means that the total angular momentum J of the photo is equal to the constant L φ and the effective potential for the photon passing close to the black hole can be written as With this effective potential, one can obtain that the impact parameter and the equation of circular photon orbits are and respectively. Here we set E = 1. The equations (29) and (30) are similar to those in the squashed KK Gödel black hole spacetime because their metric have similar forms in the equatorial plane, but they are more complex than those in the usual spherical symmetric black hole spacetime. As the rotation parameter b → 0, we find that the function H(ρ) → 0, which yields that the impact parameter (29) and the equation of circular photon orbits (30) reduce to those in the usual Schwarzschild squashed KK black hole spacetime [33].
The biggest real root external to the horizon of equation (30) defines the marginally stable circular radius of photon. For the rotating squashed KK black hole spacetime (19), the equation of circular photon orbits takes the form where the variable ρ ′ is related to ρ by and the coefficients are Obviously, this equation depends on both the rotation parameter b and the scale of transition ρ 0 . The presence of the rotation parameter b makes the equation more complex so that it is impossible to get an analytical form for the marginally circular photon orbit radius in this case. As b → 0, we can find that since the coefficients D, E and F vanish the Eq.(31) reduces to a quadratic equation and the marginally circular photon orbit radius becomes ρ ps = , which is consistent with that in the Schwarzschild squashed KK black hole [33]. As ρ 0 → 0, one can get ρ ps = 1 , which decreases with the rotation parameter b and tends to 3 2 ρ M as b disappears. Moreover, in the limit ρ 0 → ∞, we have with Obviously, it also decreases with the rotation parameter b. In Fig.(1 The deflection angles φ and ψ for the photon coming from infinite in a rotating KK black hole spacetime can be expressed as respectively. The quantities I φ (ρ s ) and I ψ (ρ s ) have the forms with where ρ s is the closest approach distance of the light ray. It is clear that both of the deflection angles increase when parameter ρ s decreases. If ρ s is equal to the marginally stable circular radius of photon ρ ps , one can find that both of the deflection angles becomes unboundedly large and the photon is captured in a circular orbit around the black hole. Let us now discuss the behavior of the deflection angles of the light rays in a rotating squashed KK black hole spacetime. It is interesting to note that the deflection angle α φ (ρ s ) is independent of whether the photon goes with or against the rotation of the black hole because the integral I φ (ρ s ) is function of the rotation parameter b 2 . However, from Eq. (39), we find that the integral I ψ (ρ s ) contains the factor b, which means that the deflection angle α ψ (ρ s ) for the photon traveling in the same direction as the rotation of the black hole is different from that traveling in converse direction. This tells us that although the black hole has the rotation paraters both in the ψ and φ directions, in the equatorial plane θ = π 2 the rotation of the black hole is really in the ψ direction rather than in the φ direction, which is also shown in the induce metric (23) where the only cross-term is dtdψ. It means that the gravitational lensing by the rotating squashed KK black hole is different from that of usual four-dimensional Kerr black hole, which could in theory help us to detect the extra dimension through the gravitational lens. When the rotation parameter b vanishes, one can find that the function H(ρ) = 0 and then the deflection angle of ψ tends to zero, which reduces to that of in the usual Schwarzschild squashed KK black hole spacetime [33].
As in [34], we will focus only on investigating the deflection angle in the φ direction when the light rays pass close to the black hole in the equatorial plane since it could be observed really by our astronomical experiments. On the other hand, it is very convenient for us to compare with the results obtained in the usual four-dimensional black hole spacetimes. As in [22,23], one can define a variable z = 1 − ρs ρ , and rewrite Eq.(38) as with and The function R(z, ρ s ) is regular for all values of z and ρ s . From Eq.(43), we find that the function f (z, ρ s ) diverges as z tends to zero, i.e., as the photon approaches the marginally circular photon orbit. Therefore, we can split the integral (41) into the divergent part I D (ρ s ) and the regular one I R (ρ s ) Following in refs. [22,23], we can expand the argument of the square root in f (z, ρ s ) to the second order in z with Comparing Eq. (30) with Eq.(46), one can find that if ρ s tends to ρ ps the coefficient p(ρ s ) vanishes. This means that the leading term of the divergence in f s (z, ρ s ) is z −1 and the integral (41) diverges logarithmically.
Thus, in the strong field region, the deflection angle in the φ direction can be approximated very well as [22]   Here the quantity D OL is the distance between observer and gravitational lens, θ = u/D OL is the angular separation between the lens and the image, the subscript "ps" represent the evaluation at ρ = ρ ps . Similarly, one can obtain the strong gravitational lensing formula for the deflection angle in the ψ direction ( α ψ (θ)), which has a similar form with the coefficients differed slightly fromā andb in Eq.(48). As ρ s tends to ρ ps , we find that the deflection angle α ψ (θ) also diverges logarithmically. Since α ψ (θ) cannot actually be observed by astronomical experiments, we do not consider it in the following discussion. for the larger one. Furthermore, in Fig.(4), we plotted the change of the deflection angle α(θ) estimated at u = u ps + 0.003 with ρ 0 and b, which tells us that in the strong field limit the deflection angles have similar properties of the coefficientā. This means that the deflection angles of the light rays are dominated by the logarithmic term in this case.

IV. OBSERVATIONAL GRAVITATIONAL LENSING PARAMETERS
In this section, we will estimate the numerical values for the observables of gravitational lensing in the strong field limit by assuming that the spacetime of the supermassive black hole at the Galactic center of Milky Way can be described by the rotatiing squashed KK black hole metric (19) and then probe the effects of the rotation parameter b and the scale parameter ρ 0 on the observables in the strong gravitational lensing.
As the source and observer are far enough from the lens, the lens equation can be approximated well as [23] where γ is the angle between the direction of the source and the optical axis. D LS is the lens-source distance and D OL is the observer-lens distance. θ = u/D OL is the angular separation between the lens and the image.
Following ref. [23], we here consider only the case in which the source, lens and observer are highly aligned. In this simplest case, the angular separation between the lens and the n−th relativistic image can be written as Here θ 0 n is the image position corresponding to α = 2nπ, and n is an integer. As n → ∞, one can find from Eqs.(51) that e n → 0, which implies that the minimum impact parameter u ps and the asymptotic position of a set of images θ ∞ obey a simple form In order to obtain the coefficientsā andb, one needs to separate the outermost image from all the others.
Following refs. [22,23], we consider here the simplest case where only the outermost image θ 1 is resolved as a single image and all the remaining ones are packed together at θ ∞ . And then the angular separation between the first image and other ones s and the ratio of the flux from the first image and those from the all other images R 0 can be simplified further as [22,23] Therefore, one can estimate the strong deflection limit coefficientsā,b and the minimum impact parameter Recently, the mass of the central object of our Galaxy is estimated to be 4.4 × 10 6 M ⊙ and its distance is around 8.5kpc [39]. This means the ratio of the mass of the central object to the distance G 4 M/D OL ≈ 2.4734 × 10 −11 . Here M is the mass of the black hole and D OL is the distance between the lens and the observer in the ρ coordination rather than that in r coordination because that in the five-dimensional spacetime the dimension of the black hole mass M is the square of that in the polar coordination r. Finally, we make a comparison among the strong gravitational lensings in the rotating squashed KK, the squashed KK Gödel and four-dimensional Kerr black hole spacetimes. For the photon moving along the equatorial plane in the four-dimensional Kerr black hole spacetime [23,24], we know that all of the photon sphere radius, the angular position of the relativistic images θ ∞ and the relative magnitudes r m in strong