1. Introduction
Recent years have witnessed a flurry of activities regarding the astronomical measures of the general relativistic effects, especially the observation of black hole shadows, e.g. for the M87 and SgA*, and the gravitational wave data from the LIGO scientific/Virgo collaborations [1-6]. The data from the Event Horizon Telescope (EHT) regarding the shadows of black holes confirmed the validity of Einstein's general relativity (GR) and established it as a firm base for any general relativistic theory of gravity [7-12]. As known from its definition, gravitational lensing is the deflection of light rays when they pass close to black holes [13-15]. Gravitational lensing is utilized as an important tool in astrophysics; for example, it is used to measure the mass of compact objects. The formation of Einstein's ring is one of the remarkable observations made during the lensing phenomenon. The ring forms when the bright source (the star), the black hole (the lens), and the observer are aligned in a perfect way. Gravitational lensing has been investigated in detail for a variety of black hole solutions in GR as well as in other theories of gravity. For reviews on both weak and strong gravitational lensings in both equatorial (θ = π/2) and non-equatorial (θ ≠ π/2) planes, we refer readers to [16-23] and the references therein. Other analytical studies of black holes have been reviewed and highlighted in GR as well as in the modified scenarios [24, 25]. The connection of the shadow and lensing properties has been explored in the context of modified gravity theories [26, 27].
From the strong lensing point of view, the light rays pass close enough to the black hole wind once or several times before leaving that region and reaching the observer. The idea that black holes act as gravitational lenses was first put forward by Darwin [13] and subsequently elaborated in [28, 29]. The most recent theoretical investigation of strong-field gravitational lensing is mainly attributed to the seminal work of Virbhadra and Ellis [30-32], where they discussed the formation of the ring, its position, and the magnification of relativistic images for the Schwarzschild black hole. Another work discussed the role of the scalar-to-mass ratio [33] in the lensing context. After this formulation, Frittelli, Kling and Newman obtained more rigorous analytical descriptions of the exact lens equation and the integral expressions for its solution [14]. A comparison of their results with those given by Virbhadra and Ellis was also presented. Later, Bozza et al [16-19] and then Tsukamoto [34, 35] proposed a completely new method for the study of strong lensing for a generic spherically symmetric static space-time. These methods are widely accepted by the scientific community and are still used as applications in the study of strong lensing phenomena. These methods were applied to study the various black hole systems, including solutions in various modified theories of gravity [36-53]. More recently, the modeling of the supermassive compact object M87* as the Schwarzschild black hole and subsequent analysis of the variations of the three different magnified images of different orders have been explored in the observational context [54]. These investigations may result in many outcomes that may be important for the operation of the generation EHT (ngEHT). Further, the ratios of the mass to distance, and the lens-source to observer-source are used to measure the compactness of the black hole lenses [55].
We take one presumption regarding the validity of the four-dimensional Ohanian lens equation in a five-dimensional scenario and we shed light into the issues concerned with the extra dimensions, examining the validity of the four-dimensional lens equations in a dimension 4 + n, where n is the number of the extra spatial dimensions. As the ratios of mass to radius for the planets in our solar system are quite small, the corrections to the gravitational lensing by planetary objects in the solar system are negligibly small and hence beyond the reach of near future experiments [56]. Moreover, with the gravitational lensing by supermassive black holes with known masses and radii, there will be potential effects of the extra dimensions in larger amounts. The future precision experiments from LIGO/Virgo or the EHT or ngEHT on the strong gravitational lensing for supermassive black holes can provide important information about the possible viable theory of gravitation apart from Einstein's GR, and hopefully about the extra dimensions [57-61]. There are many modified theories of gravity, such as tensor-scalar theories or theories with extra spatial dimensions apart from the usual four space-time dimensions. It is legitimate to speculate on ideas about these alternative gravity theories-which may be consistent with precision tests using observational data-and find an ultimate theory of gravity. Different types of gravitational interactions may arise in the modified theories, in addition to the existing interactions in Einstein's GR counted as the correction terms. We hope that the experimental measurements can put strong constraints on other theories or even rule out some of the theories. In this paper we study gravitational lensing in theories with extra dimensions along the same line of thought as outlined above. As a result, we can observe the various observable parameters such as the deflection angle, the angular position, the angular separation, the image magnifications, and also the relativistic time-delay effects. All of these parameters are highly affected due to the introduction of the extra dimensions.
The string theory is a promising candidate for the unified theory, where gravity is unified with other forces of nature. The prediction of extra dimensions and its relation to the physical observable attracted a lot of attention because it borrowed the signature of the string and hence made string theory a correct one. The search for extra dimensions had been a challenging task and a long-standing problem. It always has been an active area of investigation after the use of quasinormal modes (QNMs) arising from the perturbed black holes space-times in higher dimensions. It may also be made possible through the analysis of the spectrum of the Hawking radiation. Extra dimensions could be detected through the investigation of QNMs’ spectra of higher-dimensional space-times [62-74], which may be probed via gravitational wave detectors in the near future. One of the possible insights may be useful when one studies the higher-dimensional black holes and black branes. Additionally, the correlation between strong gravitational lensing and QNMs in the high momentum or eikonal approximation could have been probed if careful investigation had been carried out. In this context, strong gravitational lensing is an effective method for investigating the presence of dimensions higher than four. The search for extra dimensions was probed through the gravitational lensing of a charged-neutral/charged Kaluza-Klein squashed black hole [75-77], the squashed Kaluza-Klein Gödel black hole for both charged and charged-neutral cases [78, 79], the black hole with extra dimensions and the Kalb-Ramond field [80, 81]. These studies help us to understand the effective length scale on which the extra dimension lies. Motivated by these investigations, in our present paper, we wish to study the strong gravitational lensing of a five-dimensional charged equally rotating black hole in the presence of a negative cosmological constant, as given in [82], when the charged scalar hair has a vanishing limit. This black hole solution was originally proposed by Cvetič, Lü and Pope [83], hereby denoted as the CLP black hole. Such a solution is considered as a coupling of the Einstein-Maxwell system that arises as the bosonic sector of the minimal-gauged five-dimensional supergravity theories. As for the special cases, such a solution reduces all the previously known cases. The solution is characterized by the mass, the charge, the angular momentum, and the cosmological constant. We shall see how the charge, the rotation parameter, and other defining parameters affect the size of the horizon, the photon sphere radius, the formation of relativistic images, the deflection angle, and subsequently other physical observables in the strong-field lensing of gravity. On the other hand, in relation to the AdS/CFT correspondence, if one is allowed to evaluate the retarded two point functions (response functions) in a gauge covariant theory, it could be made possible by studying their fluctuations around any asymptotically AdS black holes. The poles of these response functions are the characteristic features of the QNMs' frequencies of the AdS black hole space-times. The black hole space-times in higher dimensions are very important in the context of string theory and its subsequent analysis through the AdS/CFT correspondence. Thus, the study of strong gravitational lensing as well as the QNMs in the higher-dimensional black holes from the mirror of the AdS/CFT correspondence may help us to extract information about extra dimensions in astronomical observations in future experiments.
The main physical quantity that has played a major role in predicting the bending of light beams around black holes in any gravity theory, whether it is the asymptotically flat or AdS space-time, is the photon sphere radius. For the study of the photon sphere in the AdS/CFT realm, our findings may be based on three important streams. Firstly, a symmetry group SL(2, R) indicates the set of QNM frequencies at the photon sphere radius of the Schwarzschild and Kerr black holes' space-times at the asymptotic flat limits [84, 85]. An extension of such a study was introduced into the search of the warped geometries and subsequent investigations were carried out [86-89]. Secondly, the QNMs' frequencies for the Schwarzschild-AdS black hole is a long-standing pathway: the physics has a very rich structure in its own right, and remarkable study of the WKB approximation has been performed for the QNMs' analyses of the various AdS black holes [90-92]. Thirdly, the photon sphere radius has emerged as a direct consequence of the Einstein ring while imaging the black holes in any gravitational theory, and the transformation of the images on the onset of the holographic CFTs was used to produce the Einstein rings of such types [93, 94].
The organization of the paper is as follows. In section 2 , we briefly discuss the solution of the five-dimensional charged equally rotating black hole solution, as given by Cvetič, Lü and Pope. We also investigate the horizon structure and parametric bounds on the charge and rotation parameters. Section 3 is devoted to discussing the formulation of the deflection angle using Bozza's method. We discuss the effective potential, the photon sphere radius, the impact parameter, and their connections. We also discuss them in the limiting cases, when j = 0. Next, we discuss the lensing observable and the formation of the relativistic images of the black hole in section 4 . Here, we model our concerned black holes as SgrA* and M87* to show the effect of the parameters. In section 5 , we discuss the time-delay effect. Finally, in section 6 , we summarize the paper and conclude our results.
2. The five-dimensional charged equally rotating black hole solution with a cosmological constant
We present a brief review of the action and the constituent space-time before we move on to the main discussion of the lensing phenomena. The expression for the action is given by a consistent truncation and is written as [82, 95]1 ), we put the cosmological constant or the curvature radius to be equal to unity. Later, we shall restore it whenever required. The variation of the action (1) with respect to the metric tensor gμν yields Einstein's field equations [82, 95]
$\begin{eqnarray}\begin{array}{rcl}S & = & \frac{1}{16\pi {G}_{5}}\displaystyle \int {{\rm{d}}}^{5}x\sqrt{-g}\left(R+12-\frac{3}{4}{F}_{\mu \nu }{F}^{\mu \nu }\right.\\ & & -\,\frac{3}{8}\left({\left|{D}_{\mu }\phi \right|}^{2}-\frac{{\partial }_{\mu }(\phi {\phi }^{* }){\partial }^{\mu }(\phi {\phi }^{* })}{4(4+\phi \phi * )}-4\phi \phi * \right)\\ & & \left.+\,\frac{1}{4\sqrt{-g}}{\epsilon }^{\alpha \beta \gamma \mu \nu }{F}_{\alpha \beta }{F}_{\gamma \mu }{A}_{\nu }\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}{D}_{\mu }\phi ={\partial }_{\mu }\phi -2i{A}_{\mu }\phi ,\ {F}_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.\end{eqnarray}$
It is worth mentioning that in equation ( $\begin{eqnarray}{R}_{\mu \nu }-\frac{1}{2}\left(R+12\right){g}_{\mu \nu }=-\frac{3}{2}{T}_{\mu \nu }^{\,\rm{EM}\,}+\frac{3}{8}{T}_{\mu \nu }^{\phi }.\end{eqnarray}$
The expressions for the energy-momentum tensor of the electromagnetic field ${T}_{\mu \nu }^{\,\rm{EM}\,}$, and the scalar field ${T}_{\mu \nu }^{\phi }$ are $\begin{eqnarray}\begin{array}{rcl}{T}_{\mu \nu }^{\,\rm{EM}\,} & = & {F}_{\mu }^{\alpha }{F}_{\nu \alpha }-\frac{1}{2}{g}_{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta },\\ {T}_{\mu \nu }^{\phi } & = & \frac{1}{2}\left[{D}_{\mu }\phi ({D}_{\nu }\phi )* +{D}_{\nu }\phi \left({D}_{\mu }\phi \right)* -\frac{1}{2}{g}_{\mu \nu }{\left|{D}_{\alpha }\phi \right|}^{2}+2\phi \phi * {g}_{\mu \nu }\right.\\ & & -\,\left.\frac{1}{4(4+\phi {\phi }^{* })}\left({\partial }_{\mu }(\phi \phi * ){\partial }_{\nu }\left(\phi \phi * \right)-\frac{1}{2}{g}_{\mu \nu }\left[\left({\partial }_{\sigma }\phi \phi * \right)\left({\partial }^{\sigma }\phi {\phi }^{* }\right)\right]\right)\right].\end{array}\end{eqnarray}$
The Maxwell and the scalar field equations are given, respectively, as [82, 95] $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}_{\nu }\,{F}_{\mu }^{\nu } & = & \frac{i}{4}\left[\phi {({D}_{\mu }\phi )}^{* }-{\phi }^{* }({D}_{\mu }\phi )\right.\\ & & +\,\left.\frac{1}{4\sqrt{-g}}{g}_{\mu \sigma }{\epsilon }^{\sigma \alpha \beta \gamma \nu }{F}_{\alpha \beta }{F}_{\gamma \nu }\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{D}_{\mu }{D}^{\mu }\phi +\left[\frac{{\left[{\partial }_{\sigma }(\phi {\phi }^{* })\right]}^{2}}{4{(4+\phi {\phi }^{* })}^{2}}-\frac{{{\rm{\nabla }}}^{2}(\phi {\phi }^{* })}{2(4+\phi {\phi }^{* })}\right]\phi =0.\end{eqnarray}$
The stationary, asymptotically AdS space-times have spherical horizon topology. The generic solutions to such a structure have two independent rotation parameters. To keep the structure simpler, we consider the doubly rotating black hole solutions, having the same magnitude of the rotation parameters but with two different orientations. For any generic gauge choice, we can write down the metric ansatz as follows [82]: $\begin{eqnarray}\begin{array}{rcl}{{\rm{d}}s}^{2} & = & -f(r){{\rm{d}}t}^{2}+g(r){{\rm{d}}r}^{2}+{{\rm{\Sigma }}}^{2}(r)\left[h(r)\left({\rm{d}}\psi \right.\right.\\ & & \left.{\left.+\,\frac{1}{2}\cos \theta {\rm{d}}\phi -{\rm{\Omega }}(r){\rm{d}}t\right)}^{2}+\frac{1}{4}{{\rm{d\Omega }}}_{2}^{2}\right],\end{array}\end{eqnarray}$
which is rewritten as $\begin{eqnarray}\begin{array}{rcl}{{\rm{d}}s}^{2} & = & \left(-f(r)+{{\rm{\Sigma }}}^{2}(r){{\rm{\Omega }}}^{2}(r)h(r)\right){{\rm{d}}t}^{2}+g(r){{\rm{d}}r}^{2}\\ & & +\,\frac{1}{4}{{\rm{\Sigma }}}^{2}(r){{\rm{d\theta }}}^{2}-{{\rm{\Sigma }}}^{2}(r){\rm{\Omega }}(r)h(r)\cos \theta {\rm{d}}\phi {\rm{d}}t\\ & & +\,\frac{{{\rm{\Sigma }}}^{2}(r)}{4}\left(h(r){\cos }^{2}\theta +{\sin }^{2}\theta \right){\rm{d}}{\phi }^{2}\\ & & +\,{{\rm{\Sigma }}}^{2}(r)h(r){\rm{d}}{\psi }^{2}-2{{\rm{\Sigma }}}^{2}(r){\rm{\Omega }}(r)h(r){\rm{d}}\psi {\rm{d}}t\\ & & +\,{{\rm{\Sigma }}}^{2}(r)h(r)\cos \theta {\rm{d}}\psi {\rm{d}}\phi .\end{array}\end{eqnarray}$
In the above ansatz, we can see that the metric on S3 is expressed as a Hopf fibration over the space ${\mathbb{C}}{{\mathbb{P}}}^{1}$ of unit radius. In this way, the coordinate ψ has the period of 2π and the θ, Φ coordinates have the coordinate ranges of the usual unit S2 sphere. General five-dimensional, charged, and equally rotating black hole solutions with a cosmological constant were first presented in [83], which is basically a coupled Einstein-Maxwell solution in the bosonic sector of minimal-gauged supergravity, and also Φ = 0 limit of the action (1). The black holes are governed by three hairs {q, j, m}, and an additional fourth parameter, the cosmological constant term. The parameters {q, j, m} are related to the conserved charges {Q, J, M}. The functions that appear in the metric (8) have the explicit forms $\begin{eqnarray}\begin{array}{rcl}{\rm{\Sigma }}(r) & = & r,\,f(r)=\frac{G(r)}{h(r)},\,g(r)=\frac{1}{G(r)},\\ h(r) & = & 1+{j}^{2}\left(\frac{m}{{r}^{4}}-\frac{{q}^{2}}{{r}^{6}}\right),\,{\rm{\Omega }}(r)=\frac{j}{h(r)}\left(\frac{m-q}{{r}^{4}}-\frac{{q}^{2}}{{r}^{6}}\right),\\ G(r) & = & \frac{1}{{r}^{4}}\left[{q}^{2}\left(1-\frac{{j}^{2}}{{L}^{2}}\right)+{j}^{2}m\right]-\frac{1}{{r}^{2}}\left[m\left(1-\frac{{j}^{2}}{{L}^{2}}\right)-2q\right]\\ & & +\,\frac{{r}^{2}}{{L}^{2}}+1.\end{array}\end{eqnarray}$
The angular momentum J and the mass M of the black hole are related to the parameters {q, j, m} as follows: $\begin{eqnarray}J=\frac{1}{2}j\left(m-q\right),\,M=\frac{1}{4}\left[m\left(3+\frac{{j}^{2}}{{L}^{2}}\right)-6q\right],\,Q=\frac{q}{2}.\end{eqnarray}$
Using the last two relations of the above equations, we can express m and q as $\begin{eqnarray}m=\frac{2{L}^{2}(2M+6Q)}{{j}^{2}+3{L}^{2}},\,q=2Q.\end{eqnarray}$
Please note that in the action, we put the AdS radius L = 1. Here, we restored it for the sake of clarity and calculation. The static and spherically symmetric part of the solution is obtained if one puts $\begin{eqnarray}{{\rm{d}}s}^{2}=-f(r){{\rm{d}}t}^{2}+g(r){{\rm{d}}r}^{2}+{{\rm{\Sigma }}}^{2}(r){{\rm{d\Omega }}}_{3}^{2},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{{\rm{d\Omega }}}_{3}^{2} & = & {\left({\rm{d}}\psi +\frac{1}{2}\cos \theta {\rm{d}}\phi \right)}^{2}+\frac{1}{4}{{\rm{d\Omega }}}_{2}^{2}\\ & = & {{\rm{d\theta }}}^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}+{\cos }^{2}\theta {\rm{d}}{\psi }^{2}.\end{array}\end{eqnarray}$
Obviously, the metric (12) is the five-dimensional Reissner-Nordström black holes with a cosmological constant, where f(r) and g(r) are given by [83] $\begin{eqnarray*}f(r)=1+\frac{{r}^{2}}{{L}^{2}}-\frac{m}{{r}^{2}}+\frac{{q}^{2}}{{r}^{4}},\ \ \ g(r)=1/f(r).\end{eqnarray*}$
The metric possesses three Killing vectors ∂t, ${\partial }_{\hat{\phi }}$, and ${\partial }_{\hat{\psi }}$, where $\hat{\phi }=\psi -\phi $, $\hat{\psi }=\psi +\phi $. For an equally rotating black hole, there are two additional Killing vectors [96, 97]
$\begin{eqnarray}\cos \hat{\phi }{\partial }_{\hat{\theta }}-\cot \hat{\theta }\sin \hat{\phi }{\partial }_{\hat{\phi }}+\frac{\sin \hat{\phi }}{\sin \hat{\theta }}{\partial }_{\hat{\psi }},\end{eqnarray}$
$\begin{eqnarray}-\,\sin \hat{\phi }{\partial }_{\hat{\theta }}-\cot \hat{\theta }\cos \hat{\phi }{\partial }_{\hat{\phi }}+\frac{\cos \hat{\phi }}{\sin \hat{\theta }}{\partial }_{\hat{\psi }},\end{eqnarray}$
where $\hat{\theta }=2\theta $.The event horizon is a well-defined boundary that is a null hypersurface and it comprises the outward null geodesics, which are not capable of hitting the null infinity in the future. The event horizon is a solution of F(r) = 0: 11 ). As j → 0, the horizon radius reduces to the five-dimensional charged AdS black hole. For both q, j → 0, we have the five-dimensional Schwarzschild-AdS black hole. In addition to q, j → 0, if we take L → ∞, we have ${r}_{h}^{2}=m$, which is the radius of the five-dimensional Schwarzschild-Tangherlini space-time. In figure 1 we plot the Cauchy horizon (blue dashed line) and the event horizon (black solid lines) with respect to the variation of the charge parameter (left figure) and the rotation parameter (right figure), respectively. To show the parametric bounds on q and j, we plot the charge parameter q as a function of the rotation parameter in figure 2. The region below the red line corresponds to the allowed region, whereas the region above the red line is forbidden for the parameters representing the black holes.
$\begin{eqnarray*}\begin{array}{l}{r}_{h}^{6}+{L}^{2}{r}_{h}^{4}+{r}_{h}^{2}\left({j}^{2}m-{L}^{2}m+2{L}^{2}q\right)\\ \quad +\,{j}^{2}{L}^{2}m-{j}^{2}{q}^{2}+{L}^{2}{q}^{2}=0,\end{array}\end{eqnarray*}$
which is a cubic equation in ${r}_{h}^{2}$. The solutions are given by $\begin{eqnarray}\begin{array}{rcl}{r}_{h,n}^{2} & = & 2\sqrt{\frac{\frac{{L}^{4}}{3}+{L}^{2}m-{j}^{2}m-2{L}^{2}q}{3}}\cos \\ & & \times \,\left[\frac{1}{3}\arccos \left(\frac{2{j}^{2}{L}^{2}m-3{j}^{2}{q}^{2}+\frac{2{L}^{6}}{9}+{L}^{4}m-2{L}^{4}q+3{L}^{2}{q}^{2}}{2{j}^{2}m-\frac{2{L}^{4}}{3}-2{L}^{2}m+4{L}^{2}q}\right)\right.\\ & & \left.\sqrt{\frac{3}{\frac{{L}^{4}}{3}+{L}^{2}m-{j}^{2}m-2{L}^{2}q}}-\frac{2\pi n}{3}\right],\ \ \ n=0,1,2,\end{array}\end{eqnarray}$
which indicates that the black hole has two horizons because of the two real positive roots. The constants (m, q) are related to (M, Q), as given in equation (
Figure 1. Plots of the Cauchy horizon (blue dashed line) and the event horizon (black solid lines) vs the charge parameter Q for j = 0.1, L = 10 (the left plot) and the rotation parameter j for Q = 0.1, L = 10 (the right plot). |
Figure 2. Plot of the black hole charge parameter Q / M vs the rotation parameter $j/\sqrt{M}$. This is the parameter space in the plane $(Q/M,j/\sqrt{M})$ for the CLP black hole. The red line represents the extremal black hole with degenerate Cauchy and the event horizons. The region above the red line is the no black hole region while the region below it corresponds to the black hole region. |
3. Deflection angle in a five-dimensional charged equally rotating black hole with a cosmological constant
In this section, we discuss the deflection angle of light rays when they pass by a black hole. We also investigate the strong-field gravitational lensing phenomena in the five-dimensional, charged equally rotating CLP black hole. We see the effect of the charge q and the rotation parameter j. For calculation simplicity, we restrict ourselves to the case of the equatorial plane, $\theta =\frac{\pi }{2}$. Therefore, equation (7 ) reduces to
$\begin{eqnarray}\begin{array}{rcl}{{\rm{ds}}}^{2} & = & \left(\right.-f(r)+{{\rm{\Sigma }}}^{2}(r){{\rm{\Omega }}}^{2}(r)h(r)\left)\right.{{\rm{dt}}}^{2}+g(r){{\rm{dr}}}^{2}\\ & & +\,{{\rm{\Sigma }}}^{2}(r)h(r){\rm{d}}{\psi }^{2}+\frac{1}{4}{r}^{2}{{\rm{\Sigma }}}^{2}(r){\rm{d}}{\phi }^{2}\\ & & -\,2{{\rm{\Sigma }}}^{2}(r)h(r){\rm{\Omega }}(r){\rm{d}}\psi {\rm{dt}}.\end{array}\end{eqnarray}$
The components of the metric tensor are written in a more compact functional form as 7 ) as
$\begin{eqnarray}\begin{array}{rcl}{g}_{tt} & = & {\partial }_{t}\cdot {\partial }_{t}=-f(r)+{{\rm{\Sigma }}}^{2}(r){{\rm{\Omega }}}^{2}(r)h(r)=-A(r),\\ {g}_{rr} & = & {\partial }_{r}\cdot {\partial }_{r}=g(r)=B(r),\\ {g}_{\phi \phi } & = & {\partial }_{\phi }\cdot {\partial }_{\phi }=\frac{1}{4}{r}^{2}{{\rm{\Sigma }}}^{2}(r)=C(r),\\ {g}_{\psi \psi } & = & {\partial }_{\psi }\cdot {\partial }_{\psi }={{\rm{\Sigma }}}^{2}(r)h(r)=D(r),\\ {g}_{\psi t} & = & {\partial }_{t}\cdot {\partial }_{\psi }=-{{\rm{\Sigma }}}^{2}(r){\rm{\Omega }}(r)h(r)={g}_{t\psi }=-H(r).\end{array}\end{eqnarray}$
We can rewrite equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{ds}}}^{2} & = & -A(r){{\rm{dt}}}^{2}+B(r){{\rm{dr}}}^{2}+C(r){\rm{d}}{\phi }^{2}\\ & & +\,D(r){\rm{d}}{\psi }^{2}-2H(r){\rm{d}}\psi {\rm{dt}}.\end{array}\end{eqnarray}$
It is worth noting that the coordinates (t, r, Φ, ψ) are now rescaled to be $t\to t/\sqrt{M}$, $r\to r/\sqrt{M}$, $\phi \to \phi /\sqrt{M}$, and $\psi \to \psi /\sqrt{M}$. Similarly, the parameters defining the black hole are rescaled in a dimensionless way as follows: Q → Q/M, $j\to j/\sqrt{M}$ and $L\to L/\sqrt{M}$. Therefore from now on, all the coordinates, including the black hole parameters, are understood as dimensionless quantities. The above ansatz has residual gauge freedom ψ → ψ + αt and Ω → Ω + α; here Ω is the angular velocity and it is symmetric for t → - t, Φ → Φ + 2π and ψ → ψ + π. The Killing vectors are ${\eta }_{t}^{\mu }={\delta }_{t}^{\mu }$, ${\eta }_{\phi }^{\mu }={\delta }_{\phi }^{\mu }$ and ${\eta }_{\psi }^{\mu }={\delta }_{\psi }^{\mu }$. The equations of motion are $\begin{eqnarray}\begin{array}{rcl}{ \mathcal E } & = & -{g}_{0\mu }\dot{{x}^{\mu }}=-{g}_{tt}\dot{t}-{g}_{t\psi }\dot{\psi }=A(r)\dot{t}+H(r)\dot{\psi },\\ {L}_{\phi } & = & {g}_{3\mu }\dot{{x}^{\mu }}={g}_{\phi \phi }\dot{\phi }=C(r)\dot{\phi },\\ {L}_{\psi } & = & {g}_{4\mu }\dot{{x}^{\mu }}={g}_{\psi \psi }\dot{\psi }+{g}_{\psi t}\dot{t}=D(r)\dot{\psi }-H(r)\dot{t}.\end{array}\end{eqnarray}$
Rearranging equation (20 ) we have 23 ) reduces to
$\begin{eqnarray}\begin{array}{rcl}\dot{\psi } & = & \frac{A(r){L}_{\psi }+H(r){ \mathcal E }}{{H}^{2}(r)+A(r)D(r)},\\ \dot{t} & = & \frac{D(r){ \mathcal E }-H(r){L}_{\psi }}{{H}^{2}(r)+A(r)D(r)},\\ \dot{\phi } & = & \frac{{L}_{\phi }}{C(r)},\end{array}\end{eqnarray}$
whereas the radial null geodesic is obtained by using ds2 = 0 $\begin{eqnarray}{\dot{r}}^{2}=\frac{1}{B(r)}\left[\frac{(D(r){ \mathcal E }-2H(r){L}_{\psi }){ \mathcal E }-A(r){L}_{\psi }^{2}}{{H}^{2}(r)+A(r)D(r)}-\frac{{L}_{\phi }^{2}}{C(r)}\right].\end{eqnarray}$
The geodesic equation of motion for θ is $\begin{eqnarray}\begin{array}{l}\frac{{C}^{{\prime} }(r)}{2C(r)}\frac{{\rm{d\theta }}}{{\rm{d\lambda }}}\frac{{\rm{d}}r}{{\rm{d\lambda }}}-\frac{H(r)\sin \theta }{2C(r)}\frac{{\rm{d}}\phi }{{\rm{d\lambda }}}\frac{{\rm{d}}t}{{\rm{d\lambda }}}\\ \,+\,\frac{\cos \theta \sin \theta (-C(r)+D(r))}{C(r)}\left(\right.\frac{{\rm{d}}\phi }{{\rm{d\lambda }}}{\left)\right.}^{2}\\ \,+\,\frac{D(r)\sin \theta }{2C(r)}\frac{{\rm{d}}\phi }{{\rm{d\lambda }}}\frac{{\rm{d}}\psi }{{\rm{d\lambda }}}=0.\end{array}\end{eqnarray}$
For θ = π/2, equation ( $\begin{eqnarray}\frac{{\rm{d}}\phi }{{\rm{d\lambda }}}\left(\right.-H(r)\frac{{\rm{d}}t}{{\rm{d\lambda }}}+D(r)\frac{{\rm{d}}\psi }{{\rm{d\lambda }}}\left)\right.=0,\end{eqnarray}$
which gives us either $\frac{{\rm{d}}\phi }{{\rm{d\lambda }}}=0$, i.e. LΦ = 0 or ${L}_{\psi }=\left(\right.-H(r)\frac{{\rm{d}}t}{{\rm{d\lambda }}}+D(r)\frac{{\rm{d}}\psi }{{\rm{d\lambda }}}\left)\right.=0$. In order to get simpler results, we shall consider the case when Lψ = 0, which means that the total angular momentum of the photon J is equal to LΦ and the effective potential ${V}_{\mathrm{eff}}=-{\dot{r}}^{2}$, where ${\dot{r}}^{2}$ under such conditions is given by $\begin{eqnarray}{\dot{r}}^{2}=\frac{1}{B(r)}\left[\frac{D(r){{ \mathcal E }}^{2}}{{H}^{2}(r)+A(r)D(r)}-\frac{{L}_{\phi }^{2}}{C(r)}\right].\end{eqnarray}$
To compute the impact parameter, we use Veff = 0, $\begin{eqnarray}u=J={L}_{\phi }/{ \mathcal E }=\sqrt{\frac{C({r}_{0})D({r}_{0})}{{H}^{2}({r}_{0})+A({r}_{0})D({r}_{0})}},\end{eqnarray}$
where u denotes the impact parameter. The variation of Veff with respect to the radial distance r is shown in figure 3 for different values of the impact parameter u at fixed Q and fixed j values. We observe that at us = usc, we have the critical impact parameter (red solid line), which depends on the parameters Q and j. At us = usc we have the unstable circular orbits along which the photon propagates. The photon orbit radius, as a function of the charge Q and the rotation parameter j, is depicted in figure 4. The photon orbit decreases quickly when plotted as a function of the charge Q. It decreases slowly for lower values of j and for higher values of j, the decrease in the photon orbit radius is enhanced. The case for the impact parameter is similar, as shown in figure 5.
Figure 3. The behavior of the effective potential Veff with respect to the dimensionless radial coordinate r for different values of the impact parameter u. |
Figure 4. The variations of the photon sphere radius rs with respect to the charge parameter Q (the left plot) and the rotation parameter j (the right plot). |
Figure 5. The variations of the impact parameter us with respect to the charge parameter Q (the left plot) and the rotation parameter j (the right plot). |
The equation of the circular orbit of a photon is obtained using ${V}_{\,\rm{eff}\,}^{{\prime} }=0$, 26 ) and (27 ) are more complex than those in the usual spherical, symmetric black hole space-time. As a limiting case, when the parameter j → 0, we have H(r) → 0 and therefore, equation (27 ) has a much simpler form,
$\begin{eqnarray}\begin{array}{l}C({r}_{0})\left[D{({r}_{0})}^{2}A^{\prime} ({r}_{0})-H{({r}_{0})}^{2}D^{\prime} ({r}_{0})+2D({r}_{0})H({r}_{0})H^{\prime} ({r}_{0})\right]\\ \quad -\,D({r}_{0})C^{\prime} ({r}_{0})\left[A({r}_{0})D({r}_{0})+H{({r}_{0})}^{2}\right]=0,\end{array}\end{eqnarray}$
where r0 is the closest of the photon spheres. Equations ( $\begin{eqnarray}\begin{array}{l}C({r}_{0})A^{\prime} ({r}_{0})-C^{\prime} ({r}_{0})A({r}_{0})={L}^{2}\left(4{q}^{2}+6qx+x(2x-3)\right)\\ \quad +\,{x}^{3}=0,\ \ x={r}_{0}^{2}.\end{array}\end{eqnarray}$
The above equation is cubic in nature and the roots are easily derivable as $\begin{eqnarray}\begin{array}{rcl}{x}_{k} & = & 2\sqrt{-\frac{{ \mathcal P }}{3}}\cos \left[\frac{1}{3}\arccos \left(\frac{3{ \mathcal Q }}{2{ \mathcal P }}\sqrt{\frac{-3}{{ \mathcal P }}}\right)-\frac{2\pi k}{3}\right],\\ k & = & 0,1,2,\end{array}\end{eqnarray}$
where ${ \mathcal P }=\left(6q-3\right){L}^{2}-\frac{4{L}^{4}}{3}$, ${ \mathcal Q }=4{l}^{2}{q}^{2}+2{L}^{4}\left(1-2q\right)-\frac{16{L}^{6}}{27}$. Obviously, for q = 0, we have only one root that corresponds to $x=\sqrt{{L}^{2}\left({L}^{2}+3\right)}-{L}^{2}$. However, we cannot determine the photon sphere radius analytically for j ≠ 0 so we have computed it numerically. The value of the impact parameter for j = 0 is computed to be $\begin{eqnarray}\begin{array}{rcl}{u}_{s}{| }_{j=0} & = & \frac{L{{ \mathcal X }}_{3}^{2}}{2\sqrt{3}{{{ \mathcal X }}_{2}}^{1/3}\left(27{L}^{2}{q}^{2}{{ \mathcal X }}_{2}+{L}^{2}{{ \mathcal X }}_{3}^{2}\left(4{L}^{2}-18q+{{{ \mathcal X }}_{2}}^{1/3}+9\right)+{{ \mathcal X }}_{3}\left(4{L}^{4}{{ \mathcal X }}_{2}^{2/3}-2{L}^{2}{{ \mathcal X }}_{2}+{{ \mathcal X }}_{2}^{4/3}\right)\right)},\\ {{ \mathcal X }}_{1} & = & \sqrt{-3{L}^{8}\left(4{q}^{2}-36q+9\right)-81{L}^{6}\left(8{q}^{2}-6q+1\right)+324{L}^{4}{q}^{4}},\\ {{ \mathcal X }}_{2} & = & -8{L}^{6}+27{L}^{4}(2q-1)-54{L}^{2}{q}^{2}+3{{ \mathcal X }}_{1},\\ {{ \mathcal X }}_{3} & = & 4{L}^{4}+{L}^{2}\left(-18q-2{{{ \mathcal X }}_{2}}^{1/3}+9\right)+{{ \mathcal X }}_{2}^{2/3}.\end{array}\end{eqnarray}$
In figures 6 and 7, the variations of the photon orbits and the impact parameter as functions of the horizon radius for a fixed Q (left) and a fixed j value (right) are shown, respectively. For the fixed Q value, the variation of the photon orbits as well as the impact parameter is almost constant, but the gap of the plots is highly sensitive to the j value. For a fixed j value, the variations of both rs and us are monotonically varying functions of the horizon radius. They increase as the horizon's radius increases. For vanishing angular momentum, i.e. for j = 0, we have the exact expression for the photon orbits, equation (29 ), and the minimum impact parameter, equation (30 ). We observe that, as in the four-dimensional Kerr black hole space-time, the value of the unstable circular orbit is dependent on the photon winding number in the direction of the black hole rotation (j > 0) or in the opposite direction of the black hole rotation (i.e. j < 0). It is also emphasized that for the Kaluza-Klein rotating black hole with squashed horizons, the unstable circular orbits are independent of the black hole rotation, as the rotation parameter appears to be an even power of j2.
Figure 6. The behavior of the photon orbit rs (in units of $\sqrt{M}$) with respect to the horizon radius r+ (in units of $\sqrt{M}$) for different values of the charge parameter with the fixed rotation parameter j = 0.1 (the left plot) and different values of the rotation parameter with the fixed charge parameter Q = 0.1 (the right plot). |
Figure 7. The behavior of the impact parameter us (in units of $\sqrt{M}$) with respect to the horizon radius r+ (in units of $\sqrt{M}$) for different values of the charge parameter with the fixed rotation parameter j = 0.1 (left) and for different values of the rotation parameter with the fixed charge parameter Q = 0.1 (right). |
Since ${\dot{r}}^{2}$ vanishes at the closest distance r = r0, from the trajectory equation we have 22 ), we have
$\begin{eqnarray}A({r}_{0}){\dot{t}}_{0}^{2}+2H{({r}_{0})}^{2}{\dot{t}}_{0}{\dot{\psi }}_{0}=C({r}_{0}){\dot{\phi }}_{0}^{2}+D({r}_{0}){\dot{\psi }}_{0}^{2}.\end{eqnarray}$
Using equation ( $\begin{eqnarray}\dot{r}=\frac{{\rm{d}}r}{{\rm{d\lambda }}}=\frac{1}{\sqrt{B(r)F({r}_{0})F(r)}}\sqrt{F({r}_{0})-\frac{C({r}_{0})F(r)}{C(r)}}.\end{eqnarray}$
The radial motion of the photon is governed by $\begin{eqnarray}\frac{{\rm{d}}\phi }{{\rm{d}}r}=\frac{\sqrt{B(r)F({r}_{0})F(r)}}{C(r)}\frac{1}{\sqrt{F({r}_{0})-\frac{C({r}_{0})F(r)}{C(r)}}}\end{eqnarray}$
and $\begin{eqnarray}\frac{{\rm{d}}\psi }{{\rm{d}}r}=\frac{H(r)}{D(r)}\sqrt{\frac{B(r)F({r}_{0})}{F(r)}}\frac{1}{\sqrt{F({r}_{0})-\frac{C({r}_{0})F(r)}{C(r)}}}\end{eqnarray}$
with $\begin{eqnarray}F(r)=\frac{{H}^{2}(r)+A(r)D(r)}{D(r)}.\end{eqnarray}$
The deflection angles ψ and Φ for the photon coming from the infinite can be expressed as [17] $\begin{eqnarray}\begin{array}{rcl}{I}_{\psi }({r}_{0}) & = & 2{\displaystyle \int }_{{r}_{0}}^{\infty }\frac{{\rm{d}}\psi }{{\rm{d}}r}{\rm{d}}r=2{\displaystyle \int }_{{r}_{0}}^{\infty }{\rm{d}}r\frac{H(r)}{D(r)}\\ & & \times \,\sqrt{\frac{B(r)F({r}_{0})}{F(r)}}\frac{1}{\sqrt{F({r}_{0})-\frac{C({r}_{0})F(r)}{C(r)}}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{I}_{\phi }({r}_{0}) & = & 2{\displaystyle \int }_{{r}_{0}}^{\infty }\frac{{\rm{d}}\phi }{{\rm{d}}r}{\rm{d}}r=2{\displaystyle \int }_{{r}_{0}}^{\infty }{\rm{d}}r\\ & & \times \,\frac{\sqrt{B(r)F({r}_{0})F(r)}}{C(r)}\frac{1}{\sqrt{F({r}_{0})-\frac{C({r}_{0})F(r)}{C(r)}}}.\end{array}\end{eqnarray}$
Except for the metric function H(r) which is an odd function of the rotation parameter, the other functions contain an even power of the rotation parameter. Therefore, the deflection angle corresponding to the Φ-angle is dependent on whether the black hole is in retrograde or in prograde motion. However, the deflection angle related to the angle ψ is an even function of the rotation parameter j, which means that it does not matter whether the photon is winding in the direction of the black hole rotation or in the converse direction, the deflection angle is always independent of the black hole rotation. Furthermore, unlike the four-dimensional case, in the equatorial plane (θ = π/2), the rotation of the black hole is in the ψ direction rather than in the Φ direction, which is evident from the expression of the space-time, as is given in equation (19 ), where the only cross-term is dtdψ. This also clarifies the fact that the gravitational lensing of the charged rotating black hole in five dimensions is not the same as the strong lensing phenomena of the usual four-dimensional Kerr or the Kerr-Newman black hole.
On defining $z=1-\frac{{r}_{0}}{r}$, we have 40 ), we find that f(z, r0) diverges as z tends to zero, i.e. as the photon approaches the marginally circular photon orbit. Therefore, we can split the integral equation (38 ) into the divergent part ID(r0) and the regular one IR(r0)43 ) and (44 ) we expanded f(z, r0) around z = 0 up to order z2. If we use equation (27 ), one obtains $F(r){C}^{{\prime} }(r)=C(r){F}^{{\prime} }(r)$ which leads to p(r0) = 0 and rs is the largest root of p(r0). This means that the leading term of the divergence in f0(z, r0) is z-1 and the integral equation (38 ) diverges logarithmically. Thus, in the strong-field region, the deflection angle in the direction can be approximated very well as [17]: 38 ). We can write 40 ) and S(z, r0) as
$\begin{eqnarray}{I}_{\phi }({r}_{0})={\int }_{0}^{1}R(z,{r}_{0})f(z,{r}_{0}){\rm{d}}z\end{eqnarray}$
with $\begin{eqnarray}R(z,{r}_{0})=2\frac{{r}^{2}}{{r}_{0}C(r)}\sqrt{B(r)F(r)C({r}_{0})}={P}_{1}\left(r,{r}_{0}\right),\end{eqnarray}$
$\begin{eqnarray}f(z,{r}_{0})=\frac{1}{\sqrt{F({r}_{0})-F(r)C({r}_{0})/C(r)}}={P}_{2}\left(r,{r}_{0}\right).\end{eqnarray}$
The function R(z, r0) is regular for all values of z and r0. From equation ( $\begin{eqnarray}{I}_{D,\phi }({r}_{0})={\int }_{0}^{1}R(0,{r}_{s}){f}_{0}(z,{r}_{0}){\rm{d}}z,\end{eqnarray}$
$\begin{eqnarray}{I}_{R,\phi }({r}_{0})={\int }_{0}^{1}[R(z,{r}_{0})f(z,{r}_{0})-R(0,{r}_{s}){f}_{0}(z,{r}_{0})]{\rm{d}}z.\end{eqnarray}$
We can expand the argument of the square root in f(z, r0) to the second order in z $\begin{eqnarray}{f}_{0}(z,{r}_{0})=\frac{1}{\sqrt{p({r}_{0})z+g({r}_{0}){z}^{2}}}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}p({r}_{0}) & = & \frac{{r}_{0}}{C({r}_{0})}\left[\right.{C}^{{\prime} }({r}_{0})F({r}_{0})-C({r}_{0}){F}^{{\prime} }({r}_{0})\left]\right.,\\ g({r}_{0}) & = & \frac{{r}_{0}^{2}}{2C({r}_{0})}\left[2{C}^{{\prime} }({r}_{0})C({r}_{0}){F}^{{\prime} }({r}_{0})-2{C}^{{\prime} }{({r}_{0})}^{2}F({r}_{0})\right.\\ & & +\,\left.F({r}_{0})C({r}_{0})C^{\prime\prime} ({r}_{0})-{C}^{2}({r}_{0})F^{\prime\prime} ({r}_{0})\right].\end{array}\end{eqnarray}$
To obtain equations ( $\begin{eqnarray}{\alpha }_{D,\phi }(\theta )=-\bar{a}{\mathrm{ln}}\left(\right.\frac{\theta {D}_{OL}}{{u}_{s}}-1\left)\right.+\bar{b}+O(u-{u}_{s}),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}\bar{a} & = & \frac{R(0,{r}_{s})}{2\sqrt{g({r}_{s})}},\\ \bar{b} & = & -\pi +{b}_{R}+\bar{a}{\mathrm{ln}}\left(\frac{{r}_{s}^{2}[C^{\prime\prime} ({r}_{s})F({r}_{s})-C({r}_{s})F^{\prime\prime} ({r}_{s})]}{{u}_{s}F({r}_{s})\sqrt{F({r}_{s})C({r}_{s})}}\right),\\ {b}_{R} & = & {I}_{R}({r}_{s}),\quad \quad {u}_{s}=\sqrt{\frac{C({r}_{s})}{F({r}_{s})}}.\end{array}\end{eqnarray}$
Here, the quantity DOL is the distance between the observer and the gravitational lens; θ = u/DOL is the angular separation between the lens and the image. The subscript ’s’ represents the evaluation at r = rs. Likewise, we can compute the corresponding ψ following equation ( $\begin{eqnarray}{I}_{\psi }({r}_{0})={\int }_{0}^{1}S(z,{r}_{0})f(z,{r}_{0}){\rm{d}}z\end{eqnarray}$
with f(z, r0) given by equation ( $\begin{eqnarray}S(z,{r}_{0})=2\frac{{r}^{2}H(r)}{{r}_{0}D(r)}\sqrt{\frac{B(r)F({r}_{0})}{F(r)}}.\end{eqnarray}$
The function S(z, r0) is regular for all values of z and r0. In contrast, f(z, r0) is divergent, as for the case of ID,Φ(r0), and we can follow the same expansion as well but replace $\bar{a}$ with ${\bar{a}}_{\psi }$, $\begin{eqnarray}{\bar{a}}_{\psi }=\frac{S(0,{r}_{s})}{2\sqrt{g({r}_{s})}},\end{eqnarray}$
$\begin{eqnarray}{I}_{D,\psi }({r}_{0})={\int }_{0}^{1}S(0,{r}_{s}){f}_{0}(z,{r}_{0}){\rm{d}}z,\end{eqnarray}$
$\begin{eqnarray}{I}_{S,\psi }({r}_{0})={\int }_{0}^{1}[S(z,{r}_{0})f(z,{r}_{0})-R(0,{r}_{s}){f}_{0}(z,{r}_{0})]{\rm{d}}z.\end{eqnarray}$
The study of the deflection angle only in the Φ direction would be focused on our recent purpose since it helps in observing real astronomical observations. One can likewise determine the integration regarding the strong lensing deflection angle in the ψ direction, which we denote by ${I}_{\psi ({r}_{0})}$. This term also has the coefficients, which likely have the form of $\bar{a}$ and $\bar{b}$ but differ slightly. We should mention that the deflection angle in the Φ-direction also has a term that diverges logarithmically. Nevertheless, it should not be used in determining the strong deflection phenomena from observational perspectives. In figure 8, we plot the deflection angle with respect to the impact parameter for different values of the rotation parameter j (the left figure) and the charge parameter Q (the right figure). We can see from the figure that the deflection angle increases monotonically with the critical impact parameter u and diverges for large u. Similarly, the variations of the coefficients $\bar{a}$ and $\bar{b}$ with respect to the rotation parameter and the charge parameter are shown in figures 9 and 10, respectively. We observe a monotonic variation for the coefficient $\bar{a}$-it increases with increasing Q and j. In contrast, the coefficient $\bar{b}$ increases with Q but decreases with j.
Figure 8. The variations of the deflection angle αD(u) with respect to the impact parameter u. We have set Q = 0.12 in the left plot and j = 0.3 in the right one. |
Figure 9. The variations of the deflection coefficient $\bar{a}$ with respect to the charge parameter Q (the left plot) and the rotation parameter j (the right plot). |
Figure 10. The variations of the deflection coefficient $\bar{b}$ with respect to the charge parameter Q (the left plot) and the rotation parameter j (the right plot). |
4. Observables and relativistic images
The description of the gravitational lensing phenomena is determined through the lens equation. Many methods for calculating the lens equation depend on the choice of parameters we are interested in. In our calculation, we choose the gravitational lens, where the black hole is placed in the middle of the configuration so that the observer is situated at one side, and the illuminating source of light is at another side. The light rays emanate from the source (S) and their actual paths are deviated due to the presence of the central black hole with high curvature. Then after making one or more windings, the light rays finally reach the observer position (O). In the picture of the black hole lens, the optical axis OL is the line connecting the black hole, the observer and the image, which will deviate at an angle θ with respect to the OL. On the other hand, the illuminating source of light is oriented at an angle β with respect to the OL. The light emanating from the source is detected by the observer by an angle αD(θ).
Among various mathematical formulations, the method given by Ohanian was mostly used in the literature as the lens equation [29] to approximate the position of the source, lens and observer: 52 ) along with equation (53 ), for the smaller values of θ, we have the lens equation 54 ) to know information about the image formation. We know that when the impact parameter u → us, the deflection angle αD(θ) becomes divergent. Therefore, for each loop formed during the winding near the event horizon of the black hole, we certainly have one us at which the light rays reach from the lens to the observer. Accordingly, we have infinitely more images that are formed on both sides of the black hole. Now equation (45 ) together with αD(θn 0) = 2nπ is rewritten as 55 ) along with the condition ∆θn = θ - θn 0, we have the difference of the deflection angle 54 ) is approximated to be [37]59 ) since its contribution is significantly less in comparison to other terms, and then by substituting ∆θn = (θ - θn 0), we have the θ-equation for the nth image as
$\begin{eqnarray}\xi =\frac{{{D}}_{{\rm{O}}{\rm{L}}}+{{D}}_{{\rm{L}}{\rm{S}}}}{{{D}}_{{\rm{L}}{\rm{S}}}}\theta -{\alpha }_{{D}}(\theta ).\end{eqnarray}$
The angle ξ ∈ [ - π, π] is used for the optical axis and the source orientations, whereas DOL is the distance connecting the source and the observer, and DLS is the line between the source and lens. The angles ξ and β are found to follow the relation [29, 37] $\begin{eqnarray}\frac{{D}_{{\rm{O}}{\rm{L}}}}{\sin (\xi -\beta )}=\frac{{D}_{{\rm{L}}{\rm{S}}}}{\sin \beta }.\end{eqnarray}$
For the completeness of the calculations and to have physical realization, the angles θ, ξ, and β are considered to be smaller because the formation of the relativistic images is a dominant feature of the strong lensing phenomena. The light emanating from the source encounters the black hole and makes many winds or loops before it leaves, such that the deflection angle α is framed as 2nπ + ∆αn, where n ∈ N is an integer and accounts for the loop counting and 0 < ∆αn ≪ 1. Therefore, following equation ( $\begin{eqnarray}\beta =\theta -\frac{{D}_{{\rm{L}}{\rm{S}}}}{{D}_{{\rm{O}}{\rm{L}}}+{D}_{{\rm{L}}{\rm{S}}}}{\rm{\Delta }}{\alpha }_{n}.\end{eqnarray}$
We can use equation ( $\begin{eqnarray}{\theta }_{n}{\,}^{0}=\frac{{u}_{s}}{{D}_{{\rm{O}}{\rm{L}}}}(1+{{\rm{e}}}_{{\rm{n}}}),\end{eqnarray}$
where $\begin{eqnarray}{{\rm{e}}}_{{\rm{n}}}={{\rm{e}}}^{\frac{\bar{b}-2n\pi }{\bar{a}}}.\end{eqnarray}$
Now a Taylor series expansion of αD(θ) around θn 0 up to first order in (θ - θn 0) may be approximated as [37] $\begin{eqnarray}{\alpha }_{D}(\theta )={\alpha }_{D}({\theta }_{n}{\,}^{0})+\frac{\partial {\alpha }_{D}(\theta )}{\partial \theta }{\left|\right.}_{{\theta }_{n}{\,}^{0}}(\theta -{\theta }_{n}{\,}^{0})+{ \mathcal O }{(\theta -{\theta }_{n}{\,}^{0})}^{2}.\end{eqnarray}$
Using equation ( $\begin{eqnarray}{\rm{\Delta }}{\alpha }_{n}=-\frac{\bar{a}{D}_{{\rm{O}}{\rm{L}}}}{{u}_{s}{{\rm{e}}}_{{\rm{n}}}}{\rm{\Delta }}{\theta }_{n}.\end{eqnarray}$
Therefore, the final lens equation ( $\begin{eqnarray}\beta ={\theta }_{n}{\,}^{0}+{\rm{\Delta }}{\theta }_{n}+\frac{{D}_{{\rm{L}}{\rm{S}}}}{{D}_{{\rm{O}}{\rm{L}}}+{D}_{{\rm{L}}{\rm{S}}}}\left(\frac{\bar{a}{D}_{{\rm{O}}{\rm{L}}}}{{u}_{s}{{\rm{e}}}_{{\rm{n}}}}{\rm{\Delta }}{\theta }_{n}\right).\end{eqnarray}$
We ignore the second term in equation ( $\begin{eqnarray}{\theta }_{n}={\theta }_{n}{\,}^{0}+\frac{{D}_{{\rm{O}}{\rm{L}}}+{D}_{{\rm{L}}{\rm{S}}}}{{D}_{{\rm{L}}{\rm{S}}}}\frac{{u}_{s}{{\rm{e}}}_{{\rm{n}}}}{\bar{a}{D}_{{\rm{O}}{\rm{L}}}}(\beta -{\theta }_{n}{\,}^{0}).\end{eqnarray}$
Next, we discuss the creation and subsequent formation of Einstein's ring. Among many exotic features of the lensing phenomenon, the formation of Einstein's ring is the most striking. When the light source, lens and observer are aligned in the most suitable way so that the whole lens system plays a role in the formation of complex multiple images [29-32]. We also have double Einstein's ring [98] formed depending on the source we have. For relativistic Einstein's ring formation, the deflection angle α must have a value greater than 2π. For perfect alignment of the lens and the observer, we must set β = 0, so that the lens is situated at the middle of the line connecting the source and the observer. Therefore, equation (60 ) reduces to [37]61 ) for Einstein's ring reduces to
$\begin{eqnarray}{\theta }_{n}^{E}=\left(1-\frac{2{u}_{s}{{\rm{e}}}_{n}}{{D}_{{\rm{O}}{\rm{L}}}\bar{a}}\right).\left(\frac{{u}_{s}}{{D}_{{\rm{O}}{\rm{L}}}}(1+{{\rm{e}}}_{n})\right).\end{eqnarray}$
As a limiting case, when DOL ≫ us, the angular radius equation ( $\begin{eqnarray}{\theta }_{n}^{E}=\frac{{u}_{s}}{{D}_{{\rm{O}}{\rm{L}}}}\left(1+{{\rm{e}}}_{n}\right),\end{eqnarray}$
where ${\theta }_{1}^{E}$ represents the angular position of the outermost Einstein's ring. In figure 11, we plot the angular position ${\theta }_{1}^{E}$ of the supermassive black holes SgrA* and M87. Likewise, we get the image magnification which mathematically represents the ratio of the solid angle formed due to the image and the source with the lens. For the nth image formed, the magnification is expressed as [17, 19] $\begin{eqnarray}{\mu }_{n}=\frac{1}{\beta }\left[\frac{{u}_{s}}{{D}_{{\rm{O}}{\rm{L}}}}(1+{{\rm{e}}}_{n})\left(\frac{{D}_{{\rm{O}}{\rm{S}}}}{{D}_{{\rm{L}}{\rm{S}}}}\frac{{u}_{s}{{\rm{e}}}_{n}}{{D}_{{\rm{O}}{\rm{L}}}\bar{a}}\right)\right].\end{eqnarray}$
As the value n increases, the magnification decreases and accordingly, the image becomes fainter. We have another important quantity, the angular separation s, which is the angular difference between the first and last image formed. Therefore, we have the important quantities of the relativistic images formed for a generic asymmetric rotating black hole as follows [17] $\begin{eqnarray}{\theta }_{\infty }=\frac{{u}_{s}}{{D}_{{\rm{O}}{\rm{L}}}},\end{eqnarray}$
$\begin{eqnarray}s={\theta }_{1}-{\theta }_{\infty }\approx {\theta }_{\infty }({{\rm{e}}}^{\frac{\bar{b}-2\pi }{\bar{a}}}),\end{eqnarray}$
$\begin{eqnarray}{r}_{\,\rm{mag}\,}=\frac{{\mu }_{1}}{{\sum }_{n=2}^{\infty }{\mu }_{n}}\approx {{\rm{e}}}^{\frac{2\pi }{\bar{a}}}.\end{eqnarray}$
These observables are plotted in a realistic scenario for supermassive black holes such as Sgr A* and M87 [99] in figures 12 and 13. The details of mass and other parameters for Sgr A* [100] are: M = 4.3 × 106M⊙ and d = 8.35 Kpc, while for M87 [12], M = 6.5 × 109M⊙ and d = 16.8 Mpc.
Figure 11. Plots of the outermost Einstein rings for the black holes at the center of nearby galaxies in the framework of Schwarzschild geometry. The left plot corresponds to the SgrA* and the right one to that of M87 [37]. |
Figure 12. Plots of the lensing variables for Sgr A* with respect to the charge parameter Q (the left plots) and the rotation parameter j (the right plots). |
Figure 13. Plots of the lensing variables for M87 with respect to the charge parameter Q (the left plots) and the rotation parameter j (the right plots). |
Concerning the computations of various strong lensing observable parameters, we follow the same procedures that are hugely applied to the four-dimensional black hole space-times in general as well as modified theories of gravity. All the observable parameters except the ones concerned with the Ohanian equation are well behaved and systematically evaluated. Regarding the five-dimensional lens equation, we simply employ the same techniques as done in the usual four-dimensional space-times [80, 81]. The justification for the usage of such lens equations may lie in the following arguments.
(i) We have rotations around the ψ and Φ directions, but when limited to the equatorial plane (θ = π/2), the black hole rotation is constrained only in the ψ direction. We cannot have rotation around the Φ direction, which can be shown in the metric, equation (19 ), where the only induced cross-term is dtdψ. This leads us to conclude that the gravitational lensing by the five-dimensional rotating black hole is unusual when compared to the usual four-dimensional Kerr black hole. This is the first signature where we could hope that the extra dimensions must have an imprint on the gravitational lensing effect. In addition, in the limit of the vanishing rotation parameter (j = 0), one can speculate that the function H(r) = 0. It eventually leads to the deflection angle of ψ tending to zero, and subsequently equation (19 ) reduces to that of the usual five-dimensional Reissner-Nordström-AdS black hole space-times.
(ii) Here we will limit our attention to investigate the deflection angle in the Φ direction for the light rays passing close to the black hole in the equatorial plane since it can be observed by astronomical experiments. Moreover, it is very convenient for us to compare it with the results obtained in the usual four-dimensional black hole space-time.
(iii) The expression for the angular position depends on the minimum impact parameter having a rigid base. The minimum impact parameter is derived when the effective potential is identically vanishing in all cases, regardless of the space-time dimensions. In our understanding, it could not have any effect in the derivation of the Ohanian equation. In the derivation of the angular position, the second term we encounter is the source-to-observer distance. It is the measured distance using the standard ruler. The parameters which might get affected in extra dimensions are the black hole parameters, not those measured using the standard rulers. The effects of the extra dimensions are imprinted in the parameters like angular momentum, charge, and the mass. These parameters receive scaling corrections when transitioning from higher to lower dimensions or vice-versa. Despite the analytical results of the five-dimensional Ohanian equations (if possible to derive), it will not have significant deviations in its mathematical form. We could have the same mathematical expressions up to some correction terms having not significant deviations. On this logical basis, we proceeded to calculate the observables using the four-dimensional Ohanian equation.
(iv) Other observable parameters like the angular separation, the image magnification and the relativistic time-delay effects are systematically obtained in terms of the coefficients $\bar{a}$ and $\bar{b}$, which are obtained in the strong-field region, when we approximate the deflection angle. These parameters are calculated independently from the lens equation. Here too, we see the effects of extra dimensions in the black hole parameters, but not in the mathematical derivation of the observable.
5. Time delay in the strong-field limit
In this section, we derive the time required for the photon or light to go from r0 to r or from r to r0. We follow an approach similar to the one reported in the previous subsection for the deflection angle, but with some subtraction strategies to treat the integrals. For more details, see [101-104].
For an observer at infinity, the time taken from the photon to travel from the source to the observer is given by
$\begin{eqnarray}T={\int }_{{t}_{0}}^{{t}_{f}}{\rm{d}}t.\end{eqnarray}$
Next, we change the integration variable from t to r and split the integral into two phases (approaching and leaving): $\begin{eqnarray}T={\int }_{{D}_{LS}}^{{r}_{0}}\frac{{\rm{d}}t}{{\rm{d}}r}{\rm{d}}r+{\int }_{{r}_{0}}^{{D}_{OL}}\frac{{\rm{d}}t}{{\rm{d}}r}{\rm{d}}t.\end{eqnarray}$
Utilizing the symmetry between approach and departure, we can combine the two integrals by extending the integration limits to infinity. This can be accomplished by subtracting two terms, given by
$\begin{eqnarray}T=2{\int }_{{r}_{0}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\right|{\rm{d}}r-{\int }_{{D}_{OL}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\right|{\rm{d}}r-{\int }_{{D}_{LS}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\right|{\rm{d}}r.\end{eqnarray}$
If we consider two photons traveling on different trajectories, the time delay between them is $\begin{eqnarray}\begin{array}{rcl}{T}_{1}-{T}_{2} & = & 2{\displaystyle \int }_{{r}_{0,1}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\left(r,{r}_{0,1}\right)\right|{\rm{d}}r-2{\displaystyle \int }_{{r}_{0,2}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\left(r,{r}_{0,2}\right)\right|{\rm{d}}r\\ & & -\,{\displaystyle \int }_{{D}_{OL}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\left(r,{r}_{0,1}\right)\right|{\rm{d}}r+{\displaystyle \int }_{{D}_{OL}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\left(r,{r}_{0,2}\right)\right|{\rm{d}}r\\ & & -\,{\displaystyle \int }_{{D}_{LS}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\left(r,{r}_{0,1}\right)\right|{\rm{d}}r+{\displaystyle \int }_{{D}_{LS}}^{\infty }\left|\frac{{\rm{d}}t}{{\rm{d}}r}\left(r,{r}_{0,2}\right)\right|{\rm{d}}r.\end{array}\end{eqnarray}$
Supposing that the observer and the source are very far from the black hole, dt/dr is effectively 1 in the last four integrals, which thus exactly cancel each other. We are therefore left with the first two integrals.
Dividing equation (21 ) by equation (25 ), we obtain 40 ). It can be seen that dt/dr tends to 1 at the large r limit, and the two integrals in equation (70 ) are separately divergent, while their difference is finite. The time delay results from the different paths the photons follow while they wind around the black hole. When the two photons are far from the black hole, dt/dr → 1 and the two integrals compensate each other. Separating the two regimes, we can write individually convergent integrals. To achieve this, we subtract and add the function ${\tilde{P}}_{1}\left(r,{r}_{0,i}\right)/\sqrt{{F}_{0,i}}$ to each integrand. Supposing r0,1 < r0,2, we can write 40 ). We can verify that equation (73 ) is similar to equation (70 ) by substituting all the formulas, but now it is expressed as the sum of independently convergent integrals.
$\begin{eqnarray}\frac{{\rm{d}}t}{{\rm{d}}r}={\tilde{P}}_{1}\left(r,{r}_{0}\right){P}_{2}\left(r,{r}_{0}\right),\end{eqnarray}$
where $\begin{eqnarray}{\tilde{P}}_{1}\left(r,{r}_{0}\right)=\sqrt{\frac{B(r)F({r}_{0})}{F(r)}},\end{eqnarray}$
and P2 is defined by equation ( $\begin{eqnarray}\begin{array}{rcl}{T}_{1}-{T}_{2} & = & \tilde{T}\left({r}_{0,1}\right)-\tilde{T}\left({r}_{0,2}\right)+2{\displaystyle \int }_{{r}_{0,1}}^{{r}_{0,2}}\frac{{\tilde{P}}_{1}\left(r,{r}_{0,1}\right)}{\sqrt{{F}_{0,1}}}{\rm{d}}r\\ & & +\,2{\displaystyle \int }_{{r}_{0,2}}^{\infty }\left[\frac{{\tilde{P}}_{1}\left(r,{r}_{0,1}\right)}{\sqrt{{F}_{0,1}}}-\frac{{\tilde{P}}_{1}\left(r,{r}_{0,2}\right)}{\sqrt{{F}_{0,2}}}\right]{\rm{d}}r,\end{array}\end{eqnarray}$
where $\begin{eqnarray}\tilde{T}\left({r}_{0}\right)={\int }_{0}^{1}\tilde{R}\left(z,{r}_{0}\right)f\left(z,{r}_{0}\right){\rm{d}}z,\end{eqnarray}$
$\begin{eqnarray}\tilde{R}\left(z,{x}_{0}\right)=\frac{2{r}^{2}}{{r}_{0}}{\tilde{P}}_{1}\left(r,{r}_{0}\right)\left(1-\frac{1}{\sqrt{{F}_{0}}f\left(z,{r}_{0}\right)}\right),\end{eqnarray}$
and $f\left(z,{r}_{0}\right)$ is defined by equation (In actuality, the time taken by the light ray to wind around the black hole is represented by the integral $\tilde{T}\left({r}_{0}\right)$. We have deducted a term from the definition of $R\left(z,{r}_{0}\right)$ that is negligible when the photon is near the black hole but cancels the integrand when the photon is distant from the black hole in order to cut off the integrands at large r values. As we shall see, the residual terms of this subtraction are often subleading with respect to ${\rm{\Delta }}\tilde{T}$ and are kept in the final two integrals of equation (73 ).
The integral equation (75 ) can be solved following the same technique of the integral (38 ), just replacing R by $\tilde{R}$. The result is 46 ) and
$\begin{eqnarray}\tilde{T}(u)=-\tilde{a}{\mathrm{ln}}\left(\frac{u}{{u}_{s}}-1\right)+\tilde{b}+O\left(u-{u}_{s}\right),\end{eqnarray}$
where us is defined by equation ( $\begin{eqnarray}\tilde{a}=\frac{\tilde{R}\left(0,{r}_{s}\right)}{2\sqrt{q({r}_{s})}},\end{eqnarray}$
$\begin{eqnarray}\tilde{b}=-\pi +{\tilde{b}}_{R}+\tilde{a}{\mathrm{ln}}\left(\frac{{r}_{s}^{2}[C^{\prime\prime} ({r}_{s})F({r}_{s})-C({r}_{s})F^{\prime\prime} ({r}_{s})]}{{u}_{s}F({r}_{s})\sqrt{F({r}_{s})C({r}_{s})}}\right)\end{eqnarray}$
with $\begin{eqnarray}{\tilde{b}}_{R}={\int }_{0}^{1}\left[\tilde{R}\left(z,{r}_{s}\right)f\left(z,{r}_{s}\right)-\tilde{R}\left(0,{r}_{s}\right){f}_{0}\left(z,{r}_{s}\right)\right]{\rm{d}}z.\end{eqnarray}$
We have shown the variation of $\tilde{T}(u)$ with respect to u for different j and Q values in figure 14.
Figure 14. Plots of the variation of $\tilde{T}(u)$ with respect to the parameter u for different values of the rotation parameter j (the left plot) and constant value of the charge parameter Q (the right plot). |
Figure 15. Plots of the variation of $\tilde{a}$ with respect to the charge parameter Q (the left plot) and the rotation parameter j (the right plot). |
Figure 16. Plots of the variation of $\tilde{b}$ with respect to the charge parameter Q (the left plot) and the rotation parameter j (the right plot). |
6. Conclusion
The strong gravitational lensing provides a very powerful tool to probe the nature of black holes in different modified theories of gravity. In addition, the extra dimension is a promising entity that helps us to determine a unified nature of the theory of everything. We have investigated the strong gravitational lensing in the equatorial plane (θ = π/2) of the five-dimensional charged equally rotating black hole in the presence of a negative cosmological constant. The presence of an additional factor of the dtdψ term in the space-time metric itself, even for the equatorial plane, accounts for the extra dimension. We observed systematically the effects of the rotation and the charge parameters on the circular photon sphere radius for a fixed value of the curvature radius. The strong deflection angle and the coefficients $\bar{a}$, $\bar{b}$ were plotted in order to show the role of the black hole parameters. We found that the coefficient $\bar{a}$ increases with the charge parameter, while it decreases or increases with the rotation parameter j, depending on the choice of the values of the charge parameter. However, the coefficient $\bar{b}$ increases with both the charge parameter and the rotation parameter. We also calculated the gravitational time delay effect and showed its variation with the charge parameter and the rotation parameter. In the near future, with the advent of new technologies, the detector EHT or square kilometer array may be operated to directly give the photon sphere radius of either the SgrA* or the M87 supermassive black hole. This may lead to putting constraints on the parameters of the five-dimensional, equally rotating black hole in the presence of a cosmological constant. This may help us fix which of the black hole solutions coming from the superstring or supergravity theory may be viable for the experimental extra dimensions to detect. Last but not least, we analyzed the gravitational time delay effect by taking into consideration the observables accountable for the strong-field gravitational effects. These parameters are highly usable when investigating the QNMs from the perspectives of the photon orbit radius and vis-á-vis the minimum impact parameter.
Gravitational wave astronomy, through both ringdown or electromagnetic observation and the detection of the silhouettes of black holes, may provide us with information on the space-time geometry near the black hole event horizon. The ringdown and the silhouettes are directly connected to a particular kind of null circular geodesics called light rings [105]. For the four-dimensional case, in the asymptotically flat limit for a generic axisymmetric rotating black hole with spherical horizon topology near the event horizon, such light rings (at least one) are formed in each rotation sense [106, 107]. For the five-dimensional case, there exists at least one light ring in the case of both the black hole and the naked singularity [108]. We hope to return to this issue for our concerned black hole in the near future. We also want to explore the time-like circular geodesics in the context of topological configurations for the massive particles, as they are important for accumulating particles around the accretion disk. Another future direction, in addition to the above analyses, should be devoted to the analysis of the QNMs in correlation with the photon orbit radius and hence the lensing parameters.