sectionSection #1 subsectionSection #1 subsubsectionSection #1
subsubsubsectionSection #1 equationEq. (#1) figureFig. #1 subfigureFig. 0#1
tableTab. #1 appendixAppendix #1

# Strong gravitational lensing of a five-dimensional charged, equally rotating
black hole with a cosmological constant

Md Sabir Ali<EMAIL_ADDRESS>Key Laboratory of Quantum Theory and
Applications of Ministry of Education, Lanzhou Center for Theoretical Physics,
Lanzhou University, Lanzhou 730000, China  Key Laboratory of Theoretical
Physics of Gansu Province, Institute of Theoretical Physics $\&$ Research
Center of Gravitation, Lanzhou University, Lanzhou 730000, China  School of
Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
Department of Physics, Mahishadal Raj College, West Bengal 721628, India
Shagun Kaushal<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Technology Delhi, Hauz Khas, New Delhi-110016, India  Yu-Xiao Liu
<EMAIL_ADDRESS>the corresponding author Key Laboratory of Quantum Theory
and Applications of Ministry of Education, Lanzhou Center for Theoretical
Physics, Lanzhou University, Lanzhou 730000, China  Key Laboratory of
Theoretical Physics of Gansu Province, Institute of Theoretical Physics $\&$
Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China
School of Physical Science and Technology, Lanzhou University, Lanzhou 730000,
China

###### Abstract

We study the lensing phenomena of the strong gravity regime of the five-
dimensional charged, equally rotating black holes with a cosmological
constant, familiarly known as the Cvetič-Lü-Pope black holes. These black
holes are characterized by three observable parameters, the mass $M$, the
charge $Q$ and the angular momentum $J$, in addition to the cosmological
constant. We investigate the strong gravitational lensing observables, mainly
the photon sphere radius, the minimum impact parameter, the deflection angle,
the angular size, and the magnification of the relativistic images. We model
the $M87$ and $SgrA^{*}$ for these observables. We also focus on the
relativistic time delay effect in the strong field regime of gravity and the
impact of the observable on it. The analytical expressions for the observables
of the relativistic images with vanishing angular momentum ($j=0$) are
discussed in some detail.

##  I Introduction

The recent year witnessed a flurry of activities regarding the astronomical
measures of the general relativistic effects, especially the observation of
the black hole shadows, e.g., for the M87 and SgA*, and the gravitational wave
data from the LIGO scientific/Virgo collaborations LIGOScientific:2017ycc ;
LIGOScientific:2016aoc ; LIGOScientific:2018dkp ; LIGOScientific:2019fpa ;
Wielgus:2022heh ; KAGRA:2023pio . The data from the Event Horizon Telescope
(EHT) regarding the shadows of the 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 Akiyama:2019cqa ; Akiyama:2019brx ;
Akiyama:2019sww ; Akiyama:2019bqs ; Akiyama:2019fyp ; Akiyama:2019eap . As is
known from its definition, gravitational lensing is the deflection of light
rays when they pass close to black holes Darwin ; Frittelli:1999yf ;
Eiroa:2010wm . 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
($\theta=\pi/2$) and non-equatorial ($\theta\neq\pi/2$) planes, we refer our
readers to Refs. Bozza:2001xd ; Bozza:2002zj ; Bozza:2007gt ; Bozza:2002af ;
Kuang:2022xjp ; Sengo:2022jif ; Bartelmann:2016dvf ; Hoekstra:2008db  and
references therein.

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 Darwin  and subsequently elaborated in Refs.
Ohanian:1987pc ; Luminet:1979nyg . The most recent theoretical investigation
of strong field gravitational lensing is mainly attributed to the seminal work
of Virbhadra and Ellis Virbhadra:1999nm ; Virbhadra:2002ju , where they
discussed the formation of the ring, its position, and the magnification of
relativistic images for the Schwarzschild black hole. Soon after this
formulation, more rigorous analytical descriptions of the exact lens equation
and the integral expressions for its solution were obtained by Frittelli,
Killing, and Newman Frittelli:1999yf . A comparison of their results with
those given by Virbhadra and Ellis was also presented. Later, Bozza
Bozza:2001xd ; Bozza:2002zj ; Bozza:2007gt ; Bozza:2002af  and then Tsukamoto
Tsukamoto:2016jzh ; Tsukamoto:2017fxq  proposed a completely new method for
the study of strong lensing for a generic spherically symmetric static
spacetime. These methods are largely 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 Ghosh:2020spb ;
Whisker:2004gq ; Abbas:2021whh ; Eiroa:2005ag ; Gyulchev:2006zg ; Ghosh:2010uw
; Gyulchev:2012ty ; Molla:2023hog ; Grespan:2023cpa ; Kumar:2022fqo ;
KumarWalia:2022ddq ; Kumar:2021cyl ; Lu:2021htd ; Ali:2021psk ; Hsieh:2021scb
.

The string theory is a promising candidate for the unified theory, where
gravity is unified with other forces of nature. The prediction of the 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 extra dimensions can have imprints in the
quasinormal modes of the higher-dimensional black holes Shen:2006pa ;
Harris:2005jx ; Casals:2005sa ; Creek:2006ia ; Kanti:2005xa ; Kanti:2006ua ;
Creek:2006je ; Konoplya:2011qq ; Kodama:2009rq ; Konoplya:2008rq ;
Nozawa:2008wf ; Chen:2007ay ; Chen:2007pu , which are believed to be tested
through gravitational wave probes in the near future. The higher-dimensional
black holes/black branes in the particle accelerators and in the cosmic rays
were studied to get an insight into the extra dimensions. Therefore,
gravitational lensing can also be used to probe the extra dimensions. In this
direction, the study of strong lensing in higher-dimensional black hole
solutions is worth investigating. The search for the extra dimensions was
probed through the gravitational lensing of a charged-neutral/charged Kaluza-
Klein squashed black hole Liu:2010wh ; Sadeghi:2012bj ; Ji:2013xua , the
squashed Kaluza-Klein Gödel black hole for both charged and charged-neutral
cases Chen:2011ef ; Sadeghi:2013ssa , the black hole with extra-dimensions and
the Kalb-Ramond field Majumdar:2006zza ; Chakraborty:2016lxo . 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 the five-dimensional charged equally
rotating black hole in the presence of a negative cosmological constant as
given in Ref. Markeviciute:2018cqs , when the charged scalar hair has
vanishing limit. This black hole solution was originally proposed by Cvetič,
Lü and Pope Cvetic:2004hs , 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.

The organization of the paper is as follows. In Section blueII, 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
blueIII 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 blueIV.
Here, we model our concerned black holes as SgrA∗ and M87∗ to show the effect
of the parameters. In Section blueV, we discuss the time delay effect.
Finally, in Section blueVI, we summarize the paper and conclude our results.

##  II The five-dimensional charged equally rotating black hole solution with
a cosmological constant

We present a brief review of the action and the constituent spacetime 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
Markeviciute:2018cqs ; Bhattacharyya:2010yg

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi G_{5}}\int
d^{5}x\sqrt{-g}\Bigg{(}R+12-\frac{3}{4}F_{\mu\nu}F^{\mu\nu}-\frac{3}{8}\left(\left|D_{\mu}\phi\right|^{2}-\frac{\partial_{\mu}(\phi\phi^{*})\partial^{\mu}(\phi\phi^{*})}{4(4+\phi\phi^{*})}-4\phi\phi^{*}\right)$
(1)
$\displaystyle+\frac{1}{4\sqrt{-g}}\epsilon^{\alpha\beta\gamma\mu\nu}F_{\alpha\beta}F_{\gamma\mu}A_{\nu}\Bigg{)},$

where

$\displaystyle
D_{\mu}\phi=\partial_{\mu}\phi-2iA_{\mu}\phi,\;F_{\mu\nu}=\partial_{\mu}{A_{\nu}}-\partial_{\nu}{A_{\mu}}.$
(2)

The variation of the action 1 with respect to the metric tensor $g_{\mu\nu}$
yields the Einstein’s field equations Markeviciute:2018cqs ;
Bhattacharyya:2010yg

$\displaystyle
R_{\mu\nu}-\frac{1}{2}\left(R+12\right)g_{\mu\nu}=-\frac{3}{2}T_{\mu\nu}^{\text{EM}}+\frac{3}{8}T_{\mu\nu}^{{\phi}}.$
(3)

The expressions for the energy-momentum tensor of the electromagnetic field
$T_{\mu\nu}^{\text{EM}}$, and the scalar field $T_{\mu\nu}^{\phi}$ are

$\displaystyle T_{\mu\nu}^{\text{EM}}$ $\displaystyle=$ $\displaystyle
F_{\mu}^{\alpha}F_{\nu\alpha}-\frac{1}{2}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta},$
$\displaystyle T_{\mu\nu}^{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}D_{\mu}\phi(D_{\nu}\phi)^{*}+D_{\nu}\phi(D_{\mu}\phi)^{*}-\frac{1}{2}g_{\mu\nu}\left|D_{\alpha}\phi\right|^{2}+2\phi\phi^{*}g_{\mu\nu}$
(4)
$\displaystyle-\frac{1}{4(4+\phi\phi^{*})}\left(\partial_{\mu}(\phi\phi^{*})\partial_{\nu}(\phi\phi^{*})-\frac{1}{2}g_{\mu\nu}\left[\left(\partial_{\sigma}\phi\phi^{*}\right)\left(\partial^{\sigma}\phi\phi^{*}\right)\right]\right)\Bigg{]}.$

The Maxwell and the scalar field equations are given, respectively, as
Markeviciute:2018cqs ; Bhattacharyya:2010yg

$\displaystyle\nabla_{\nu}F_{\mu}^{\nu}=\frac{i}{4}\left[\phi(D_{\mu}\phi)^{*}-\phi^{*}(D_{\mu}\phi)+\frac{1}{4\sqrt{-g}}g_{\mu\sigma}\epsilon^{\sigma\alpha\beta\gamma\nu}F_{\alpha\beta}F_{\gamma\nu}\right],$
(5) $\displaystyle
D_{\mu}D^{\mu}{\phi}+\left[\frac{\left[\partial_{\sigma}(\phi\phi^{*})\right]^{2}}{4(4+\phi\phi^{*})^{2}}-\frac{\nabla^{2}(\phi\phi^{*})}{2(4+\phi\phi^{*})}\right]\phi=0.$
(6)

The stationary, asymptotically anti-de Sitter spacetimes 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 Markeviciute:2018cqs
:

$\displaystyle
ds^{2}=-f(r)dt^{2}+g(r)dr^{2}+\Sigma^{2}(r)\Big{[}h(r)\big{(}d\psi+\frac{1}{2}\cos{\theta}d\phi-\Omega(r)dt\big{)}^{2}+\frac{1}{4}d\Omega_{2}^{2}\Big{]},$
(7)

which is rewritten as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Big{(}-f(r)+\Sigma^{2}(r)\Omega^{2}(r)h(r)\Big{)}dt^{2}+g(r)dr^{2}+\frac{1}{4}\Sigma^{2}(r)d\theta^{2}-\Sigma^{2}(r)\Omega(r)h(r)\cos\theta
d\phi dt$ (8) $\displaystyle+$
$\displaystyle\frac{\Sigma^{2}(r)}{4}\Big{(}h(r)\cos^{2}{\theta}+\sin^{2}{\theta}\Big{)}d\phi^{2}+\Sigma^{2}(r)h(r)d\psi^{2}-2\Sigma^{2}(r)\Omega(r)h(r)d\psi
dt$ $\displaystyle+$ $\displaystyle\Sigma^{2}(r)h(r)\cos{\theta}d\psi d\phi.$

In the above $ansatz$, we can see that the metric on $S^{3}$ is expressed as a
Hopf fibration over the space $\mathbb{C}\mathbb{P}^{1}$ of unit radius. In
this way, the coordinate $\psi$ has the period of $2\pi$ and the
$\theta,\;\phi$ coordinates have the coordinate ranges of the usual unit
$S^{2}$ sphere. General five-dimensional, charged, and equally rotating black
hole solutions with a cosmological constant were first presented in Ref.
Cvetic:2004hs , which is basically a coupled Einstein-Maxwell solution in the
bosonic sector of minimal gauged supergravity, and also $\phi=0$ limit of the
action (blue1). 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 appear in the metric (blue8) have the explicit forms

$\displaystyle\Sigma(r)$ $\displaystyle=$ $\displaystyle
r,~{}~{}~{}f(r)=\frac{G(r)}{h(r)},~{}~{}~{}g(r)=\frac{1}{G(r)},$
$\displaystyle h(r)$ $\displaystyle=$ $\displaystyle
1+j^{2}\Big{(}\frac{m}{r^{4}}-\frac{q^{2}}{r^{6}}\Big{)},~{}~{}~{}\Omega(r)=\frac{j}{h(r)}\Big{(}\frac{m-q}{r^{4}}-\frac{q^{2}}{r^{6}}\Big{)},$
(9) $\displaystyle G(r)$ $\displaystyle=$
$\displaystyle\frac{1}{r^{4}}\Big{[}q^{2}\Big{(}1-\frac{j^{2}}{L^{2}}\Big{)}+j^{2}m\Big{]}-\frac{1}{r^{2}}\Big{[}m\Big{(}1-\frac{j^{2}}{L^{2}}\Big{)}-2q\Big{]}+\frac{r^{2}}{L^{2}}+1.$

The angular momentum $J$ and the mass $M$ of the black hole are related to the
parameters $\\{q,j,m\\}$ as follows:

$\displaystyle
J=\frac{1}{2}j\left(m-q\right),~{}~{}~{}M=\frac{1}{4}\left[m\left(3+\frac{j^{2}}{L^{2}}\right)-6q\right],~{}~{}~{}Q=\frac{q}{2}.$
(10)

Using the last two relations of the above equations we can express $m$ and $q$
as

$\displaystyle m=\frac{2L^{2}(2M+6Q)}{j^{2}+3L^{2}},~{}~{}~{}q=2Q.$ (11)

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

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+g(r)dr^{2}+\Sigma^{2}(r)d\Omega_{3}^{2},$ (12)
$\displaystyle d\Omega_{3}^{2}$ $\displaystyle=$
$\displaystyle(d\psi+\frac{1}{2}\cos\theta
d\phi)^{2}+\frac{1}{4}d\Omega_{2}^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}+\cos^{2}\theta d\psi^{2}.$ (13)

Obviously, the metric (blue12) is the five-dimensional Reissner-
Nordstr$\ddot{o}$m black holes with a cosmological constant, where $f(r)$ and
$g(r)$ are given by Cvetic:2004hs

$f(r)=1+\frac{r^{2}}{L^{2}}-\frac{m}{r^{2}}+\frac{q^{2}}{r^{4}},\;\;\;g(r)=1/f(r).$

The metric possesses three Killing vectors $\partial_{{t}}$,
$\partial_{\hat{\phi}}$, 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 Frolov:2003en ; Frolov:2002xf

$\displaystyle\cos{\hat{\phi}}{\partial_{\hat{\theta}}}-\cot{\hat{\theta}}\sin{\hat{\phi}}{\partial_{\hat{\phi}}}+\frac{\sin{\hat{\phi}}}{\sin{\hat{\theta}}}\partial_{\hat{\psi}},$
(14)
$\displaystyle-\sin{\hat{\phi}}{\partial_{\hat{\theta}}}-\cot{\hat{\theta}}\cos{\hat{\phi}}{\partial_{\hat{\phi}}}+\frac{\cos{\hat{\phi}}}{\sin{\hat{\theta}}}\partial_{\hat{\psi}},$
(15)

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 $F(r)=0$:

$r_{h}^{6}+L^{2}r_{h}^{4}+r_{h}^{2}\left(j^{2}m-L^{2}m+2L^{2}q\right)+j^{2}L^{2}m-j^{2}q^{2}+L^{2}q^{2}=0,$

which is a cubic equation in $r_{h}^{2}$. The solutions are given by

$\displaystyle r_{h,n}^{2}$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{\frac{L^{4}}{3}+L^{2}m-j^{2}m-2L^{2}q}{3}}\cos\Bigg{[}\frac{1}{3}\arccos{\left(\frac{2j^{2}L^{2}m-3j^{2}q^{2}+\frac{2L^{6}}{9}+L^{4}m-2L^{4}q+3L^{2}q^{2}}{2j^{2}m-\frac{2L^{4}}{3}-2L^{2}m+4L^{2}q}\right)}$
(16)
$\displaystyle\sqrt{\frac{3}{\frac{L^{4}}{3}+L^{2}m-j^{2}m-2L^{2}q}}-\frac{2\pi
n}{3}\Bigg{]},\;\;\;n=0,1,2,$

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 11.
As $j\to 0$, the horizon radius reduces to the five-dimensional charged AdS
black hole. For both $q,j\to 0$, we have the five-dimensional Schwarzschild
AdS black hole. In addition to $q,j\to 0$, if we take $L\to\infty$, we have
$r_{h}^{2}=m$, which is the radius of the five-dimensional Schwazschild-
Tangherlini spacetime. We plot in 1, 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 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.

Figure 1: The plot 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: The 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.

##  III 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, 7 reduces to

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Big{(}-f(r)+\Sigma^{2}(r)\Omega^{2}(r)h(r)\Big{)}dt^{2}+g(r)dr^{2}+\Sigma^{2}(r)h(r)d\psi^{2}+\frac{1}{4}r^{2}\Sigma^{2}(r)d\phi^{2}$
(17) $\displaystyle-2\Sigma^{2}(r)h(r)\Omega(r)d\psi dt.$

The components of the metric tensor are written in a more compact functional
form as

$\displaystyle g_{tt}$ $\displaystyle=$
$\displaystyle\partial_{t}\cdot\partial_{t}=-f(r)+\Sigma^{2}(r)\Omega^{2}(r)h(r)=-A(r),$
$\displaystyle g_{rr}$ $\displaystyle=$
$\displaystyle\partial_{r}\cdot\partial_{r}=g(r)=B(r),$ $\displaystyle
g_{\phi\phi}$ $\displaystyle=$
$\displaystyle\partial_{\phi}\cdot\partial_{\phi}=\frac{1}{4}r^{2}\Sigma^{2}(r)=C(r),$
(18) $\displaystyle g_{\psi\psi}$ $\displaystyle=$
$\displaystyle\partial_{\psi}\cdot\partial_{\psi}=\Sigma^{2}(r)h(r)=D(r),$
$\displaystyle g_{\psi t}$ $\displaystyle=$
$\displaystyle\partial_{t}\cdot\partial_{\psi}=-\Sigma^{2}(r)\Omega(r)h(r)=g_{t\psi}=-H(r).$

We can rewrite 7 as

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)d\phi^{2}+D(r)d\psi^{2}-2H(r)dtd\psi.$ (19)

It is to be mentioned here that the coordinate $(t,r,\phi,\psi)$ 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: $Q\to Q/M$, $j\to j/\sqrt{M}$
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 $\psi\to\psi+\alpha t$ and
$\Omega\to\Omega+\alpha$, here $\Omega$ is the angular velocity and it is
symmetric for $t\to-t$, $\phi\to\phi+2\pi$ and $\psi\to\psi+\pi$. Killing
vectors are $\eta^{\mu}_{t}=\delta^{\mu}_{t}$,
$\eta^{\mu}_{\phi}=\delta^{\mu}_{\phi}$ and
$\eta^{\mu}_{\psi}=\delta^{\mu}_{\psi}$. Equations of motion are

$\displaystyle\mathcal{E}$ $\displaystyle=$ $\displaystyle-
g_{0\mu}\dot{x^{\mu}}=-g_{tt}\dot{t}-g_{t\psi}\dot{\psi}=A(r)\dot{t}+H(r)\dot{\psi},$
$\displaystyle L_{\phi}$ $\displaystyle=$ $\displaystyle
g_{3\mu}\dot{x^{\mu}}=g_{\phi\phi}\dot{\phi}=C(r)\dot{\phi},$ (20)
$\displaystyle L_{\psi}$ $\displaystyle=$ $\displaystyle
g_{4\mu}\dot{x^{\mu}}=g_{\psi\psi}\dot{\psi}+g_{\psi
t}\dot{t}=D(r)\dot{\psi}-H(r)\dot{t}.$

Rearranging 20 we have

$\displaystyle\dot{\psi}$ $\displaystyle=$
$\displaystyle\frac{A(r)L_{\psi}+H(r)\mathcal{E}}{H^{2}(r)+A(r)D(r)},$
$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\frac{D(r)\mathcal{E}-H(r)L_{\psi}}{H^{2}(r)+A(r)D(r)},$ (21)
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{L_{\phi}}{C(r)},$

whereas the radial null geodesic is obtained by using $ds^{2}=0$

$\dot{r}^{2}=\frac{1}{B(r)}\Bigg{[}\frac{(D(r)\mathcal{E}-2H(r)L_{\psi})\mathcal{E}-A(r)L^{2}_{\psi}}{H^{2}(r)+A(r)D(r)}-\frac{L^{2}_{\phi}}{C(r)}\Bigg{]}.$
(22)

The geodesic equation of motion for $\theta$ is

$\frac{C^{{}^{\prime}}(r)}{2C(r)}\frac{d\theta}{d\lambda}\frac{dr}{d\lambda}-\frac{H(r)\sin{\theta}}{2C(r)}\frac{d\phi}{d\lambda}\frac{dt}{d\lambda}+\frac{\cos{\theta}\sin{\theta}(-C(r)+D(r))}{C(r)}\Big{(}\frac{d\phi}{d\lambda}\Big{)}^{2}+\frac{D(r)\sin{\theta}}{2C(r)}\frac{d\psi}{d\lambda}\frac{d\phi}{d\lambda}=0.$
(23)

For $\theta=\pi/2$, 23 reduces to

$\frac{d\phi}{d\lambda}\Big{(}-H(r)\frac{dt}{d\lambda}+D(r)\frac{d\psi}{d\lambda}\Big{)}=0,$
(24)

which gives us either $\frac{d\phi}{d\lambda}=0$, i.e., $L_{\phi}=0$ or
$L_{\psi}=\Big{(}-H(r)\frac{dt}{d\lambda}+D(r)\frac{d\psi}{d\lambda}\Big{)}=0$.
In order to get the simpler results, we shall consider the case when
$L_{\psi}=0$, which means the total angular momentum of the photon $J$ is
equal to $L_{\phi}$ and the effective potential $V_{eff}=-\dot{r}^{2}$, where
$\dot{r}^{2}$ under such conditions is given by

$\dot{r}^{2}=\frac{1}{B(r)}\Bigg{[}\frac{D(r)\mathcal{E}^{2}}{H^{2}(r)+A(r)D(r)}-\frac{L^{2}_{\phi}}{C(r)}\Bigg{]}.$
(25)

To compute the impact parameter, we use $V_{eff}=0$,

$u=J=L_{\phi}/\mathcal{E}=\sqrt{\frac{C(r_{0})D(r_{0})}{H^{2}(r_{0})+A(r_{0})D(r_{0})}},$
(26)

where $u$ denotes the impact parameter. The variation of $V_{eff}$ with
respect to radial distance $r$ is shown in 3 for different values of the
impact parameter $u$ at fixed $Q$ and fixed $j$ values. We observe that at
$u_{s}=u_{sc}$, we have the critical impact parameter (red solid line), which
depends on the parameters $Q$ and $j$. At $u_{s}=u_{sc}$ 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 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. Similar is the case for
the impact parameter, as shown in 5.

Figure 3: The behaviour of the effective potential $V_{eff}$ 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 $r_{s}$ 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 $u_{s}$ 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
$V^{\prime}_{\text{eff}}=0$,

$\text{C}(r_{0})\left[\text{D}(r_{0})^{2}A^{\prime}(r_{0})-H(r_{0})^{2}\text{D}^{\prime}(r_{0})+2\text{D}(r_{0})H(r_{0})H^{\prime}(r_{0})\right]-\text{D}(r_{0})\text{C}^{\prime}(r_{0})\left[A(r_{0})\text{D}(r_{0})+H(r_{0})^{2}\right]=0.$
(27)

where $r_{0}$ is the closest of the photon spheres. Eqs. (blue26) and (blue27)
are more complex than those in the usual spherical, symmetric black hole
spacetime. As a limiting case, when the parameter $j\to 0$, we have $H(r)\to
0$ and therefore, 27 has a much simpler form,

$\text{C}(r_{0})A^{\prime}(r_{0})-\text{C}^{\prime}(r_{0})A(r_{0})=L^{2}\left(4q^{2}+6qx+x(2x-3)\right)+x^{3}=0,\;\;x=r_{0}^{2}.$
(28)

The above equation is cubic in nature and the roots are easily derivable as

$\displaystyle
x_{k}=2\sqrt{-\frac{\mathcal{P}}{3}}\cos\left[\frac{1}{3}\arccos{\left(\frac{3\mathcal{Q}}{2\mathcal{P}}\sqrt{\frac{-3}{\mathcal{P}}}\right)}-\frac{2\pi
k}{3}\right],\;\;\;k=0,1,2,$ (29)

where $\mathcal{P}=\left(6q-3\right)L^{2}-\frac{4L^{4}}{3}$,
$\mathcal{Q}=4l^{2}q^{2}+2L^{4}\left(1-2q\right)-\frac{16L^{6}}{27}$.
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 $j\neq 0$ so we have computed it
numerically. The value of the impact parameter for $j=0$ is computed to be

$\displaystyle u_{s}|_{j=0}$ $\displaystyle=$
$\displaystyle\frac{L\mathcal{X}_{3}^{2}}{2\sqrt{3}{\mathcal{X}_{2}}^{1/3}\left(27L^{2}q^{2}\mathcal{X}_{2}+L^{2}\mathcal{X}_{3}^{2}\left(4L^{2}-18q+{\mathcal{X}_{2}}^{1/3}+9\right)+\mathcal{X}_{3}\left(4L^{4}\mathcal{X}_{2}^{2/3}-2L^{2}\mathcal{X}_{2}+\mathcal{X}_{2}^{4/3}\right)\right)},$
$\displaystyle\mathcal{X}_{1}$ $\displaystyle=$
$\displaystyle\sqrt{-3L^{8}\left(4q^{2}-36q+9\right)-81L^{6}\left(8q^{2}-6q+1\right)+324L^{4}q^{4}},$
$\displaystyle\mathcal{X}_{2}$ $\displaystyle=$
$\displaystyle-8L^{6}+27L^{4}(2q-1)-54L^{2}q^{2}+3\mathcal{X}_{1},$
$\displaystyle\mathcal{X}_{3}$ $\displaystyle=$ $\displaystyle
4L^{4}+L^{2}\left(-18q-2{\mathcal{X}_{2}}^{1/3}+9\right)+\mathcal{X}_{2}^{2/3}.$
(30)

Figure 6: The behaviour of the photon orbit $r_{s}$ (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 behaviour of the impact parameter $u_{s}$ (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).

In 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 $r_{s}$ and $u_{s}$ 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, 29 and the minimum impact parameter, III. We observe that,
as in the four-dimensional Kerr black hole spacetime, 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 $j^{2}$.

Since $\dot{r}^{2}$ vanishes at the closest distance $r=r_{0}$, from the
trajectory equation we have

$A(r_{0})\dot{t}_{0}^{2}+2H(r_{0})^{2}\dot{t}_{0}\dot{\psi}_{0}=C(r_{0})\dot{\phi}_{0}^{2}+D(r_{0})\dot{\psi}_{0}^{2},$
(31)

Using 22, we have

$\dot{r}=\frac{dr}{d\lambda}=\frac{1}{\sqrt{B(r)F(r_{0})F(r)}}\sqrt{F(r_{0})-\frac{C(r_{0})F(r)}{C(r)}}.$
(32)

The radial motion of the photon is governed by

$\frac{d\phi}{dr}=\frac{\sqrt{B(r)F(r_{0})F(r)}}{C(r)}\frac{1}{\sqrt{F(r_{0})-\frac{C(r_{0})F(r)}{C(r)}}}$
(33)

and

$\frac{d\psi}{dr}=\frac{H(r)}{D(r)}\sqrt{\frac{B(r)F(r_{0})}{F(r)}}\frac{1}{\sqrt{F(r_{0})-\frac{C(r_{0})F(r)}{C(r)}}}$
(34)

with

$F(r)=\frac{H^{2}(r)+A(r)D(r)}{D(r)}.$ (35)

The deflection angles $\psi$ and $\phi$ for the photon coming from the
infinite can be expressed as Bozza:2002zj

$\displaystyle I_{\psi}(r_{0})$ $\displaystyle=$ $\displaystyle
2\int^{\infty}_{r_{0}}\frac{d\psi}{dr}dr=2\int^{\infty}_{r_{0}}dr\frac{H(r)}{D(r)}\sqrt{\frac{B(r)F(r_{0})}{F(r)}}\frac{1}{\sqrt{F(r_{0})-\frac{C(r_{0})F(r)}{C(r)}}},$
(36) $\displaystyle I_{\phi}(r_{0})$ $\displaystyle=$ $\displaystyle
2\int^{\infty}_{r_{0}}\frac{d\phi}{dr}dr=2\int^{\infty}_{r_{0}}dr\frac{\sqrt{B(r)F(r_{0})F(r)}}{C(r)}\frac{1}{\sqrt{F(r_{0})-\frac{C(r_{0})F(r)}{C(r)}}}.$
(37)

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 $\phi$-angle
is dependent on whether the black hole is in retrograde or in prograde motion.
However, the deflection angle related to the angle $\psi$ 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 $(\theta=\pi/2)$, the rotation of the black hole is in the
$\psi$ direction rather than in the $\phi$ direction, which is evident from
the expression of the spacetime as is given in 19, where the only cross-term
is $dtd\psi$. 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

$I_{\phi}(r_{0})=\int_{0}^{1}R(z,r_{0})f(z,r_{0})dz$ (38)

with

$\displaystyle R(z,r_{0})$ $\displaystyle=$ $\displaystyle
2\frac{r^{2}}{r_{0}C(r)}\sqrt{B(r)F(r)C(r_{0})}=P_{1}\left(r,r_{0}\right),$
(39) $\displaystyle f(z,r_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{F(r_{0})-F(r)C(r_{0})/C(r)}}=P_{2}\left(r,r_{0}\right).$
(40)

The function $R(z,r_{0})$ is regular for all values of $z$ and $r_{0}$. From
40, we find that $f(z,r_{0})$ diverges as $z$ tends to zero, i.e., as the
photon approaches the marginally circular photon orbit. Therefore, we can
split the integral 38 into divergent part $I_{D}(r_{0})$ and the regular one
$I_{R}(r_{0})$

$\displaystyle I_{D,\phi}(r_{0})$ $\displaystyle=$
$\displaystyle\int^{1}_{0}R(0,r_{s})f_{0}(z,r_{0})dz,$ (41) $\displaystyle
I_{R,\phi}(r_{0})$ $\displaystyle=$
$\displaystyle\int^{1}_{0}[R(z,r_{0})f(z,r_{0})-R(0,r_{s})f_{0}(z,r_{0})]dz.$
(42)

We can expand the argument of the square root in $f(z,r_{0})$ to the second
order in $z$

$f_{0}(z,r_{0})=\frac{1}{\sqrt{p(r_{0})z+g(r_{0})z^{2}}}$ (43)

with

$\begin{split}p(r_{0})&=\frac{r_{0}}{C(r_{0})}\Big{[}C^{\prime}(r_{0})F(r_{0})-C(r_{0})F^{\prime}(r_{0})\Big{]},\\\
g(r_{0})&=\frac{r_{0}^{2}}{2C(r_{0})}\Big{[}2C^{\prime}(r_{0})C(r_{0})F^{\prime}(r_{0})-2C^{\prime}(r_{0})^{2}F(r_{0})+F(r_{0})C(r_{0})C^{\prime\prime}(r_{0})-C^{2}(r_{0})F^{\prime\prime}(r_{0})\Big{]}.\end{split}$
(44)

To obtain 43 and 44 we expanded $f(z,r_{0})$ around $z=0$ upto order $z^{2}$.
If we use 27, one obtains $F(r)C^{\prime}(r)=C(r)F^{\prime}(r)$ which leads to
$p(r_{0})=0$ and $r_{s}$ is the largest root of $p(r_{0})$. This means that
the leading term of the divergence in $f_{0}(z,r_{0})$ is $z^{-1}$ and the
integral 38 diverges logarithmically. Thus, in the strong field region, the
deflection angle in the direction can be approximated very well as
Bozza:2002zj :

$\alpha_{D,\phi}(\theta)=-\bar{a}\log\Big{(}\frac{\theta
D_{OL}}{u_{s}}-1\Big{)}+\bar{b}+O(u-u_{s}),$ (45)

where

$\begin{split}\bar{a}&=\frac{R(0,r_{s})}{2\sqrt{g(r_{s})}},\\\
\bar{b}&=-\pi+b_{R}+\bar{a}\log\frac{r^{2}_{s}[C^{\prime\prime}(r_{s})F(r_{s})-C(r_{s})F^{\prime\prime}(r_{s})]}{u_{s}F(r_{s})\sqrt{F(r_{s})C(r_{s})}},\\\
b_{R}&=I_{R}(r_{s}),\quad\quad
u_{s}=\sqrt{\frac{C(r_{s})}{F(r_{s})}}.\end{split}$ (46)

Here the quantity $D_{\text{OL}}$ is the distance between observer and
gravitational lens, $\theta=u/D_{\text{OL}}$ is the angular separation between
the lens and the image. The subscript “$s$” represents the evaluation at
$r=r_{s}$. Likewise, we can compute the corresponding $\psi$ following 38. We
can write

$I_{\psi}(r_{0})=\int_{0}^{1}S(z,r_{0})f(z,r_{0})dz$ (47)

with $f(z,r_{0})$ given by 40 and $S(z,r_{0})$ as

$S(z,r_{0})=2\frac{r^{2}H(r)}{r_{0}D(r)}\sqrt{\frac{B(r)F(r_{0})}{F(r)}}.$
(48)

The function $S(z,r_{0})$ is regular for all values of $z$ and $r_{0}$.
Whereas $f(z,r_{0})$ is divergent as for the case of $I_{D,\phi}(r_{0})$, and
we can follow the same expansion as well but replace $\bar{a}$ with
$\bar{a}_{\psi}$,

$\displaystyle\bar{a}_{\psi}=\frac{S(0,r_{s})}{2\sqrt{g(r_{s})}},$ (49)
$\displaystyle I_{D,\psi}(r_{0})=\int^{1}_{0}S(0,r_{s})f_{0}(z,r_{0})dz,$ (50)
$\displaystyle
I_{S,\psi}(r_{0})=\int^{1}_{0}[S(z,r_{0})f(z,r_{0})-R(0,r_{s})f_{0}(z,r_{0})]dz.$
(51)

Figure 8:  The variation of the deflection angle $\alpha_{D}(u)$ 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).

The study of the deflection angle only in the $\phi$ 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 $\psi$ 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 $\phi$-driection also has a term that diverges
logarithmically. Nevertheless, it should not be used in determining the strong
deflection phenomena from observational perspectives. In 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 9
and 10, respectively. We observe a monotonic variation for the coefficient
$\bar{a}$ it increases with increasing $Q$ and $j$, whereas the coefficient
$\bar{b}$ increases with $Q$ but decreases with $j$.

##  IV 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
$\theta$ with respect to OL. On the other hand, the illuminating source of
light is oriented at an angle $\beta$ with respect to OL. The light emanating
by the source is detected by the observer by an angle $\alpha_{D}(\theta)$.

Among various mathematical formulations, the method given by Ohanian was
mostly used in the literature as the lens equation (Ohanian:1987pc, ) to
approximate the position of the source, lens and observer:

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{D_{OL}+D_{LS}}{D_{LS}}\theta-\alpha_{D}(\theta).$ (52)

The angle $\xi\in[-\pi,\pi]$ is used for the optical axis and the source
orientations, whereas $D_{OL}$ is the distance connecting the source and the
observer, and $D_{LS}$ is the line between the source and lens. The angles
$\xi$ and $\beta$ are found to follow the relation Ohanian:1987pc ;
Ghosh:2020spb

$\displaystyle\frac{D_{OL}}{\sin(\xi-\beta)}$ $\displaystyle=$
$\displaystyle\frac{D_{LS}}{\sin\beta}.$ (53)

For the completeness of the calculations and to have physical realization, the
angles $\theta$, $\xi$, and $\beta$ are considered to be smaller because the
formation of the relativistic images are dominant feature of the strong
lensing phenomena. The light emanating from the source encounters the black
hole and makes many winding or loops before it leaves such that the deflection
angle $\alpha$ is framed as $2n\pi+\Delta\alpha_{n}$, where $n\in N$ is an
integer and is accounted for the loop counting and $0<\Delta\alpha_{n}\ll 1$.
Therefore, following 52 along with 53, for the smaller values of $\theta$, we
have the lens equation

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\theta-\frac{D_{LS}}{D_{OL}+D_{LS}}\Delta\alpha_{n}.$ (54)

We can use 54 to know information about the image formation. We know that when
the impact parameter $u\to$ $u_{s}$, the deflection angle $\alpha_{D}(\theta)$
becomes divergent. Therefore, for each loop formed during the winding near the
event horizon of the black hole, we certainly have one $u_{s}$ at which the
light rays reach from the lens to the observer. Accordingly, we have
infinitely many images that are formed on both sides of the black hole. Now 45
together with $\alpha_{D}(\theta_{n}{{}^{0}})=2n\pi$ is rewritten as

$\displaystyle\theta_{n}{{}^{0}}$ $\displaystyle=$
$\displaystyle\frac{u_{s}}{D_{OL}}(1+e_{n}),$ (55)

where

$\displaystyle e_{n}$ $\displaystyle=$ $\displaystyle
e^{\frac{\bar{b}-2n\pi}{\bar{a}}}.$ (56)

Now a Taylor series expansion of $\alpha_{D}(\theta)$ around
$\theta_{n}{{}^{0}}$ up to first order in $(\theta-\theta_{n}{{}^{0}})$ may be
approximated as Ghosh:2020spb

$\displaystyle\alpha_{D}(\theta)$ $\displaystyle=$
$\displaystyle\alpha_{D}(\theta_{n}{{}^{0}})+\frac{\partial\alpha_{D}(\theta)}{\partial\theta}\Bigg{|}_{\theta_{n}{{}^{0}}}(\theta-\theta_{n}{{}^{0}})+\mathcal{O}(\theta-\theta_{n}{{}^{0}})^{2}.$
(57)

Using 55 along with the condition
$\Delta\theta_{n}=\theta-\theta_{n}{{}^{0}}$, we have the difference of the
deflection angle

$\displaystyle\Delta\alpha_{n}$ $\displaystyle=$
$\displaystyle-\frac{\bar{a}D_{OL}}{u_{s}e_{n}}\Delta\theta_{n}.$ (58)

Therefore, the final lens equation 54 is approximated to be Ghosh:2020spb

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\theta_{n}{{}^{0}}+\Delta\theta_{n}+\frac{D_{LS}}{D_{OL}+D_{LS}}\Bigg{(}\frac{\bar{a}D_{OL}}{u_{s}e_{n}}\Delta\theta_{n}\Bigg{)}.$
(59)

We ignore the second term in 59 since its contribution is very less in
comparison to other terms, and then by substituting
$\Delta\theta_{n}=(\theta-\theta_{n}{{}^{0}})$, we have the $\theta$-equation
for the $n$-th image as

$\displaystyle\theta_{n}$ $\displaystyle=$
$\displaystyle\theta_{n}{{}^{0}}+\frac{D_{OL}+D_{LS}}{D_{LS}}\frac{u_{s}e_{n}}{\bar{a}D_{OL}}(\beta-\theta_{n}{{}^{0}}).$
(60)

Figure 11: Plots of the outermost Einstein rings for the black holes at the
centre of nearby galaxies in the framework of the Schwarzschild geometry. The
left plot corresponds to the $SgrA^{*}$ and the right one to that of $M87$
Ghosh:2020spb .

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 one. When the light source, lens, and
observer are aligned in a most suitable way so that the whole lens system
plays a role in the formation of complex multiple images Ohanian:1987pc ;
Virbhadra:1999nm ; Virbhadra:2002ju . We also have double Einstein’s ring
Gavazzi:2008  formed depending on the source we have. For relativistic
Einstein’s ring formation, the deflection angle $\alpha$ must have a value
greater than $2\pi$. For perfect alignment of the lens and the observer, we
must set $\beta=0$, so that the lens is situated at the middle of the line
connecting the source and the observer. Therefore, 60 reduces to Ghosh:2020spb

$\displaystyle\theta_{n}^{E}$ $\displaystyle=$
$\displaystyle\left(1-\frac{2u_{s}e_{n}}{D_{OL}\bar{a}}\right).\left(\frac{u_{s}}{D_{OL}}(1+e_{n})\right).$
(61)

As a limiting case when $D_{OL}\gg u_{s}$, the angular radius 61 for
Einstein’s ring reduces to

$\displaystyle\theta_{n}^{E}=\frac{u_{s}}{D_{OL}}\left(1+e_{n}\right),$ (62)

where $\theta_{1}^{E}$ represents the angular position of the outermost
Einstein’s ring. In 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 $n$th image
formed, the magnification is expressed as Bozza:2002zj ; Bozza:2002af

$\displaystyle\mu_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\Bigg{[}\frac{u_{s}}{D_{OL}}(1+e_{n})\Bigg{(}\frac{D_{OS}}{D_{LS}}\frac{u_{s}e_{n}}{D_{OL}\bar{a}}\Bigg{)}\Bigg{]}.$
(63)

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
Bozza:2002zj

$\displaystyle\theta_{\infty}$ $\displaystyle=$
$\displaystyle\frac{u_{s}}{D_{OL}},$ (64) $\displaystyle s$ $\displaystyle=$
$\displaystyle\theta_{1}-\theta_{\infty}\approx\theta_{\infty}(e^{\frac{\bar{b}-2\pi}{\bar{a}}}),$
(65) $\displaystyle r_{\text{mag}}$ $\displaystyle=$
$\displaystyle\frac{\mu_{1}}{\sum{{}_{n=2}^{\infty}}\mu_{n}}\approx
e^{\frac{2\pi}{\bar{a}}}.$ (66)

These observables are plotted in a realistic scenario for supermassive black
holes such as Sgr A∗ and M87 Kormendy:2013  in 12 and 13. The details of mass
and other parameters for Sgr A∗ Do:2019vob  are: $M=4.3\times 10^{6}M_{\odot}$
and $d=8.35$ Kpc, while for M87 Akiyama:2019eap , $M=6.5\times
10^{9}M_{\odot}$ and $d=16.8$ Mpc.

Figure 12:  Plot of the lensing variables for Sgr A* with respect to the
charge parameter $Q$ (the left plot) and the rotation parameter $j$ (the right
plot).

Figure 13:  Plot of the lensing variables for M87 with respect to the charge
parameter $Q$ (the left plot) and the rotation parameter $j$ (the right plot).

##  V Time delay in the strong field limit

In this section, we derive the time required for the photon or light to go
from $r_{0}$ to $r$ or from $r$ to $r_{0}$. 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
Refs. Bozza:2003cp ; Hsieh:2021rru ; Lu:2016gsf .

For an observer at infinity, the time taken from the photon to travel from the
source to the observer is given by

$T=\int_{t_{0}}^{t_{f}}dt.$ (67)

Next, we change the integration variable from $t$ to $r$ and split the
integral into two phases (approaching and leaving):

$T=\int_{D_{LS}}^{r_{0}}\frac{dt}{dr}dr+\int_{r_{0}}^{D_{OL}}\frac{dt}{dr}dr.$
(68)

Utilising 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

$T=2\int_{r_{0}}^{\infty}\left|\frac{dt}{dr}\right|dr-\int_{D_{OL}}^{\infty}\left|\frac{dt}{dr}\right|dr-\int_{D_{LS}}^{\infty}\left|\frac{dt}{dr}\right|dr.$
(69)

If we consider two photons travelling on different trajectories, the time
delay between them is

$\begin{split}T_{1}-T_{2}=2\int_{r_{0,1}}^{\infty}\left|\frac{dt}{dr}\left(r,r_{0,1}\right)\right|dr-2\int_{r_{0,2}}^{\infty}\left|\frac{dt}{dr}\left(r,r_{0,2}\right)\right|dr\\\
-\int_{D_{OL}}^{\infty}\left|\frac{dt}{dr}\left(r,r_{0,1}\right)\right|dr+\int_{D_{OL}}^{\infty}\left|\frac{dt}{dr}\left(r,r_{0,2}\right)\right|dr\\\
-\int_{D_{LS}}^{\infty}\left|\frac{dt}{dr}\left(r,r_{0,1}\right)\right|dr+\int_{D_{LS}}^{\infty}\left|\frac{dt}{dr}\left(r,r_{0,2}\right)\right|dr.\end{split}$
(70)

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 thus left with the first two integrals.

Dividing 21 by 25, we obtain

$\frac{dt}{dr}=\tilde{P}_{1}\left(r,r_{0}\right)P_{2}\left(r,r_{0}\right),$
(71)

where

$\tilde{P}_{1}\left(r,r_{0}\right)=\sqrt{\frac{B(r)F(r_{0})}{F(r)}},$ (72)

and $P_{2}$ is defined by 40. It can be seen that $dt/dr$ tends to one at
large $r$ limit, and the two integrals in 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, $dt/dr\rightarrow 1$ 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 $r_{0,1}<r_{0,2}$, we can write

$\begin{split}T_{1}-T_{2}=\tilde{T}\left(r_{0,1}\right)-\tilde{T}\left(r_{0,2}\right)+2\int_{r_{0,1}}^{r_{0,2}}\frac{\tilde{P}_{1}\left(r,r_{0,1}\right)}{\sqrt{F_{0,1}}}dr\\\
+2\int_{r_{0,2}}^{\infty}\left[\frac{\tilde{P}_{1}\left(r,r_{0,1}\right)}{\sqrt{F_{0,1}}}-\frac{\tilde{P}_{1}\left(r,r_{0,2}\right)}{\sqrt{F_{0,2}}}\right]dr,\end{split}$
(73)

where

$\displaystyle\tilde{T}\left(r_{0}\right)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\tilde{R}\left(z,r_{0}\right)f\left(z,r_{0}\right)dz,$
(74) $\displaystyle\tilde{R}\left(z,x_{0}\right)$ $\displaystyle=$
$\displaystyle\frac{2r^{2}}{r_{0}}\tilde{P}_{1}\left(r,r_{0}\right)\left(1-\frac{1}{\sqrt{F_{0}}f\left(z,r_{0}\right)}\right),$
(75)

and $f\left(z,r_{0}\right)$ is defined by 40. We can verify that 73 is similar
to 70 by substituting all the formulas, but now it is expressed as the sum of
independently convergent integrals.

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 in a moment, the residual terms of this subtraction
are often subleading with respect to $\Delta\tilde{T}$ and are kept in the
final two integrals of 73.

The integral 75 can be solved following the same technique of the integral
(blue38), just replacing $R$ by $\tilde{R}$. The result is

$\tilde{T}(u)=-\tilde{a}\log\left(\frac{u}{u_{s}}-1\right)+\tilde{b}+O\left(u-u_{s}\right),$
(76)

where $u_{s}$ is defined by 46 and

$\tilde{a}=\frac{\tilde{R}\left(0,r_{s}\right)}{2\sqrt{q(r_{s})}},$ (77)
$\tilde{b}=-\pi+\tilde{b}_{R}+\tilde{a}\log\frac{r^{2}_{s}[C^{\prime\prime}(r_{s})F(r_{s})-C(r_{s})F^{\prime\prime}(r_{s})]}{u_{s}F(r_{s})\sqrt{F(r_{s})C(r_{s})}}$
(78)

with

$\tilde{b}_{R}=\int_{0}^{1}\left[\tilde{R}\left(z,r_{s}\right)f\left(z,r_{s}\right)-\tilde{R}\left(0,r_{s}\right)f_{0}\left(z,r_{s}\right)\right]dz.$
(79)

We have shown the variation of $\tilde{T}(u)$ with respect to $u$ for
different $j$ and $Q$ values in 14.

Figure 14: Plot 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).

Next, the variations of $\tilde{a}$ and $\tilde{b}$ with respect to the
parameters $Q$ and $j$ are shown in 15 and 16, respectively.

Figure 15: Plot 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:  Plot of the variation of $\tilde{b}$ with respect to the charge
parameter $Q$ (the left plot) and the rotation parameter $j$ (the right plot).

##  VI 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 in determining a
unified nature of the theory of everything. We have investigated the strong
gravitational lensing in the equatorial plane ($\theta=\pi/2$) of the five-
dimensional charged equally rotating black hole in the presence of a negative
cosmological constant. The presence of an additional factor of $dtd\psi$ term
in the spacetime 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 upon 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.

Gravitational wave astronomy through both the ringdown or electromagnetic
observations and the detection of the silhouettes of black holes may provide
us hitherto information on the spacetime geometry near the black hole event
horizon. The ringdown and the silhouettes are directly connected to a
particular kind of null circular geodesics called the light rings
Cardoso:2016rao . 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 Cunha:2020azh ; Wu:2023eml . For the five-dimensional
case, there exists at least one light ring in the case of both the black hole
and the naked singularity Tavlayan:2022hzl . We hope to return to this issue
for our concerned black hole in the near future. We also want to explore the
timelike circular geodesics in the context of topological configurations for
the massive particles as they are important for accumulating particles around
the accretion disk.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China
(Grants No. 12347177, No. 11875151 and No. 12247101), the 111 Project under
(Grant No. B20063) and Lanzhou City’s scientific research funding subsidy to
Lanzhou University, the Gansu Province Major Scientific and Technological
Special Project.

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