††thanks: Corresponding author<EMAIL_ADDRESS>
# Holographic study on Einstein ring for a charged black hole in conformal
gravity

Xin-Yun Hu  College of Economic and Management, Chongqing Jiaotong University,
Chongqing 400074, China  Xiao-Xiong Zeng<EMAIL_ADDRESS>Department of
Mechanics, Chongqing Jiaotong University, Chongqing 400074, China  Li-Fang Li
Center for Gravitational Wave Experiment, National Microgravity Laboratory,
Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China.
Peng Xu  Center for Gravitational Wave Experiment, National Microgravity
Laboratory, Institute of Mechanics, Chinese Academy of Sciences, Beijing
100190, China.  Lanzhou Center of Theoretical Physics, Lanzhou University, No.
222 South Tianshui Road, Lanzhou 730000, China

###### Abstract

Under the help of AdS/CFT correspondence, the Einstein ring of a charged black
hole in conformal gravity has been studied. Imposing an oscillating Gauss
source on one side of the AdS boundary which propagates in the bulk, the
response function are derived on the other side of the boundary. Given the
proposed wave optics system, the Einstein rings are observed as expected. All
these results reveal that when the observer locates at the north pole, there
always exists the Einstein ring and surrounding concentric stripes. And such
ring becomes into a luminosity-deformed ring or light spot while the observer
departs away from the north pole. We also investigate the effect of
temperature $T$, chemical potential $u$ and gravity-related parameters $c_{0}$
on the ring radius. We find the ring radius increases with the decrease of the
temperature, increase of the chemical potential, and increase of the gravity-
related parameters respectively. To check the results in the framework of
holography, we investigate the ingong angle of photon at the photon ring via
geometric optics and find it is consistent with the angle of Einstein ring
obtained via holography.

##  I Introduction

Over the last century, Einstein’s General Relativity (GR) had passed through
numerous tests Giesler2019 ; Isi2019 ; Isi2021 ; Isi_2021 ; Franchini2023 ,
and acquired significant observational supports. Therefore, GR becomes the
theoretical foundations of a wide range of research fields, including
cosmology, astrophysics, gravitational wave detection, unified field theory
etc. Bamber2021 ; Cardoso2022 ; Barack2019 ; Meszaros2019_astro . While,
challenges still exist from the observations at larger, galactic scale. The
long-standing theoretical issues, like the resolutions of singularities and
searches for quantum theory of gravity Ashtekar2014 ; Pawlowski2021 , turn out
to be much severer and more critical at the beginning of this new century of
relativity.

As we all know, shadows are expected to be found in black holes Akiyama2019 ;
Akiyama2019Lett ; Akiyama2019Astrophys ; Akiyama2019_Lett ; Akiyama2019_ring ;
Akiyama2019_875 . To image this phenomenon, the Event Horizon Telescope, a
global long baseline interference array observing at a wavelength of 1.3 mm,
is able to reconstruct event-horizon-scale images of the supermassive black
hole candidate in the center of the giant elliptical galaxy M87⋆. And the
black hole shadow and its observation along with other significant properties
have also been analyzed in Gralla2019 ; Zeng2023 ; Zeng2020 ; Zeng:2020vsj .
In these days, a number of researchers have devoted themselves to exploring
the physics of black hole images.

In the papers Hashimoto:2018okj ; Hashimoto:2019jmw , they proposed a direct
procedure to construct holographic images of the black hole in the bulk from a
given response function of the QFT on the boundary with the AdS/CFT duality,
which establishes a connection between gravity and field theories. As we know,
the realization of the holographic principle is motivated by string theory.
This duality in theoretical physics connects two seemingly different theories:
the gravitational theory in a (d+1)-dimensional Anti-de Sitter space and a
d-dimensional conformal field theory. With this AdS/CFT correspondence, the
response function with respect to an external source corresponds to the
asymptotic data of the bulk field generated by the source on the AdS boundary.
Using this response function, the holographic images gravitationally lensed by
the black hole can be constructed for a thermal state on two-dimensional
sphere which is dual to Schwarzschild AdS4 black hole. The obtained results
are consistent with the size of the photon sphere of the black hole calculated
through the geometrical optics method. Following these breakthroughs
Liu:2022cev ; Zeng:2023zlf ; Zeng_Li2023 ; CEJM ; Hu:2023eow , the results
showed that these holographic images exist in different gravitational
background. But according to the specific bulk dual geometry, the photon
sphere varies and the detailed behavior of Einstein ring varies. Therefore, in
this paper, we are tempted to investigate the behavior of the lensed response
for the charged black hole in conformal gravity. On one hand, we want to
investigate the effect of the chemical potential on the Einstein ring. As
stressed in Liu:2022cev , the real quantum materials are engineered at a
finite chemical potential, the chemical potential plays an important role in
strongly coupled systems. Therefore, investigation on the effect of the
chemical potential on the Einstein ring is also important and necessary. On
the other hand, we intend to explore the effect of the gravity-related
parameter in conformal gravity on the Einstein ring. And whether we can
distinguish the conformal gravity from Einstein gravity via holographic images
of black holes. The action of conformal gravity is defined by the Weyl tensor
square, which allows for more solutions than Einstein gravity. Especially, it
was regarded as a desirable UV culmination of gravity 1  and useful setting up
of supergravity theories 2 . Moreover in conformal gravity, dark matter and
dark energy are not necessary in explaining the galaxy rotation curves 3 ; 4 .
Therefore, conformal gravity is an important alternative gravity and thus
investigations on the Einsteins ring in this gravity are also important and
interesting.

This paper is organized as follows. In section II, we review the charged black
hole solution in conformal gravity for a spherically symmetric AdS black hole.
In section III, we give a holographic setup of such model and analyze the
corresponding lensed response function, explicitly. Given the optical system,
we indeed observe the Einstein ring in our model and compare our results with
the optical approximation in section IV. Our results shows that the position
of photon ring obtained from the geometrical optics is full consistence with
that of the holographic ring. Section VI is our conclusions.

##  II Review of the holographic construction of Einstein ring in AdS black
holes

Conformal pure gravity is constructed from the Weyl-squared term, and its
conformal symmetry is preserved when it couples minimally to the Maxwell
field. Its Lagrangian is Li:2012gh

$h^{-1}\mathcal{L}=-\frac{1}{4}C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}-\frac{1}{6}F^{2},$
(1)

here $h=\sqrt{g}$ and $g$ is the determinant of the metric. $F=dA$ and
$C_{\mu\nu\rho\sigma}$ is the Weyl tensor. The corresponding equations of
motion are

$\nabla^{\mu}F_{\mu\nu}=\frac{1}{2}(2\nabla^{\rho}\nabla^{\sigma}+R^{\rho\sigma})C_{\mu\rho\sigma\nu}+\frac{1}{3}(F^{2}_{\mu\nu}-\frac{1}{4}F^{2}g_{\mu\nu})=0.$
(2)

The static charged AdS black hole solution in conformal gravity is Li:2012gh

$ds^{2}=\text{}-F(r)dt^{2}+\text{}\frac{1}{F(r)}dr^{2}\text{}+r^{2}d\Omega_{2,\epsilon}^{2},$
(3)

with

$\displaystyle F(r)=c_{0}+c_{1}r-\frac{1}{3}\Lambda r^{2}+\frac{d}{r},$ (4)

and the Maxwell field

$\displaystyle A=-\frac{Q}{r}\text{}dt.$ (5)

The parameter $\epsilon$ can take three discrete values, -1, 0, 1, which
correspond to the hyperbolic, planar and spherical geometry respectively. The
solutions with $\epsilon=-1,0$ are the topological black holes, which exist
only in AdS backgrounds. In this paper, we are interested in the case
$\epsilon=1$. The integration constants $Q,c_{0},c_{1},d,\Lambda$ obey the
constraint condition

$\displaystyle 3c_{1}d+\epsilon^{2}+Q^{2}-c_{0}^{2}=\text{}0,$ (6)

in which $c_{1}$ is the massive spin -2 hair, $Q$ is regarded as the electric
charge and $d$ is regarded as the mass. Note that in Eq.(4), the charge $Q$
does not appear as the usual charged black hole. It affects the geometry of
the spacetime only implicitly through the constraint condition shown in
Eq.(6). In addition $Q$ appears only in squared form in Eq.(6), the spacetime
geometry does not discriminate positive and negative values of $Q$. In this
paper, we will consider exclusively the case $Q\geq 0$ as in Li:2012gh . For
the case $\Lambda<0$, there is only a nonzero real positive root of equation
$F(r)=0$, which corresponds to the event horizon of the back hole. The
solution describes a charged AdS black hole in this case, and we set
$1/l^{2}=-\frac{1}{3}\Lambda$, where $l$ is the Ads radius.

To reduce the parameters, we will keep $c_{0}$ while drop $c_{1}$. From
Eq.(6), we can get

$\displaystyle c_{1}=\frac{c_{0}^{2}-Q^{2}-1}{3\text{}d}.$ (7)

Substituting this condition into Eq.(4), we have

$\displaystyle
F(r)=c_{0}+\frac{c_{0}^{2}-Q^{2}-1}{3d}\text{}r-\frac{1}{3}\Lambda
r^{2}+\frac{d}{r},$ (8)

where we have set $l=1$. The Hawking temperature in this case is

$\displaystyle
T=\frac{\kappa}{2\pi}=\frac{{F(r)}^{\prime}}{4\pi}\mid_{r_{h}}=\frac{6r_{h}^{3}-\sqrt{3}\sqrt{r_{h}^{2}\left(-c_{0}^{2}+6c_{0}r_{h}^{2}+3r_{h}^{4}+4Q^{2}+4\right)}}{12\pi\text{}r_{h}^{2}},$
(9)

in which $\kappa$ is the surface gravity and $r_{h}$ is the event horizon of
the black hole.

Next we focus on the spherical metric (8). With a new definition $r_{*}=1/r$,
we have $F(r)=r_{*}^{-2}F(r_{*})$. We rewrite the metric (3) in the new
coordinate $(t,r_{*},\theta,\phi)$ and derive

$ds^{2}=\frac{1}{r_{*}^{2}}[-F(r_{*})dt^{2}+\text{}\frac{dr_{*}^{2}}{F(r_{*})}+d\Omega^{2}],$
(10)

here $d\Omega^{2}=d\theta^{2}+\sin\theta^{2}d\phi^{2}$. We will take Eq.(10)
as our background.

The complex scalar field $\Phi$ in such background is determined by the
following Klein-Gordan equation Liu:2022cev

$D_{b}D^{b}\Phi\text{}-M^{2}\Phi=0,$ (11)

here $D_{a}=\nabla_{a}-ieA_{a}$ and we will take $M^{2}=-2$ in the following.
In order to solve the above Klein-Gordon equation easily, the ingoing
Eddington coordinate is introduced, which is expressed as

$\nu=t-\int\frac{\text{}dr_{*}}{F(r_{*})}\text{}.$ (12)

With the above new coordinate, the metric (10) is further rewritten as

$ds^{2}=\frac{1\text{}}{r_{*}^{2}}[-F(r_{*})d\nu^{2}\text{}-2dr_{*}d\nu+d\Omega^{2}].$
(13)

The asymptotic solution of the complex scalar field $\Phi$ near the AdS
boundary becomes Liu:2022cev

$\text{}\text{}\text{}\text{}\Phi(\nu,r_{*},\theta,\phi)=r_{*}J_{\mathcal{O}}(\nu,\theta,\phi)+r_{*}^{2}\text{}\langle\mathcal{O}\rangle+\mathrm{O}(r_{*}^{3}).\text{}\text{}\text{}\text{}\text{}$
(14)

According to the AdS/CFT correspondence, $J_{\mathcal{O\text{}}}$ is the
source for the boundary field theory. And the corresponding expectation value
of dual operator is

$\langle\text{}\mathcal{O}\rangle_{J_{\mathcal{O}}}=\langle\mathcal{O}\rangle-(\partial_{\nu}\text{}-ieu)J_{{\mathcal{O}}},$
(15)

here $\langle\mathcal{O}\rangle_{J_{\mathcal{O}}}$ is also named the response
function, in which $u=Q/r_{h}$ is the chemical potential. Obviously,
$\langle\mathcal{O}\rangle$ corresponds to the expectation value of the dual
operator with the source turned off.

Here we would like to study the holographic Einstein images of the charged
black hole in conformal gravity based on the AdS/CFT correspondence. We choose
an oscillatory Gaussian wave source
$J_{\mathcal{O}\text{}}(\text{}\nu,\theta)$ on one side of the AdS boundary,
and the scalar waves generated by the source propagate in the bulk which is
shown in Fig. 1. When the bulk of scalar waves reach the other side of the AdS
boundary, the response function
$\langle\text{}\mathcal{O}\rangle_{J_{{\mathcal{O}}}\text{}}$ is generated on
the opposite side of the source. Here we take a monochromatic and axisymmetric
Gaussian wave packet which is centered on the south pole
$\theta_{0}\text{}=\pi\text{}$ as the source

$\displaystyle\text{}J_{\mathcal{O}}(\nu,\text{}\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\sigma^{2}}e^{-i\omega\text{}\nu}\exp{[-\frac{(\pi-\theta)^{2}}{2\sigma^{2}}]}$
(16) $\displaystyle=$ $\displaystyle
e^{-i\omega\nu}\sum_{l=0}^{\infty}C_{l0}Y_{l0}(\theta),$

$\sigma$ is the wave width which is produced by Gaussian source. $Y_{l0}$ is
the spherical harmonics function. If we take the size $\sigma<<\pi$, the
corresponding coefficients of $Y_{l0}$ is expressed as

$\displaystyle
C_{l0}=(-1)^{l}(\frac{l+1/2}{2\pi})^{1/2}\exp{[-\frac{(l+1/2)^{2}\tau^{2}}{2}]}.$
(17)

Considering the spacetime symmetry, the complex scalar field
$\Phi(\nu,r_{*},\theta,\phi)$ is further decomposed as

$\displaystyle\Phi(\nu,r_{*},\theta,\phi)=e^{-i\omega\nu}\sum_{l=0}^{\infty}C_{l0}r_{*}\mathcal{Z}_{l}(r_{*})Y_{l0}(\theta),$
(18)

where $\mathcal{Z}_{l}$ satisfies the equation of motion

$r_{*}^{2}f\mathcal{Z}_{l}^{\prime\prime}+r_{*}^{2}[f^{\prime}+2i(\omega-
eA)]\mathcal{Z}_{l}^{\prime}+[(2-\text{}2f)+r_{*}f^{\prime}-r_{*}^{2}\text{}l(l+1)-ier_{*}^{2}A^{\prime}]\mathcal{Z}_{l}=0,$
(19)

where $e$ is the charge of the complex scalar field. The asymptotic behavior
of $\text{}\mathcal{Z}_{l}$ near the AdS boundary goes like

$\text{}\mathcal{Z}_{l}=1+r_{*}\langle\mathcal{O}_{l}\text{}\rangle+O(r_{*}^{2}).$
(20)

And the response function $\langle\mathcal{O}\rangle_{\text{}J_{\mathcal{O}}}$
is also expressed as

$\langle\mathcal{O\text{}}\rangle_{J_{\mathcal{O}}}=e^{-i\omega\text{}\nu}\sum_{l=0}^{\infty}C_{l0}\langle\mathcal{O}\text{}\rangle_{J_{\mathcal{O}l}}Y_{l0}(\theta).$
(21)

Then we have

$\langle\mathcal{O}\rangle_{J_{\mathcal{O}l}}=\langle\mathcal{O}\rangle_{l}+i\hat{\omega},$
(22)

here $\hat{\omega}=\text{}\omega+eu$.

Figure 1: A monochromatic Gaussian source is at one side of the AdS boundary,
and its response is observed at another side on the same boundary.

Under the help of the pseudo-spectral method, we derive the numerical solution
Hashimoto:2018okj ; Hashimoto:2019jmw  for $\mathcal{Z}_{l}$ and
$\mathcal{O}\text{}_{l}$ straightforwardly. With $\mathcal{O\text{}}_{l}$, we
obtain the total response by Eq.(21). The typical profiles of the total
response are shown in Fig. 2 to Fig. 6. We indeed see the interference pattern
produced from the diffraction of our scalar field by the black hole.
Explicitly in Fig. 2, we plot the amplitude of $\langle O\text{}\rangle$ for
different $T$ with $Q=0.5$, $\omega=80$, $c_{0}=0.5$ and $e=1$. We clearly see
that when the temperature increases, while the amplitude of the total response
function decreases. Also we plot the amplitude of $\text{}\langle O\rangle$
for different $c_{0}$ with $r_{*h}=0.5$, $Q=0.5$, $\omega=80$ and $e=1$ in
Fig. 3. The amplitude of the total response function raises with the
increasing of the values of parameter $c_{0}$. Then we plot the amplitude of
$\langle O\rangle$ for different $\omega$ with $r_{*h}=0.5$, $c_{0}=0.5$,
$Q=0.5$ and $e=1$ in Fig. 4. The oscillation period of the wave is maximum
when $\omega=30$ and the amplitude decreases as $\omega$ increases to
$\omega=50$ and $\omega=70$. The results mean the frequency $\omega$ of the
Gaussian source can increase the wave width and reduce the wave period. These
in turn shows the total response function depends closely on the Gaussian
source. We further plot the amplitude of $\langle O\rangle$ for different
chemical potential $u$ with $r_{*h}=0.5$, $\text{}\omega=80$,
$\text{}c_{0}=0.5$ and $e=1$ in Fig. 5. When the chemical potential is small,
$u=0.05\text{}$ , the amplitude of $\langle O\rangle$ is also small. When the
chemical potential gradually increases to $u=1$, the amplitude also gradually
increases. At last, we show the amplitude of $\langle O\rangle$ for different
charge $e$ with parameters $Q=0.5$, $\text{}\omega=80$, $c_{0}=0.5$ and
$r_{*h}=0.5$ in Fig. 6. We see that the change of the amplitude $\langle
O\rangle$ in this case seems not obvious as in other cases, although we can
observe that the larger the value of $e$, the smaller the value of amplitude
$\langle O\rangle$.

Figure 2: The amplitude of $\langle\mathcal{O}\rangle$ with different
temperature $T$ for $Q=0.5$, $\omega=80$, $c_{0}=0.5$ and $e=1$. Figure 3: The
amplitude of $\langle\mathcal{O}\rangle$ with different $c_{0}$ for
$r_{*h}=0.5$, $Q=0.5$, $\omega=80$ and $e=1$. Figure 4: The amplitude of
$\langle\mathcal{O}\rangle$ with different $\omega$ for $r_{*h}=0.5$,
$c_{o}=0.5$, $Q=0.5$ and $e=1$. Figure 5: The amplitude of
$\langle\mathcal{O}\rangle$ with different $u$ for $r_{*h}=0.5$, $\omega=80$,
$c_{0}=0.5$ and $e=1$. Figure 6: The amplitude of $\langle\mathcal{O}\rangle$
with different $e$ for $Q=0.5$, $\omega=80$, $c_{0}=0.5$, and $r_{*h}=0.5$.

##  III The holographic ring formation setup

As stated in Liu:2022cev , we need an optical system to observe the response
function which is shown in Fig. 7. An optical apparatus, a convex lens and a
spherical screen are needed. Here we suppose the lens is infinitely thin. The
size of the lens is much smaller than the focal length $f$. And the lens is
regarded as a “converter” between plane and spherical waves. The right side of
the optical system is a screen and the received image from the transmitted
wave is depicted on this screen.

Figure 7: The optical system with a convex lens and a telescope.

Here we consider a wave $\Psi(\hat{x})$ irradiated to the lens from the left
side. Such wave is converted to $\Psi_{T}(\hat{x})$ which converges at the
focus $f$. The relationship between $\Psi(\hat{x})$ and $\Psi_{T}(\hat{x})$ is

$\Psi_{T}(\hat{x})=e^{-\frac{i{\omega}|\hat{x}|^{2}}{2f}}\Psi(\hat{x}).$ (23)

Fixing a spherical screen at a cartesian coordinate system
$(x,y,z)=(x_{s},y_{s},z_{s})$ with $x_{s}^{2}+y_{s}^{2}+z_{s}^{2}=f^{2}$
Hashimoto:2018okj ; Hashimoto:2019jmw , the transmitted wave function
$\Psi_{T}(\hat{x})$ imaging onto the screen is changed into a new function
$\Psi_{s}(\hat{x})$, which can be expressed as

$\displaystyle\Psi_{s}(\hat{x})$ $\displaystyle=$
$\displaystyle\int_{|\hat{x}|\leq d}\Psi_{T}(\hat{x})e^{i{\omega}L}d^{2}x$
(24) $\displaystyle=$ $\displaystyle\int_{|\hat{x}|\leq
d}\Psi(\hat{x})e^{-\frac{i{\omega}\hat{x}\cdot\hat{x_{s}}}{f}}d^{2}x$
$\displaystyle=$
$\displaystyle\int\mathrm{w}(\hat{x})\Psi(\hat{x})e^{-\frac{i{\omega}\hat{x}\cdot\hat{x_{s}}}{f}}d^{2}x,$

where $d$ is the length radius. $L$ is the distance between the lens point
$(x,y,0)$ and the screen point $(x_{s},y_{s},z_{s})$. The above window
function $\mathrm{w}(\hat{x})$ is given by

$\displaystyle\mathrm{w}(\hat{x})=\left\\{\begin{aligned}
0&,&0\leq|\hat{x}|\leq d\\\ 1&,&|\hat{x}|\geq d\end{aligned}\right.$ (25)

Figure 8: The diagram of the ingoing angle and outgoing angle for a photon
moving around the black hole.

From a wave-optical method, we also have a formula which help us convert the
response function $\langle O(\hat{x})\rangle$ to the image of the dual black
hole $|\Psi_{s}(\hat{x}_{s})|^{2}$ on a virtual screen shown as follows

$\displaystyle\Psi_{s}(\hat{x}_{s})$ $\displaystyle=$
$\displaystyle\int_{|\hat{x}|\leq d}\langle O(\hat{x})\rangle
e^{-\frac{i{\omega}\hat{x}\cdot\hat{x}_{s}}{f}}d^{2}x$ (26) $\displaystyle=$
$\displaystyle\int\mathrm{w}(\hat{x})\langle O(\hat{x})\rangle
e^{-\frac{i{\omega}\hat{x}\cdot\hat{x}_{s}}{f}}d^{2}x.$

Next, we will employ Eq.(26) to construct the Einstein ring and investigate
how the wave source and spacetime affect the ring.

##  IV Holographic Einstein ring in ADS black hole

As predicted on the AdS boundary, we observe different image profiles. In
order to study the impact of wave source, we give the holographic Einstein
images for various $\omega$ with fixed $Q=0.5$, $r_{*h}=0.5$, $c_{0}=0.5$ and
$e=0.01$ in Fig. 9 when the position of the observer is at the north pole of
the AdS boundary, which means $\theta_{obs}=0$. We clearly see a series of
axis-symmetric concentric circular rings appear. And as we increase the
frequency $\omega$, the corresponding ring becomes sharper. This means that
the geometric optics approximation better describes the image in the high-
frequency limit. For a comprehensive understanding of Fig. 9, we also plot the
profiles of the lensed response function in Fig. 10. It is clear that
different frequencies lead to changes in the brightness and also the location
of brightness peaks.

(a) $\omega=80$ (b) $\omega=60$ (c) $\omega=40$ (d) $\omega=20$

Figure 9: The images of the lensed response observed for different $\omega$
are shown at the observation angle $\theta_{obs}=0$. We see $Q=0.5$,
$r_{*h}=0.5$, $c_{0}=0.5$, $e=0.01$.

(a) $\omega=80$ (b) $\omega=60$ (c) $\omega=40$ (d) $\omega=20$

Figure 10: The brightness of the lensed response are shown on the screen for
different $\omega$ with $Q=0.5$, $z_{h}=0.5$, $c_{0}=0.5$, $e=0.01$.

(a) $c_{0}=1$,$\theta_{obs}=0$ (b) $c_{0}=1$,$\theta_{obs}=\pi/6$ (c)
$c_{0}=1$,$\theta_{obs}=\pi/3$ (d) $c_{0}=1$,$\theta_{obs}=\pi/2$

(e) $c_{0}=3$,$\theta_{obs}=0$ (f) $c_{0}=3$,$\theta_{obs}=\pi/6$ (g)
$c_{0}=3$,$\theta_{obs}=\pi/3$ (h) $c_{0}=3$,$\theta_{obs}=\pi/2$

(i) $c_{0}=5$,$\theta_{obs}=0$ (j) $c_{0}=5$,$\theta_{obs}=\pi/6$ (k)
$c_{0}=5$,$\theta_{obs}=\pi/3$ (l) $c_{0}=5$,$\theta_{obs}=\pi/2$

(m) $c_{0}=7$,$\theta_{obs}=0$ (n) $c_{0}=7$,$\theta_{obs}=\pi/6$ (o)
$c_{0}=7$,$\theta_{obs}=\pi/3$ (p) $c_{0}=7$,$\theta_{obs}=\pi/2$

Figure 11: The images of the lensed response are shown for different $c_{0}$
observed at various observation angles with $Q=4$, $z_{h}=4$, $\omega=80$,
$e=0.01$.

Also, the images of the lensed response observed at various observation angles
for different $c_{0}$ with $Q=4$, $r_{h*}=4$, $\omega=80$ and $e=0.01$ are
plotted in Fig. 11. At first, the observer is at the position
$\theta_{obs}=0^{\circ}$. This means the observation location is at the north
pole of the AdS boundary. From the left-most column of Fig. 11, a series of
axisymmetric concentric rings appear and they tend to move away from the
center as $c_{0}$ increases. Next we fix our observer to $\theta=\pi/6$ (the
second column from the left shown in Fig. 11). As the parameter $c_{0}$
increases, the same phenomena exists as in the $\theta=0^{\circ}$ case.
Explicitly, there still exists a series of axisymmetric concentric rings. And
the brightness on the right side of the ring is less than the brightness on
the left side of the ring. When the observer moves to $\theta=\pi/3$, the
luminosity rings disappear. All left are just bright light arcs or light spots
which are consistent with Zeng:2023zlf ; Zeng_Li2023 . And these light spots
or light arcs exist only on the left side. When the parameter $c_{0}$
increases, the bright light spots change to the light arcs. At last, we move
to $\theta=\pi/2$. Only bright spots are shown on the right-most column of
Fig. 11. As the parameter $c_{0}$ increases, the bright spot is far away from
the center. To see this better, in Fig. 12, we plot the brightness for
different $c_{0}$ at $\theta_{obs}=0$. Obviously, the peak of the curves
corresponds to the radius of rings. From these subfigures, we see that the
ring radius increases correspondingly as $c_{0}$ increases.

The images of the lensed response observed at the observation angle
$\theta_{obs}=0$ for different chemical potentials $u$ with a fixed
temperature $T=0.5$ are shown in Fig. 13. Here we set $\omega=80$, $c_{0}=0.5$
and $e=0.01$. These figures clearly show that as the potential increases from
$u=0.1$ to $u=1.8$, the positions of the brightest ring vary slightly.
However, as the chemical potential $u$ increases to $u=2.5$, the brightest
ring is further away from the center. These phenomena are also seen in Fig. 14
which shows the profile of the lensed response. The peaks of these curves
change slowly from $u=0.1$ to $u=1.8$, but position of the brightness peak is
far away from the center for $u=2.5$. Nevertheless, from both Fig. 13 and Fig.
14, we know that as $u$ increases, the ring radius increases, too.

(a) $c_{0}=1$ (b) $c_{0}=3$ (c) $c_{0}=5$ (d) $c_{0}=7$

Figure 12: The brightness of the lensed response for different $c_{0}$
observed at the observation angle $\theta_{obs}=0$ with $Q=4$, $z_{h}=4$,
$\omega=80$, $e=0.01$.

(a) $u=0.1$ (b) $u=1.1$ (c) $u=1.8$ (d) $u=2.5$

Figure 13: The images of the lensed response for different chemical potential
$u$ observed at the observation angle $\theta_{obs}=0$ with a fixed
temperature $T=0.5$. Here $\omega=80$ and $c_{0}=0.5$, $e=0.01$.

(a) $u=0.1$ (b) $u=1.1$ (c) $u=1.8$ (d) $u=2.5$

Figure 14: The brightness of the lensed response for different chemical
potential $u$ are shown on the screen with a fixed temperature $T=0.5$. Here
$\omega=80$ and $c_{0}=0.5$, $e=0.01$.

(a) $T=0.914$ (b) $T=0.369$ (c) $T=0.299$ (d) $T=0.296$

Figure 15: The images of the lensed response for different $T$ observed at the
observation angle $\theta_{obs}=0$ with a fixed $u=1$. Here, $\omega=80$,
$c_{0}=0.5$, $e=0.01$ and from (a) to (d), the charges correspond to
$Q=0.1,0.3,0.5,0.7$ respectively.

(a) $T=0.914$ (b) $T=0.369$ (c) $T=0.299$ (d) $T=0.296$

Figure 16:  The brightness of the lensed response are shown on the screen for
different $T$ with a fixed $u=1$. Here, $\omega=80$, $c_{0}=0.5$, $e=0.01$ and
from (a) to (d), the charges correspond to $Q=0.1,0.3,0.5,0.7$ respectively.

We also investigate the effect of the horizon temperature $T$ on the profiles
of the lensed response shown in Fig. 15, which is observed at $\theta_{obs}=0$
with fixed parameters $u=1$, $\omega=80$, $c_{0}=0.5$ and $e=0.01$. We
depicted the observational image of the dual black hole for increasing values
of temperature $T$. When the charges $Q=0.1,0.3,0.5,0.7$, the corresponding
temperatures $T=0.914,0.369,0.299,0.296$ respectively. We see one particular
bright ring in the region. When the temperature decreases from $T=0.914$ to
$T=0.296$, the brightest ring moves far away from the center, which is also
seen from the curves in Fig. 16, where the peak is also far away from the
center as the temperature $T$ decreases.

##  V The comparison between the holographic results and optical results

The above analysis shows that the brightest ring exists at the position of the
photon sphere of the black holes. We will verify this bright ring existing
here from the optical geometry.

It is well-known that in the Eikonal approximation, the Klein-Gordon equation
reduced to the following Hamilton-Jacoboi equation

$g^{\mu\nu}(\partial_{\mu}S-eA_{t})(\partial_{\nu}S-eA_{t})-M^{2}=0,$ (27)

in which $S$ is the action. For a spacetime with metric in Eq.(3), This
equation can be expressed as

$-\frac{1}{F(r)}\left(\partial_{t}S-eA_{t}\right)^{2}+F(r)\partial_{r}S^{2}+\frac{1}{r^{2}}\partial_{\theta}^{2}+\frac{1}{r^{2}\sin^{2}\theta}\partial_{\phi}^{2}-M^{2}=0.$
(28)

Next we only consider $\theta=\pi/2$, so that the photon orbit lying on the
equatorial plane. For the static spherically symmetric space time, there are
two killing vectors, so that the action can be written as

$S(t,r,\phi)=-\omega t+L\phi+\int\frac{1}{F(r)}\sqrt{R}dr,$ (29)

in which

$R=(\omega-eA_{t})^{2}-F(r)\left(\frac{L^{2}}{r^{2}}-2\right),$ (30)

and

$\frac{\partial S}{\partial t}=-\omega,\frac{\partial
S}{\partial\phi}=L,\frac{\partial S}{\partial r}=\frac{\sqrt{R}}{F(r)}.$ (31)

Figure 17: The ingoing angle and outgoing angle of the photon at the photon
sphere.

The photon will propagate along the trajectory of geodesics satisfying the
above conditions.

According to the definition Hashimoto:2018okj ; Hashimoto:2019jmw

$\displaystyle\text{}\cos\theta_{in}$ $\displaystyle=$
$\displaystyle\frac{g_{ij}u^{i}n^{j}}{|u||n|}|_{r=\infty},$ (32)

the ingoing angle $\theta_{in}$ with the normal vector of boundary
$\text{}n^{b}=\frac{\partial}{\partial r^{b}}$ and four-velocity
$\text{}u^{a}=(\frac{\partial}{\partial\lambda})^{a}$ satisfies

$\displaystyle\text{}\sin\theta_{in}=\frac{{L}}{\hat{\omega}}\text{},$ (33)

which is shown in Fig. 17. The above relation is still valid when the photon
is located at the photon ring. Label the angular momentum as $L_{p}$, which is
determined by the following conditions Liu:2022cev

$R=0,\frac{d{R}}{dr}=0.$ (34)

In the geometrical optics, the angle $\theta_{in}$ gives the angular distance
of the image of the incident ray from the zenith if an observer located on the
AdS boundary looking up into the AdS bulk. When two end points of the geodesic
and the center of the black hole are in alignment, the observer watches a ring
with a radius which corresponds to the incident angle $\theta_{in}$ because of
axial symmetry Hashimoto:2018okj ; Hashimoto:2019jmw .

Figure 18: The diagram between the ring angle $\theta_{R}$ and ring radius
$r_{R}$.

(a) $c_{0}=-0.5$ (b) $c_{0}=0.5$ (c) $c_{0}=1.5$

Figure 19: Comparison between the results obtained by wave optics and
geometric optics for different $c_{0}$ with a fixed temperature $T=0.5$.

(a) $c_{0}=-0.5$ (b) $c_{0}=0.5$ (c) $c_{0}=1.5$

Figure 20: Comparison between the results obtained by wave optics and
geometric optics for different $c_{0}$ with a fixed chemical potential $u$.

Furthermore, with Fig. 20, the angle of the Einstein ring is

$\text{}\sin\theta_{R}=\frac{r_{R}}{f}\text{}.$ (35)

Following Hashimoto:2018okj ; Hashimoto:2019jmw , we have
$\sin\theta_{R}=\sin\theta_{in}$ for a sufficiently large $l$. Then we obtain

$\text{}\frac{r_{R}}{f}=\frac{L_{p}}{\hat{\omega}},\text{}$ (36)

where $L_{p}$ is the angular momentum at the photon sphere. This relation also
can be further proved numerically. In Fig. 19 and Fig. 20, we fit
$\frac{r_{R}}{f}$ with $\frac{L_{p}}{\hat{\omega}}$ for different chemical
potential and horizon respectively. Both radii of the black hole horizon and
the circular orbit are also exhibited simply for curiosity’s sake. From both
figures, we can see that the Einstein ring angle derived by our holographic
method is consistent well with the ingoing angle obtained by geometric optics.

Figure 21:  The dependence of the correction term $h$ on the temperature and
the chemical potential for $m^{2}=10$.

In fact, the response function or the Einstein ring can be calculated directly
by the Green function

$\text{}\text{}\text{}\langle\mathcal{O}\rangle_{J_{\mathcal{O}}}=\int_{0}^{2\pi}\text{}\text{}\text{}\varphi^{\prime}\int_{0}^{\pi}\theta^{\prime}\sin\theta^{\prime}G(t,\theta,\varphi;t^{\prime},\theta^{\prime},\varphi^{\prime})J_{\mathcal{O}}\text{}\text{}\text{}(t^{\prime},\theta^{\prime}\varphi^{\prime})=\sum_{l=0}^{\infty}e^{-i\omega
t}G_{l0}(\omega)c_{l0}Y_{l0}(\theta).\text{}\text{}\text{}$ (37)

For a weakly interacting quantum field theory at finite temperature and finite
chemical potential, the retarded Green function can be expressed as
Liu:2022cev

$\text{}\text{}G_{lm}(\omega)=\frac{1}{\hat{\omega}^{2}-m_{T}^{2}-l(l+1)},\text{}$
(38)

where the thermal mass is given by $m_{T}^{2}=m^{2}+\frac{\gamma}{2}h(T,u)$,
in which

$h=\sum_{l}\frac{1}{\omega_{l}}[\frac{1}{e^{\beta(\omega_{o}+u)}-1}+\frac{1}{e^{\beta(\omega_{o}-u)}-1}],$
(39)

where $\beta$ is the inverse of the temperature and
$\omega_{o}=\sqrt{m^{2}+l(l+1)}$. Principally, as we know the variation
tendency of $h$, we can know the response or ring radius qualitatively. In
Fig. 21, we plot $h$ with respect to $T$ and $u$. From (a), we know that for
the low temperature, $h$ changes little. While for high temperature, we find
the higher the $u$, the larger the $h$. In other words, the response or ring
radius increases as the potential $u$ increases. This result is consistent
with that obtained by Fig. 14. From (b), we know that for the small $u$, the
higher the $T$ , the larger the $h$. That is to say, as the temperature $T$
raises, the ring radius increases too. This conclusion is inconsistent with
that obtained by Fig. 16, but consistent with reference Liu:2022cev .

##  VI Conclusions

As well known, the black holes are one of the incredible predictions of
General Relativity. And this gives us an opportunity to explore the formation
of black hole shadows. The Event Horizon Telescope collaboration obtained an
amazing success by releasing the first-ever image of a black hole. The Event
Horizon Telescope is important for deeper understanding and explanation of the
observed phenomena, such as Einstein ring. Therefore, in this paper, with the
holographic dual, we study the Einstein ring of the charged black hole in
conformal gravity.

Usually, the real quantum materials are engineered at a finite chemical
potential. To realize this in the framework of holography, we should turn on
the bulk electromagnetic field. Therefore, the black hole should be charged.
We investigated firstly the effect of the chemical potential on the amplitude
of $\langle O\rangle$. We found that when the chemical potential increases,
the amplitude of $\langle O\rangle$ increases correspondingly. In addition, we
also investigated the effect of the chemical potential on the ring radius, and
it was found that the ring radius increases as the chemical potential
increases though it changes a little for the low chemical potential. Moreover,
the effect of the temperature on the amplitude of $\langle O\rangle$ and ring
radius are discussed too. We found the higher the temperature, the smaller the
amplitude of $\langle O\rangle$, and the higher the temperature, the smaller
the ring radius. Though the amplitude of $\langle O\rangle$ does not describe
the Einstein ring, the change tendency is the same as that of the ring radius.

Conformal gravity is an important theories of gravity partly. On one hand, its
on-shell equivalence to Einstein gravity and its power counting
renormalizability. On the other hand, its spherically symmetric solution
contains a term linear in radial coordinate, which may explain the galaxy
rotation curves. We hence also investigated the effect of the gravity-related
parameter $c_{0}$ on the amplitude of $\langle O\rangle$ as well as the ring
radius, and found they grows as $c_{0}$ increases.

All the results are obtained in the framework of holography. To verify the
correctness of these results, we also obtained the radius of photon ring and
the corresponding ingoing angle via geometric optics. For different
temperatures and chemical potentials, we found the ingoing angles are
consistent with the angles of Einstein ring. The wave optics in the framework
of holography thus is reliable.

In Liu:2022cev , Einstein rings of a charged black hole in Einstein gravity
were investigated. They found the chemical potential has little effect on the
ring radius. They considered maybe only the small chemical potential. As the
chemical potential is large enough, its effect on the ring radius will be
obvious. In addition, in Liu:2022cev , it was found that the higher the
temperature, the smaller the ring radius. Obviously, our result is opposite to
this result. The reason for this is that in conformal gravity, there is no
charge in the metric, but it is given by the constraint relation. Therefore,
the holographic images play an important role in differentiating the geometric
features of different black holes for fixed wave source and optical system.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grants No. 11675140, No. 11705005, and No. 12375043), and Innovation and
Development Joint Foundation of Chongqing Natural Science Foundation (Grant
No. CSTB2022NSCQ-LZX0021) and Basic Research Project of Science and Technology
Committee of Chongqing (Grant No. CSTB2023NSCQ-MSX0324).

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