# Holographic Einstein Ring of a Charged Rastall AdS Black Hole with Bulk
Electromagnetic Field

M. Israr Aslam<EMAIL_ADDRESS>Department of Mathematics, COMSATS
University Islamabad, Lahore Campus, Lahore-$54000$ Pakistan.  Xiao-Xiong Zeng
<EMAIL_ADDRESS>State Key Laboratory of Mountain Bridge and Tunnel
Engineering, Chongqing Jiaotong University, Chongqing $400074$, China
Department of Mechanics, Chongqing Jiaotong University, Chongqing $400074$,
China  Rabia Saleem<EMAIL_ADDRESS>Department of Mathematics,
COMSATS University Islamabad, Lahore Campus, Lahore-$54000$ Pakistan.  Xin-Yun
Hu<EMAIL_ADDRESS>College of Economic and Management, Chongqing Jiaotong
University, Chongqing $400074$, China

###### Abstract

In this paper, we mainly study the Einstein images of a charged Rastall AdS
black hole (BH) within the framework of AdS/CFT correspondence. Considering
the holographic setup, we first analyze the amplitude of the total response
function for various values of model parameters. With an increase in parameter
$\lambda$ and temperature $T$, the amplitude of the response function is
decreased, while it is increased with the increase of the electric charge $e$
and chemical potential $\mu$. The influence of frequency $\omega$ also plays
an important role in the bulk field, as it is found that the decreasing
$\omega$ leads to an increase in the periods of the waves, which means that
the amplitude of the response function also depends on the wave source as
well. The relation between the temperature $T$ to the inverse of the horizon
$r_{h}$ and for the parameter $\lambda$ is also interpreted under fixed values
of other involved parameters. These, in turn, affect the behavior of the
response function and the Einstein ring, which may be used to differentiate
the present work from previous studies. With the help of a special optical
system, we construct the holographic images of the BH in bulk. The obtained
results show that the Einstein ring always appears with concentric stripes at
the position of the north pole, and this ring transforms into the luminosity-
deformed ring or bright light spot when the distant observer lies away from
the north pole. Moreover, from the brightness profiles, it is observed that as
$\lambda$ grows, the shadow radius is significantly decreased, while the
brightness of the ring is negligibly changed. Besides this, we further
investigate the influence of temperature $T$ and chemical potential $\mu$ on
the ring radius. These results reveal that the increasing values of
temperature $T$ lead to an increase in the shadow radius and decrease as the
values of chemical potential $\mu$ are growing. Finally, we finish this work
with a discussion on the ingoing angle of the photon orbit, which is based on
geometrical optics and observed that it is consistent with the results
obtained through the holographic framework.
Keywords: Rastall Gravity; AdS/CFT Correspondence; Black Hole; Einstein Rings.

##  I Introduction

The expanding nature of our universe can be defined as the study of evolution,
the vast number of experimental and productive changes that have been
performed during all time and across all space from the big bang to humankind.
In the last few years, General Relativity (GR) R1  has prevailed as the most
successful description of gravitational interaction. Indeed, Einstein’s GR
possesses astonishing predictions that have been observed throughout the years
such as the discovery of super-massive BHs at the center of galaxies R2 ; R3 ;
R4 ; R5  and the detection of gravitational waves R6 ; R7 , which has
fundamentally altered the way we perceive and test the accelerated expansion
of our universe. Nevertheless, GR has also been shown to have some
shortcomings observed directly from the observational data, such as the
discovery of the late-time accelerated expansion of the universe found through
Type Ia supernovae R8 ; R9  or from a theoretical point of view R10 . For the
last twenty years, scientists have focused on theoretical and observational
efforts so that the understanding of humanity regarding this phenomenon of the
universe may be improved. From a theoretical point of view, it is well-known
that this theory entails some open questions such as the cosmological constant
problem R11  or the coincidence problem R12 , which triggered us to find a
more general theory of gravity. Moreover, GR encapsulates singularities i.e.,
primordial Big Bang singularity, which are still puzzling to the research
community, and need to be a quantum description of gravity for a comprehensive
analysis R10 .

To tackle the above mentioned problems, one can consider the modified theories
of gravity (MTG) that provide a foundation for the current understanding of
the physical phenomena happening in the universe R13 . An extensive study
regarding MTG conducted by many researchers showed that these theories are not
only capable of dealing with dark energy (DE) problems but also with the
inflationary epoch R14 . Some of the MTG include the $f(R)$, $f(R,T)$,
$f(\mathcal{T})$ and $f(\mathcal{Q})$ gravity (where $R$ is the Ricci scalar,
$T$ is the trace of energy-momentum tensor, $\mathcal{T}$ is the torsion
scalar and $\mathcal{Q}$ is the non-metricity scalar) R15 ; R16 ; R17 ; R18 ;
R19  and many others have been suggested in the last few decades. In GR, the
coupling between space-time geometry and matter is minimal, leading to the
conservation law of energy-momentum tensor, which is valid only for flat
space-time. In this scenario, an interesting modification of GR has been
proposed by Rastall in $1972$ R20  and nowadays it has become a prominent
theory of gravity because it is presented as a MTG with non-conserved energy-
momentum tensor and measures the affinity of the space-time geometry to couple
with matter field in a non-minimal way. In this proposal, Rastall confronted
the usual conservation law of energy-momentum tensor (i.e.,
$\nabla_{\alpha}T^{\alpha\beta}=0$ (where $\nabla_{\alpha}$ is the covariant
derivative)) in curved space-time.

In fact, this conservation law ($\nabla_{\alpha}T^{\alpha\beta}=0$) may be
violated in the curved space-time. By considering this restriction of
conservation law, Rastall introduced a special amendment in GR, by improving
the usual conservation law as $\nabla_{\alpha}T^{\alpha\beta}=\lambda
R^{,\beta}$, where $\lambda$ is a coupling parameter of Rastall gravity (RG)
R20 . For the case $\lambda=0$, the Einstien’s GR can be recovered. Another
important feature of RG is that the field equations are quite simpler than the
other MTG and hence easier to handle. However, the field equations of RG are
quite simple. However, it appears to be consistent with some observational
data on the Hubble parameter and age of the universe R21 , leads that it could
be free of the entropy and the problems of the standard cosmology R22 . Also,
RG provides good agreement with experimental observations of the matter-
dominated phase against Einstein’s GR R23 . Observational data associated with
helium nucleosynthesis also satisfy the RG R24 . More studies on the
cosmological features of the RG including its consistency with numerous cosmic
phases can be discussed in R25 . In addition, RG has been discussed in the
framework of the Gödel-type universe with a perfect fluid matter and it was
found that the geodesics of particles do not alter R26 .

Besides, some attention has been focused on a debate such as, whether RG is
equivalent to GR. In this context, Visser claimed that RG is a rearrangement
of the matter sector of Einstein’s GR R27 . In the context of non-equilibrium
thermodynamics (for homogeneous and isotropic flat Friedmann-Lemaître-
Robertson-Walker background model), Das et al. R28  concluded that generalized
RG is equivalent to Einstein’s GR. On the contrary, Darabi and his colleagues
R29  disagree with Visser’s claim and they concluded that Visser
misinterpreted the matter-geometry coupling term which led him to the wrong
conclusion. In this perspective, they further showed that by applying Visser’s
method to $f(R)$ theory, one may derive that RG is equivalent to GR, which is
not true. In addition, they indicated that RG is an “open” theory in
comparison to GR and is more compatible with observational cosmology. Hansraj
et al. R30  also investigated this dispute and their results are in good
agreement with the result obtained by Darabi et al. R29 . They concluded that
RG could satisfy the basic conditions for a physically viable model, where GR
doesn’t meet the requirements, for details see Refs R30 . Hence, RG is not
equivalent to GR, which has been proven through many studies R31 ; R32 ; R33 ,
and it is worth studying because it faces a challenge from cosmological as
well as astrophysical observations.

To reveal the geometrical optics of BH, the newly born field of multi-
messenger astronomy uses its full weaponry, plasma radiations, gravitational
waves, and neutrinos R34 . Historically, the deflection of light due to strong
gravitational force around compact objects has been a powerful tool in the
verification of GR predictions R35 . Since the light rays travel across the
null geodesics of the background metric, the solution of the geodesic equation
provides realistic information to unveil the geometry of anybody compact
enough to hold a critical curve. This is an unstable circular orbit for mass-
less particles and corresponds to the backtrack of the light trajectory from
the observer’s screen that asymptotically approaches a bound photon orbit R36
. In astrophysics, a BH or a compact object will be typically surrounded by
its accretion disk, which provides the fundamental source of illumination R37
. The term BH shadow has come to interpret the interior of the critical curve.
The size and shape of the critical curve are determined by the background
geometry, whereas the optical appearance of a compact object is heavily
influenced by the astrophysics of the accretion disk. If the disk is optically
thin, one expects the optical appearance to be dominated by a central
brightness depression, the so-called shadow, surrounded by strongly lensed
light rays that have turned many times around the BH close to the critical
curve, and which revealed superimposed through the direct emission close the
boundary of the shadow R38 .

The details of the BH shadow image were beautifully confirmed by the Event
Horizon Telescope (EHT) R2 ; R3 ; R4 ; R5  in $2019$, where the tracking of
the plasma radiations around the super-massive BH in the center of the Messier
(M)$87$ elliptical galaxy and modulated by the strong magnetic fields,
released a bright ring-shaped lump of radiations surrounding a circular dark
region. This picture convincingly confirms the existence of BHs in our
Universe. Furthermore, the results obtained from EHT exhibit the existence of
a magnetic field around M$87$, prompting us to better understand and explain
the launching of energetic jets from its core. It carries the information
about the geometry of the electromagnetic fields responsible for the
synchrotron emission and found that magnetically arrested accretion disks
surrounded the M$87$ R39 ; R40 . An astrophysical BH provides a constant
space-time structure, illuminated by a time-varying emission region by some
external sources of the luminous accretion material, leading the BH to have a
variety of shapes and emit a variety of colors. Since, due to different
interesting scenarios of matter accretions around the BHs, the study of BH
shadows and their physical characteristics has reached a peak position among
the scientific community. Hence the study of BH shadow configurations not only
enables us to comprehend the geometric structure of space-time but also helps
us to explore various gravity models more deeply.

The anti-de Sitter (AdS)/conformal field theory (CFT) consists of a solid
apparatus where strongly coupled field theories are investigated. Any given
field theory, including finite temperature, has a hydrodynamical description
in the infrared (IR) limit, corresponding to long-length scales. The AdS/CFT
correspondence as a concrete realization of the holographic principle
explicitly identifies that the theory of quantum gravity in AdS bulk space is
equivalent to a dual CFT on the boundary R41 . The dual pair is the most
prominent example between the type IIB string theory on the Ad$S_{5}\times$S5
and the maximally super-symmetric gauge theory in four dimensions, $N=4$ super
Yang-Mills R42 ; R43  theory. Since then, the holographic principle of gravity
obtained the peak position among the various fields of physics because it is
not only used to indirectly test the correspondence related to quantum physics
but also provides solutions to some problems faced by strong coupling systems.
In this view, the AdS/CFT correspondence is successfully applied in various
areas along with the analysis of the strong coupling dynamics of quantum
chromodynamics (QCD) and the electroweak theories, BHs and quantum gravity,
relativistic hydrodynamics or different applications in condensed matter
physics, superfluidity and superconductivity, which shed some light for
understanding high-temperature superconducting physics R44 ; R45 ; R46 ; R47 ;
R48 ; R49 ; R50 ; R51 ; R52 . So far, regarding the theoretical aspects of the
holographic principle, it has been of special interest to scientists and
numerous publications have been devoted to this subject in the context of MTG
R53 ; R54 ; R55 ; R56 ; R57 ; R58 ; R59 . Recently, in the context of AdS/CFT
correspondence, Kaku et al. kaku  proposed a method to create a star orbiting
in an asymptotically AdS space-time and then they demonstrate the angular
position of the star with the help of lensed response function. Therefore, the
AdS/CFT correspondence strongly supports a better understanding of various
physical topics.

Because of the above discussion, apart from the study of the shape and size of
shadows, the possible observational characteristics of the BHs shadow
surrounded by different accretion flows were studied for a long time. In the
context of the RG, Guo et al. R60  investigated the shadows and photons sphere
of the charged BH surrounded by a perfect fluid radiation field with the
static/infalling accretion flow models and found that the shadow luminosity of
this BH with static spherical accretion is brighter than that of the infalling
spherical accretion one. In R61 , authors studied the shadow cast of non-
commutative BH in RG and observed that the luminosity of the resulting shadow
closely depends on the non-commutative parameter. The shadow of a rotating BH
surrounded by an anisotropic fluid field in the RG is investigated by Kumar et
al. R62 . They found that the rotating Rastall BH leads to a shadow with a
smaller size than the Kerr BH for a given value of the rotating parameter.
Saleem and Aslam R63  investigated the shadow and observable signatures of
non-commutative charged Kiselev BH within different accretion flow models.
They observed that the specific intensity of this BH profile was darker in the
case of infalling gas due to the Doppler effect as compared to static one. He
et al. R64  discussed the BH shadow within the framework of the clouds of
strings and quintessence and found that the brightness distribution of the
photon sphere is a normal function of the string attenuation factor. Since
then, the study of BH shadows and their relevant phenomenological consequences
have been investigated in several publications within the framework of MTG as
well as in RG (for example see Refs. R64 ; R65 ; R66 ; R67 ; R68 ; R69 ; R70
).

Although the mentioned literature provides a realistic description of the
shadow dynamics simulations of the BH, there is still a thrust to further
understand the geometrical properties and their associated deeper
phenomenological consequences need to be addressed through effective
implementations. Therefore, in the framework of the AdS/CFT correspondence,
the holographic image of AdS BH in the bulk was constructed when the scalar
wave emitted by the source at the AdS boundary enters the bulk and then
propagates in the bulk R71 ; R72 . Particularly, they observed the Einstein
ring in the framework of holographic and the size of the ring is consistent
with the size of the BH photon sphere, which is observed through geometrical
optics. Motivated by this, the authors in R73 ; R74 ; R75 ; R76 ; R77 ; R78
constructed the holographic images of AdS BHs in the context of different MTG
backgrounds. In all these studies, they investigated the distinct features of
Einstein’s ring structure in holographic quantum matter and concluded that
holographic images can be used as an effective tool to distinguish different
types of BHs for fixed wave sources and optical systems. Following the idea
given in R71 ; R72 , in the present paper, we further investigate the Einstein
ring structure for the lensed response of the complex scalar field as a probe
wave on the charged Rastall AdS BH solution within the framework of AdS/CFT
correspondence. We analyze the possible effect of model parameters on the
resulting Einstein ring, which will further confirm that the appearance of
such an Einstein ring structure may be used as a strong signal for the
existence of its gravity dual. To reach this goal, the rest of the paper is
organized as follows.

In section II, we briefly define the background of charged Rastall AdS BH and
the holographic setup with a finite chemical potential. Section III is
dedicated to constructing the special optical system and studying the Einstein
ring images under the various values of the involved model parameters. The
comparison between the results obtained by the geometric optics approximation
and the wave optics are dealt with in Section IV. Finally, Section V is
devoted to discussion and concluding remarks.

##  II Charged AdS Black Hole in Rastall Gravity and Holographic Setup

According to Rastall’s proposal, the energy-momentum conservation law is
revised as R20

$\nabla_{\alpha}T^{\alpha\beta}=\lambda R^{,\beta},$ (1)

and the field equations of the RG are formulated as

$G_{\alpha\beta}+k\lambda g_{\alpha\beta}R=kT_{\alpha\beta},$ (2)

where $G_{\alpha\beta}$, $T_{\alpha\beta}$ and $k$ are Einstein tensor,
energy-momentum tensor and Rastall gravitational coupling constant,
respectively. In the last few years, the RG has begun a new era in
gravitational physics and many works on various BH solutions and related
thermodynamics have been investigated in R79 ; R80 ; R81 ; R82 . In the
framework of RG, we consider the static spherically symmetric BH metric with
the electromagnetic field in the cosmological constant background as given
below R79

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$ (3)

where

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3-12\lambda}r^{2},$
(4)

where $f(r)$ is the metric function, which is determined in terms of mass $M$
and charge $Q$, and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$. The
cosmological constant $\Lambda$ is related to the AdS radius $l$, which is
defined as $\Lambda=-\frac{3}{l^{2}}$. For simplicity, we hereafter set $l=1$.
Further, to analyze the holographic images of the charged AdS BH solution in
RG within the framework of AdS/CFT correspondence, we first define the
holographic setup of the dual BH images from the response function of the
boundary quantum field theory with external sources. Closely followed by R71 ;
R72 , for the source $\mathcal{J}_{\mathcal{O}}$, we employ a time-periodic
localized Gaussian source with the frequency $\omega$, on one side of the AdS
boundary and scalar waves generated by the source can propagate in the bulk.
The scalar wave propagates inside the BH space-time and reaches the other side
of AdS boundary, the corresponding response function will be generated, such
as $\langle O\rangle$, which gives the information about the bulk structure of
the BH space-time. The schematic diagram of this setup is shown in Fig. 1.

Figure 1: The schematic picture of imaging a dual BH.

By using a special optical system, we can convert the extracted response
function $\langle O\rangle$, into the holographic image, which can be seen on
a virtual screen. The ($2+1$)-dimensional boundary CFT on a $2$-sphere $S^{2}$
is naturally dual to a BH in the AdS4 space-time or the massless scalar field
in the space-time. To achieve the goal, we use a new definition as $v=1/r$,
then we have $f(r)=v^{-2}f(v)$. In this perspective, we revised the metric (3)
as following

$ds^{2}=\frac{1}{v^{2}}[-f(v)dt^{2}+\frac{dv^{2}}{f(v)}+d\Omega^{2}].$ (5)

The value of $v=0$ corresponds to the AdS boundary, where the dual quantum
system lies. The Hawking temperature $T$ of the BH is related to its surface
gravity, which is calculated as R79

$T=\frac{3+v^{2}_{h}+4\lambda v^{2}_{h}-Q^{2}v^{4}_{h}+4\lambda
Q^{2}v^{4}_{h}}{4\pi v_{h}(1-4\lambda)},$ (6)

where $v_{h}$ is the inverse of the horizon radius of BH.

Figure 2: The relation between the temperature $T$ and $v_{h}$ for
$Q=0.5,\lambda=0.01$ are shown in the left panel. Whereas the relation between
the temperature $T$ and $\lambda$ for $Q=v_{h}=0.5$ are shown in the right
panel.

The behavior of temperature $T$ versus the inverse of the horizon $v_{h}$ and
the behavior of temperature $T$ versus $\lambda$ are shown in the left and
right panels of Fig. 2, respectively. From Fig. 2 (left panel), we see that
when $Q=0.5,\lambda=0.01$, the temperature $T$ has the maximum value at point
($v_{h}=0.0822522,T=3.02991$) and then gradually decreases with the increasing
of the inverse of the horizon $v_{h}$. The right panel of Fig. 2 shows that
when $Q=v_{h}=0.5$, the temperature $T$ has lowest value at point
($\lambda=0.01,T=0.534661$) and then sharply increases as the strength of
Rastall parameter $\lambda$ increases. This feature may be used as a method to
distinguish the charged Rastall AdS BH solution from the previous studies R73
; R75 ; R76 ; R77 ; R78  and may affect the following response function, which
is derived through the holographic setup. Now, we consider the complex scalar
field as a probe field in the charged Rastall AdS background. In this regard,
the corresponding dynamics is described by the following Klein-Gordon equation
as R83

$D^{\nu}D_{\nu}\Phi-\mathcal{M}^{2}\Phi=0,$ (7)

where $D_{\nu}\equiv\nabla_{\nu}-ieA_{\nu}$ is the covariant derivative
operator, $A$ is associated with electromagnetic four-potential, $\Phi$
interprets the complex scalar field with $e$ its electric charge and
$\mathcal{M}$ stands for the mass of the scalar field. Further, to solve it by
numeric in a more convenient manner, we prefer to define the ingoing Eddington
coordinate as R73

$\displaystyle u=t+v_{\star}=t-\int\frac{1}{f(v)}dv,$ (8)

and then the non-vanishing bulk background fields are transformed into the
following form.

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}[-f(v)du^{2}-2dvdu+d\Omega^{2}],$ (9)
$\displaystyle A_{\nu}$ $\displaystyle=$ $\displaystyle-A(v)(du)_{\nu},$ (10)

where $A(v)=Q(v-v_{h})$. In this case, the gauge transformation is also
applied to the electromagnetic four-potential. By holography, $\mu=Qv_{h}$ is
denoted as the chemical potential of the boundary system. Here we take
$\mathcal{M}^{2}=-2$ for definiteness. With $\Phi=v\phi$, the asymptotic
behavior of the scalar field close to the AdS boundary can be expressed as

$\phi(u,v,\theta,\varphi)=\mathcal{J}_{O}(u,\theta,\varphi)+\langle O\rangle
v+\mathcal{O}(v^{2}).$ (11)

Based on the holographic dictionary, $\mathcal{J}_{O}(u,\theta,\varphi)$
denotes the external source for the boundary field theory, and the
corresponding expectation value of the dual operator, so-called the response
function, which is defined as R85

$\langle O\rangle_{\mathcal{J}_{O}}=\langle
O\rangle-(\partial_{u}-ie\mu)\mathcal{J}_{O},$ (12)

where $\langle O\rangle$ corresponds to the expectation value of the dual
operator when the source is turned off. As defined in R71 ; R72 , we choose an
axisymmetric and monochromatic oscillating Gaussian wave packet source, and
fixed it at the south pole of the boundary ${S}^{2}$, i.e., $\theta_{0}=\pi$
as the source. So, we have

$\displaystyle\mathcal{J}_{O}(u,\theta)=e^{-i\omega
u}\frac{1}{2\pi\eta^{2}}\exp\big{[}\frac{-(\pi-\theta)^{2}}{2\eta^{2}}\big{]}=e^{-i\omega
u}\sum_{n=0}^{\infty}c_{n0}Y_{n0}(\theta),$ (13)

where $\eta$ is the width of Gaussian wave source and $Y_{n0}$ is the
spherical harmonics function. We ignore the Gaussian tail safely due to its
smallest value and hence we only consider the case $\eta<<\pi$. Further, the
coefficients of the spherical harmonics $Y_{n0}$ can be obtained as

$c_{n0}=(-1)^{n}\bigg{(}\frac{(n+1/2)}{2\pi}\bigg{)}^{\frac{1}{2}}\exp\bigg{[}-\frac{1}{2}(n+1/2)^{2}\eta^{2}\bigg{]}.$
(14)

Based on Eq. (13), the corresponding bulk solution takes the following form

$\phi(u,v,\theta)=e^{-i\omega
u}\sum_{n=0}^{\infty}c_{n0}V_{n}(v)Y_{n0}(\theta),$ (15)

where $V_{n}$ satisfies the equation of motion as

$v^{2}f(v)V^{\prime\prime}_{n}+v^{2}[f^{\prime}(v)+2i(\omega-
eA)]V^{\prime}_{n}+[(2-2f(v))+vf^{\prime}(v)-v^{2}(ieA^{\prime}+n(n+1))]V_{n}=0,$
(16)

where the asymptotic behavior of $V_{n}$ can be expressed as in the following
form

$V_{n}=1+{\langle O\rangle}_{n}v+\mathcal{O}(v^{2}).$ (17)

Similarly, the resulting response ${\langle O\rangle}_{\mathcal{J}_{O}}$ can
be defined as

${\langle O\rangle}_{\mathcal{J}_{O}}=e^{-i\omega
u}\sum_{n=0}^{\infty}c_{n0}{\langle
O\rangle}_{\mathcal{J}_{O}n}Y_{n0}(\theta).$ (18)

And then we have

${\langle O\rangle}_{\mathcal{J}_{O}n}={\langle O\rangle}_{n}+i\hat{\omega},$
(19)

where $\hat{\omega}=\omega+e\mu$. Now, our main purpose is to solve the radial
Eq. (16) with the following boundary condition such as $V_{n}(0)=1$ at the AdS
boundary and the horizon boundary condition on the BH event horizon. From this
perspective, we obtain the efficient numerical solution of Eq. (16) through
the pseudo-spectral method R73  and derive the corresponding numerical
solution for $V_{n}$ and extract ${\langle O\rangle}_{n}$. Then, the total
response function ${\langle O\rangle}$ can be found with the help of extracted
${\langle O\rangle}_{n}$ and through Eq. (18). Here, we choose some proper
values of the considering BH space-time and lens parameters as examples to
clearly show the optical appearance arises from the diffraction of the scalar
field of the BH, which can be seen in Figs. 3-5. The amplitude of the total
response function is plotted for different values of $\lambda$ with
$Q=v_{h}=0.5,\omega=80,e=1$, and for different $\omega$ with
$\lambda=0.01,Q=v_{h}=0.5,e=1$ as shown in the left and right panels of Fig.
3, respectively. Clearly, the left panel of Fig. 3 shows that the amplitude is
maximum when $\lambda$ has smaller values and then decreases with the larger
values of $\lambda$. Whereas the right panel of Fig. 3 interpreted that the
period of the scalar wave is maximum when $\omega=30$, and it gradually
decreases with the increasing values of $\omega$. The left panel of Fig. 4
illustrates the amplitude of the total response function for different values
of electric charge $e$ with $\lambda=0.01,Q=v_{h}=0.5,\omega=80$. Similarly,
the right panel of Fig. 4 depicted the amplitude of the total response
function for different values of $\mu$ with
$\lambda=0.01,v_{h}=0.5,\omega=80,e=1$, where $Q=0.1,0.5$ and $0.9$
corresponds to $\mu=1/5,1$ and $9/5$, respectively. This figure shows that the
absolute amplitude of the total response function increases when both $e$ and
$\mu$ have larger values. Figure 5 also depicted the behavior of the amplitude
by varying the temperature $T$ of the boundary system for
$\lambda=0.01,Q=0.5,\omega=80,e=1$, where $v_{h}=0.4,0.5$ and $0.6$
corresponds to $T=0.652,0.535$ and $0.458$, respectively. From this figure,
one can see that the amplitude of the total response function significantly
changes with temperature $T$, for instance, when $T=0.458$, the amplitude
reaches a peak position and moves down at $T=0.535$ and $0.652$ nicely, see
Fig. 5. This implies that the amplitude of the total response function
increases with the decreasing values of $T$. All these results imply that the
amplitude of the total response function closely depends on the Gaussian
source and the space-time geometry. In the next section, we transformed this
response function as the observed images on the screen, which may be useful to
reflect the distinct features of the space-time geometry.

Figure 3: The absolute amplitude of total response function for different
values of $\lambda$ with $Q=v_{h}=0.5,\omega=80,e=1$ (left panel) and for
different $\omega$ with $\lambda=0.01,Q=v_{h}=0.5,e=1$ (right panel).

Figure 4: The absolute amplitude of the total response function for different
values of $e$ for $\lambda=0.01,Q=v_{h}=0.5,\omega=80$ (left panel) and for
different $\mu$ for $\lambda=0.01,v_{h}=0.5,\omega=80,e=1$ (right panel).
Further, in the right panel, from top to bottom, the values of $\mu$
correspond to $Q=0.1,0.5,0.9$, respectively. Figure 5: The absolute amplitude
of total response function for different values of $T$ with
$\lambda=0.01,Q=0.5,\omega=80,e=1$. From top to bottom, the values of $T$
correspond to $v_{h}=0.4,0.5,0.6$.

##  III The Optical System and Holographic Rings of Charged AdS Black Hole in
Rastall Gravity

After analysis of the physical interpretation of the extracted response
function ${\langle O\rangle}$, we will use it to directly interpret the image
of the BH on the screen. As stated in R71 ; R72 , to observe the extracted
response function ${\langle O\rangle}$, we need to introduce a special optical
system, which is composed of an extremely thin convex lens and the spherical
system as shown in Fig. 6. In the middle position, there is a convex lens, and
we assume the lens is infinitely thin and the size of the lens is much smaller
than the focal length $f$, which is regarded as a transform of the plane wave
into the spherical waves. Imagine that, from the left side of Fig. 6, the
incident wave is irradiated at the lens, and the wave will convert to the
transmitted wave at the focus, which will be depicted on the screen, see the
right side of Fig. 6. With the help of this apparatus, we analyze the visual
appearance of the holographic Einstein ring image under the suitable values of
model parameters, which may help us to deeply understand the actual
astrophysical situation of space-time structure and its associated deeper
phenomenological consequences.

Figure 6: The structure of image formation system on the screen.

Let us define an observational point at
($\theta,\varphi)=(\theta_{\text{obs}},~{}0$) on the AdS boundary, where an
observer is surrounded by a small white circle as shown in the left side of
Fig. 6. We introduce new polar coordinates as
($\theta^{\prime},\varphi^{\prime}$) in such a way that

$\sin\theta^{\prime}\cos\varphi^{\prime}+i\cos\theta^{\prime}=e^{i\theta_{\text{obs}}}(\sin\theta\cos\varphi+i\cos\theta),$
(20)

($\theta^{\prime}=0,~{}\varphi^{\prime}=0$), which corresponds to the center
of the observation region. For simplicity, we define cartesian coordinates
($x,y,z$) with
$(x,y)=(\theta^{\prime}\cos\varphi^{\prime},~{}\theta^{\prime}\sin\varphi^{\prime})$
in the observation region. To develop an optical setup, in the middle
position, we fix the convex lens on the ($x,y$)-plane. The focal length and
radius of the convex lens are presented by $f$ and $d$, respectively.
Moreover, we adjust a spherical screen having coordinates as
$\vec{x}_{s}=(x,y,z)=(x_{s},y_{s},z_{s})$, satisfying
$x^{2}_{s}+y^{2}_{s}+z^{2}_{s}=f^{2}$ R71 ; R72 , and consider
$\Psi_{in}(\vec{x})$ as the incident wave with frequency $\omega$, which is
obtained through the convex lens, then the transmitted wave
$\Psi_{tr}(\vec{x})$ can be expressed as

$\Psi_{tr}(\vec{x})=e^{-i\omega\frac{|\vec{x}|^{2}}{2f}}\Psi_{in}(\vec{x}),$
(21)

where $\vec{x}=(x,y,0)$ is the coordinate position on the AdS boundary, where
the convex lens is placed in the observation range. Then this wave is
considered as the spherical wave and will transform into the observed wave
$\Psi_{s}(\vec{x}_{s})$, when it is reached on the screen as

$\displaystyle\Psi_{s}(\vec{x}_{s})=\int_{|\vec{x}|\leq
d}d^{2}x\Psi_{tr}(\vec{x})e^{i\omega\varpi}\propto\int_{|\vec{x}|\leq
d}d^{2}x\Psi_{in}(\vec{x})e^{-i\frac{\omega}{f}\vec{x}.\vec{x}_{s}}=\int
d^{2}x\Psi_{in}(\vec{x})\Xi(\vec{x})e^{-i\frac{\omega}{f}\vec{x}.\vec{x}_{s}},$
(22)

where $\varpi$ is the distance from the lens point $(x,y,0)$ to the screen
point ($x^{2}_{s},~{}y^{2}_{s},~{}z^{2}_{s}$) and $\Xi(\vec{x})$ is the window
function, read as

$\Xi(\vec{x})\equiv\begin{cases}\text{$1,$ \quad$0\leq|\vec{x}|\leq d$},\\\
\text{$0$,\quad~{}$|\vec{x}|>d$}.\end{cases}$ (23)

Now we obtain a formula through the wave optics method, which converts the
extracted response function ${\langle O(\vec{x})\rangle}$ to the image of the
dual BH $|\Psi_{s}(\vec{x}_{s})|^{2}$ on a virtual screen as R71 ; R72

$\displaystyle\Psi_{s}(\vec{x}_{s})=\int_{|\vec{x}|\leq
d}d^{2}x\Psi_{in}(\vec{x})e^{-i\frac{\omega}{f}\vec{x}.\vec{x}_{s}}=\int
d^{2}x{\langle
O(\vec{x})\rangle}\Xi(\vec{x})e^{-i\frac{\omega}{f}\vec{x}.\vec{x}_{s}}.$ (24)

Based on Eq. (24), we now plot the holographic Einstein ring image on the
screen for the different values of model parameters and illustrate how the
wave source and space-time geometry affect the Einstein ring.

(a) $\lambda=0.01,~{}\theta_{\text{obs}}=0^{o}$

(b) $\lambda=0.01,~{}\theta_{\text{obs}}=30^{o}$

(c) $\lambda=0.01,~{}\theta_{\text{obs}}=60^{o}$

(d) $\lambda=0.01,~{}\theta_{\text{obs}}=90^{o}$

(e) $\lambda=0.06,~{}\theta_{\text{obs}}=0^{o}$

(f) $\lambda=0.06,~{}\theta_{\text{obs}}=30^{o}$

(g) $\lambda=0.06,~{}\theta_{\text{obs}}=60^{o}$

(h) $\lambda=0.06,~{}\theta_{\text{obs}}=90^{o}$

(i) $\lambda=0.11,~{}\theta_{\text{obs}}=0^{o}$

(j) $\lambda=0.11,~{}\theta_{\text{obs}}=30^{o}$

(k) $\lambda=0.11,~{}\theta_{\text{obs}}=60^{o}$

(l) $\lambda=0.11,~{}\theta_{\text{obs}}=90^{o}$

(m) $\lambda=0.16,~{}\theta_{\text{obs}}=0^{o}$

(n) $\lambda=0.16,~{}\theta_{\text{obs}}=30^{o}$

(o) $\lambda=0.16,~{}\theta_{\text{obs}}=60^{o}$

(p) $\lambda=0.16,~{}\theta_{\text{obs}}=90^{o}$

Figure 7: The density maps of the lensed response on the screen at various
observation angles under the suitable values of $\lambda$ with
$Q=v_{h}=0.5,~{}\omega=80,~{}e=1$.

To understand this, we illustrate the density maps of the holographic Einstein
ring images on the screen for different values of parameter $\lambda$ and
observational angles of the AdS boundary with
$Q=v_{h}=0.5,~{}\omega=80,~{}e=1$, see Fig. 7. From this figure, one can see
that when $\lambda=0.01$ and the distant observer is located at
$\theta_{obs}=0^{o}$, there is a bright ring with a series of concentric
striped patterns appears in the image, which is shown in Fig. 7 (a). From Fig.
7 (a) to 7 (d), we observe that as the value of the observational angle is
increased, the bright ring transforms into a luminosity-deformed ring, instead
of a strict axisymmetric ring. Particularly, when $\theta_{obs}=60^{o}$ (see
Fig. 7 (c)), bright light arcs appear, rather than a bright ring and further
transform into bright light points when $\theta_{obs}=90^{o}$, which is shown
in the right side of the screen (see Fig. 7 (d)). When we increase the
parameter $\lambda$, such as $\lambda=0.06$, the same phenomena exist as for
$\theta_{obs}=0^{o}$. And when $\theta_{obs}=30^{o}$, we observe that there
are two large light arcs appear, in which the right arc is brighter as
compared to the left one (see Fig. 7 (f)). Further, when
$\theta_{obs}=60^{o}$, now the large arcs are changed into small arcs and
further converted into two light spots, when $\theta_{obs}=90^{o}$, as shown
in Fig. 7 (g) and 7 (h), respectively. These findings are also consistent with
R71 ; R72 . Next, when we fix $\lambda=0.11$ and vary the positions of the
distant observer (see Fig.7 (i) to 7 (l)), we notice that the optical
appearance of the ring behaves almost the same as we discussed in the previous
case when $\lambda=0.06$. However, a significant difference is found such as
the radius of the ring gradually moving towards the center of the screen,
which is more prominent in this case. When the strength of parameter $\lambda$
further increases, i.e., $\lambda=0.16$, we see the series of axisymmetric
concentric rings at $\theta_{obs}=0^{o}$, and these rings are closer to the
center of the screen, as shown in Fig.7 (m). And when $\theta_{obs}=30^{o}$,
we see bright light arcs and these arcs are changed into pairs of bright light
points when $\theta_{obs}$ changed from $\theta_{obs}=60^{o}$ to
$\theta_{obs}=90^{o}$. Further from top to bottom, we note that as the value
of RG parameter $\lambda$ increases, the overall brightness of the ring
slightly increases, which is hard to observe.

(a) $\lambda=0.01$

(b) $\lambda=0.06$

(c) $\lambda=0.11$

(d) $\lambda=0.16$

Figure 8: Changes in brightness of the lensed response on the screen for
different $\lambda$ at $Q=v_{h}=0.5,~{}\omega=80,~{}e=1$.

And more as shown in Fig. 8, we further analyze the influence of the parameter
$\lambda$ on the brightness curves with fixed values of
$Q=v_{h}=0.5,~{}\omega=80$ and $e=1$. The $x$ and $-x$ intercept correspond to
the radius of rings and the vertical axis shows the brightness of the lensed
response on the screen. From Fig. 8 (a), when $\lambda=0.01$, it is observed
that the peak curves lie close to the boundary, which means that the ring
radius has maximum value such as the curves lie at the points $-1$ and $1$ on
the $x$-axis. When parameter $\lambda$ grows, i.e., $\lambda=0.06$, the peak
curves gradually move towards the center and lie at the points $-0.8$ and
$0.8$ on the $x$-axis, meaning the ring’s radius decreases. However, the
brightness of the lensed response slightly increases in this case, see the
vertical axis of Fig. 8 (b), similarly, in Figs. 8 (c) and 8 (d), we see that
the peak curves gradually move towards the center with increasing values of
parameter $\lambda$. Moreover, a significant difference is found in Fig. 8 (d)
as compared to previous cases, when $\lambda=0.16$, in the middle of the
screen, there is a peak curve, which shows the maximum value as compared to
boundary curves. This effect can also be seen in Fig. 7 (m), where the bright
light spot in the center of the screen corresponds to the peak curve of Fig. 8
(d). All these results imply the increasing values of $\lambda$, decrease the
ring’s radius and slightly affect the brightness of shadow.

Now, we discuss the influence of wave source on the holographic Einstein ring
image, which is observed at the position of the north pole, i.e.,
$\theta_{\text{obs}}=0^{o}$ as depicted in Fig. 9. We selected the values of
parameters such as $\lambda=e=0.01,~{}Q=v_{h}=0.5$ and plotted four different
density maps according to some specific values of $\omega$ as an example. To
analyze the effect of the wave source, we fix $\eta=0.05$ and $d=0.6$ for the
convex lens and note that as the value of the frequency increases, the
corresponding ring becomes sharper. Further, for a better understanding of the
description of the wave source, we depicted the corresponding profiles of the
lensed response function in Fig. 10 under the same set of parameters as
mentioned in Fig. 9. From this figure, we observe that the decreasing values
of frequency $\omega$ lead to enhancing the gap between the brightness curves
as well as increasing the brightness. Based on these profiles, we concluded
that the geometric optics approximation provided better configurations about
the dual image of the BH in high frequency limit.

(a) $\omega=80$

(b) $\omega=60$

(c) $\omega=40$

(d) $\omega=20$

Figure 9: The density maps of the lensed response on the screen for different
$\omega$ at the observational angle $\theta_{\text{obs}}=0^{o}$ with
$\lambda=e=0.01,~{}Q=v_{h}=0.5$.

(a) $\omega=80$

(b) $\omega=60$

(c) $\omega=40$

(d) $\omega=20$

Figure 10: Changes in brightness of the lensed response on the screen for
different $\omega$ with $\lambda=e=0.01,~{}Q=v_{h}=0.5$.

(a) $T=0.0249$

(b) $T=0.0746$

(c) $T=0.1243$

(d) $T=0.174$

Figure 11: The density maps of the lensed response on the screen for different
$T$ at the observational angle $\theta_{\text{obs}}=0^{o}$ with a fixed value
of chemical potential $\mu=1$ and $\lambda=e=0.01,~{}\omega=80$. From left to
right, the values of temperature $T$ corresponds to
$Q=0.1,~{}0.3,~{}0.5,~{}0.7$, respectively.

Now, we are interested in defining the possible effect of the horizon
temperature $T$ on the profiles of the lensed response function, which is
depicted in Fig. 11 at the observational angle $\theta_{\text{obs}}=0^{o}$
with a fixed value of chemical potential $\mu=1$ and
$\lambda=e=0.01,~{}\omega=80$. We obtained the values of temperature
$T=0.0249,~{}0.0746,~{}0.1243,~{}0.174$ corresponds to charge
$Q=0.1,~{}0.3,~{}0.5,~{}0.7$, respectively. In particular, when $T=0.0249$,
the corresponding ring lies close to the center. As the temperature $T$ grows
such as $T=0.0746$, the resulting ring moves away from the center. And
similarly, when the temperature further grows such as $T=0.174$, the
corresponding bright ring is further shifted towards the boundary of the
screen, see Fig. 11 (d). This effect can also be seen by going left-to-right
on the sequence of images in Fig. 12, where the peak curves of the brightness
are gradually shifted towards the boundary with the increasing of the
temperature $T$. Furthermore, when $T=0.0249,~{}0.0746,~{}0.1243$, we see
fewer brightness curves in the center of the large curves (see Fig. 12), and
corresponding to these curves, we see the bright light spot in the screen, as
shown in Fig. 11. However, when $T=0.174$, we see a very dim light spot in the
center of the screen, which is difficult to detect, see Fig. 11 (d).

(a) $T=0.0249$

(b) $T=0.0746$

(c) $T=0.1243$

(d) $T=0.174$

Figure 12: Changes in brightness of the lensed response on the screen for
different $T$ with a fixed value of chemical potential $\mu=1$ and
$\lambda=e=0.01,~{}\omega=80$. From left to right, the values of temperature
$T$ corresponds to $Q=0.1,~{}0.3,~{}0.5,~{}0.7$, respectively.

(a) $\mu=0.05$

(b) $\mu=0.35$

(c) $\mu=0.65$

(d) $\mu=0.95$

Figure 13: The density maps of the lensed response on the screen for different
chemical potential $\mu$ at the observational angle
$\theta_{\text{obs}}=0^{o}$ with a fixed value of temperature $T=0.45$ and
$\lambda=e=0.01,~{}\omega=80$. From left to right, the values of chemical
potential $\mu$ correspond to $Q=0.0099,~{}0.0599,~{}0.0706,~{}0.0165$,
respectively.

Consequently, we will further observe the influence of chemical potential
$\mu$ on the Einstein ring image of AdS BH at the observational angle
$\theta_{\text{obs}}=0^{o}$ with a fixed value of temperature $T=0.45$ and
$\lambda=e=0.01,~{}\omega=80$, as shown in Fig. 13. From Fig. 13, we noted
that when $\mu=0.05$ and $0.35$, the radius of the Einstein ring does not
change and lies almost at the same positions on the screen from the center,
indicating that here the effect of the chemical potential $\mu$ on the
Einstein ring is negligible. However, as we increase the value of $\mu$ such
as $\mu=0.65$ as shown in Fig. 13 (c), the position of the Einstein ring
slightly varies and moves towards the center. But when we fixed $\mu=0.95$, we
see that the position of the Einstein ring is dramatically changed and comes
closer to the center of the screen sharply, see Fig. 13 (d). These results are
also depicted in Fig. 14, in which we show the changes of the lensed response
function on the screen under the same set of parameters as mentioned in Fig.
13. We observe that the peaks of the curves are slightly changed from
$\mu=0.05$ to $\mu=0.95$, but the positions of the peak curves significantly
move towards the center, particularly, when $\mu=0.95$. In a word, we say that
the smaller values of chemical potential $\mu$ have a negligible contribution
to changing the position of the bright ring, however, the large values of
$\mu$ significantly affected it. In this view, we concluded that under this
framework, the increasing values of $\mu$ leading to decrease the radius of
the bright ring.

(a) $\mu=0.05$

(b) $\mu=0.35$

(c) $\mu=0.65$

(d) $\mu=0.95$

Figure 14: Changes in brightness of the lensed response on the screen for
different chemical potential $\mu$ with a fixed value of temperature $T=0.45$
and $\lambda=e=0.01,~{}\omega=80$. From left to right, the values of chemical
potential $\mu$ corresponds to $Q=0.0099,~{}0.0599,~{}0.0706,~{}0.0165$,
respectively.

##  IV Comparison Between the Holographic Results and Geometrical Optics

In wave optics, we observe that at the position of the photon orbit, there is
the brightest ring in the image. Now, we will analyze this brightest ring in
the image through the optical geometry. So, to understand the motion of
photons around the BH, we need to define the geodesic equations by considering
Hamilton-Jacobi formulation R86 . The dynamics of BH shadows is widely studied
by the Lagrangian and the Hamiltonian formalism in the context of different
MTG. And the usual procedures give two conserved quantities such as the energy
$E$ and the angular momentum $L$ along the axis of symmetry. Despite
presenting the motivation for Hamilton-Jacobi formulation, we require the
Lagrangian for deriving the relation between the constants of motion. In this
view, the motions of photon in the vicinity of a charged AdS BH are described
by a Lagrangian formalism as

$\displaystyle\mathcal{L}=\frac{1}{2}g_{\mu\nu}(\dot{x}^{\mu}-eg^{\mu\nu}A_{\nu})(\dot{x}^{\nu}-eg^{\mu\nu}A_{\mu})=\frac{1}{2}\bigg{(}-f(r)\bigg{(}\dot{t}+\frac{eA_{t}}{f(r)}\bigg{)}^{2}+\frac{\dot{r}^{2}}{f(r)}+r^{2}(\dot{\theta}^{2}+\sin^{2}\theta\dot{\varphi}^{2})\bigg{)}.$
(25)

where $\dot{x}^{\mu}$ is the four-velocity of the photon and “.” is the
derivative with respect to the affine parameter $\sigma$ along the geodesics.
Since we only consider the photons that move on the equatorial plane, we apply
the initial conditions as $\theta=\pi/2$ and $\dot{\theta}=0$. Further, the
Lagrangian is independent explicitly on time $t$ and azimuthal angle $\phi$,
and hence one can obtain the expressions of two conserved quantities as

$\displaystyle
E=\frac{\partial\mathcal{L}}{\partial\dot{t}}=f(r)\dot{t}+eA_{t},\quad
L=\frac{\partial\mathcal{L}}{\partial\dot{\varphi}}=r^{2}\dot{\varphi}.$ (26)

Since the Eq. (26) will be used in further calculations to analyze the motion
of photons in a particular space-time through the corresponding geodesics
equations. Hence, this equation will help in converting the system in terms of
the conserved quantities. Now in the background of charged AdS BH, the Klein-
Gordon equation is reduced to the following Hamilton-Jacobi equation as R73 ;
R87

$\frac{1}{2}\mathcal{M}^{2}=-\frac{1}{2}g^{\mu\nu}\bigg{(}\frac{\partial\mathcal{S}}{\partial
x^{\mu}}-eA_{\mu}\bigg{)}\bigg{(}\frac{\partial\mathcal{S}}{\partial
x^{\nu}}-eA_{\nu}\bigg{)},$ (27)

in which $\mathcal{S}$ is the action. The Hamilton-Jacobi equation is
separable, and it possesses the solution of the form

$\mathcal{S}=-Et+L\varphi+\int\frac{\sqrt{\widehat{R}(r)}}{f(r)}dr,$ (28)

where $t$ denotes the time-like coordinate, $\varphi$ parameterizes the orbits
of the space-like Killing field and $\widehat{R}(r)$ is defined as

$\widehat{R}(r)=(E-eA_{t})^{2}-f(r)\bigg{(}\frac{L^{2}}{r^{2}}-2\bigg{)}.$
(29)

The trajectory of geodesics can be further obtained by considering the partial
derivatives of $\mathcal{S}$ with respect to $t$, $\varphi$ and $r$, as

$\displaystyle\frac{\partial\mathcal{S}}{\partial
t}=-E,\quad\frac{\partial\mathcal{S}}{\partial\varphi}=L,\quad\frac{\partial\mathcal{S}}{\partial
r}=\frac{\sqrt{\widehat{R}(r)}}{f(r)}.$ (30)

The ingoing angle $\theta_{in}$ of the light ray with the normal vector of
boundary $u^{b}=\frac{\partial}{\partial r^{b}}$ is defined as R72

$\displaystyle\cos\theta_{in}=\frac{g_{ab}v^{a}u^{b}}{|v||u|}\bigg{|}_{r=\infty}=\sqrt{\frac{\dot{r}^{2}/f(r)}{\dot{r}^{2}/f(r)+L^{2}/r^{2}}}\bigg{|}_{r=\infty},$
(31)

where $v^{a}$ is the spatial component of $4$-velocity of the geodesic,
$g_{ab}$ is the induced metric on the $t=$constant, $|v|$ and $|u|$ are the
norms of $v^{a}$ and $u^{b}$ with respect to $g_{ab}$. Moreover, Eq. (31) has
the equivalent relation as

$\displaystyle\sin\theta^{2}_{in}=1-\cos\theta^{2}_{in}=\frac{L^{2}}{\widehat{E}^{2}},$
(32)

where $\widehat{E}$ is considered as the energy of the single particle. Hence,
the incident angle $\theta_{in}$ of the photon orbit, which comes from the
infinite boundary satisfies the following relation as

$\sin\theta_{in}=\frac{L}{\widehat{E}},$ (33)

which is depicted in Fig. 15. This relation is still applicable when the
photon is located in the photon sphere. In this scenario, we denote the
angular momentum as $L_{\rho}$, which is calculated as

$\displaystyle\widehat{R}(r)=0,\quad\frac{d\widehat{R}}{dr}=0.$ (34)

Figure 15: Schematic picture showing the particular ingoing and the outgoing
angle of the photon revolves around the BH once.

In geometrical optics, when a distant observer near the AdS boundary looks up
into AdS bulk, angle $\theta_{in}$ provides the information about the angular
distance of the image of the ray from the zenith. Since there is axisymmetry
when both endpoints of the geodesic and the center of the BH are coinciding,
the observer will see the circular-shaped image with a radius equal to the
incident angle $\theta_{in}$ R71 ; R72 . Further, as presented in Fig. 16, one
can calculate the angle of the Einstein ring, which is displayed on the screen
with radius $r_{R}$ as

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

In addition, when the angular momentum is sufficiently large such as
$\sin\theta_{in}=\sin\theta_{R}$, then we have the following relation as R72

$\frac{L_{\rho}}{\widehat{E}}=\frac{r_{R}}{f}.$ (36)

Figure 16: Schematic picture showing the relation between $\theta_{R}$ and
$r_{R}$.

The incident angle of the photon and the angle of the photon ring illustrate
the position at which any distant observer can see the clear picture of the
photon ring. Next, we will numerically analyze the visual appearance of the
corresponding results, which should be essentially equal. In Fig. 17, we show
the radii of the BH horizon, the location of the photon orbit, Einstein radius
of the photon orbit and the Einstein ring radius in the unit of $f$ as a
function of BH horizon $r_{h}$ for different values of $\lambda$.

(a) $\lambda=0.01$

(b) $\lambda=0.06$

(c) $\lambda=0.11$

(d) $\lambda=0.16$

Figure 17: Plots show the BH horizon $r_{h}$, location of photon orbit
$r_{c}$, Einstein radius of the photon orbit (black line) and Einstein ring
radius obtained by wave optics (discrete red points) in the unit of $f$ for
different values of $\lambda$ with a fixed value of $\mu=1,~{}e=0.01$ and
$\omega=80$.

In Fig. 17 (a), when $\lambda=0.01$, we see that the location of the photon
ring continuously changes with the increasing of the horizon $r_{h}$, (see
blue curve). Further, we see that the Einstein ring radius appreciably
increases at lower values of the horizon $r_{h}$ and then starts to flatten
out with respect to the BH horizon $r_{h}$, see the black curve of Fig. 17
(a). Last, we see that the Einstein ring radius obtained by our holographic
method fits well with geometric optics, as discrete red points always lie on
the black curve or its vicinity. Similarly, in Figs. 17 (b)-(d), the
dependence of the radii of BH horizons $r_{h}$, the location of the photon
orbit and Einstein ring radius obtained by both wave optics and geometric
optics on the parameter $\lambda$ are presented. In these graphs, we have
noticed that with increasing values of parameter $\lambda$, the radii of BH
horizon $r_{h}$, the location of the photon orbit $r_{c}$ increases and
similar behavior is found as discussed in Fig. 17 (a). However, a significant
difference is observed, as the values of parameter $\lambda$ are increasing
from left to right, the Einstein ring radius (see black curve) slightly
increases at the smaller values of the BH horizon $r_{h}$ and then starts to
flatten out throughout the horizon $r_{h}$. This effect can be seen more
clearly, when $\lambda=0.16$, where the Einstein ring radius starts to flatten
out at very smaller values of the horizon $r_{h}$, as compared to previous
cases. Lastly, in all cases, the discrete red points presented the Einstein
ring radius obtained by our considering holographic method, as expected,
always lie on the black curve or its surroundings.

(a) $\lambda=0.01$

(b) $\lambda=0.06$

(c) $\lambda=0.11$

Figure 18: Plots show the BH horizon $r_{h}$, location of photon orbit
$r_{c}$, Einstein radius of the photon orbit (black line) and Einstein ring
radius obtained by wave optics (discrete red points) in the unit of $f$ for
different values of $\mu$ with a fixed values of $T=0.37,~{}e=0.01$ and
$~{}\omega=80$.

For a comprehensive analysis, we further plot the trajectories of BH horizon
radii $r_{h}$, location of the photon orbit $r_{c}$, Einstein radius of the
photon orbit (black curve) and the Einstein ring radius in the unit of $f$
(discrete red points) as a function of chemical potential $\mu$ for different
values of $\lambda$ in Fig. 18. From Fig. 18 (a), when $\mu$ has smaller
values, the radii of the BH horizon show the maximum value, and then drop
sharply with the increasing values of $\mu$. Similarly, the location of the
photon orbit and Einstein ring radius gradually changes with respect to the
increasing values of $\mu$. Moreover, the Einstein ring radius which is
obtained through the holographic method (see the discrete red points), closely
lies on the black curve or its surroundings, indicating that our results
obtained by wave optics fit well with those by geometric optics. In Fig. 18
(b) and 18 (c), we observe that the radii of BH horizon and the location of
the photon orbit behave similarly as defined in the previous case. However, it
is observed that when $\lambda=0.06$, at smaller values of $\mu$, the discrete
red points are slightly shifted away from the black curve and exactly lie on
the curve when $\mu\sim 0.7$ and then slightly shifted away from the black
curve. Similarly, when $\lambda=0.11$, this effect can be seen more clearly
such as at initial values of $\mu$, the discrete red points are significantly
shifted away from the black curve but when $\mu\sim 0.5$, these points closely
lie on the black curve or its surroundings. As expected, all these results
imply that the Einstein ring radius obtained by our wave optics is almost
consistent with that by geometric optics.

##  V Conclusion

The study of BH space-times is a hot topic among the research community in GR
and MTG as well. Significant information on gravitation, thermodynamics and
quantum effects in curved space-time can be revealed through BHs. In the past
few years, several progress in theoretical and observational explorations of
BHs have been witnessed. These findings opened a new era in gravitational
physics triggered by the leap in quantity, quality, and variety of
observational data from different probes. Recent collaborations such as the
EHT have shown that it is feasible to carry out observations of super-massive
BHs with near horizon scale resolution. This allows us to directly study the
gravitational lensing effect at work in the strong gravity regime, where we
expect the peculiarities of different space-times to become apparent in the
images. To this end, strong gravitational lensing is an important
observational tool for probing the space-time geometries around massive
objects, where the optical appearance of the photon sphere provides a concrete
description of space-time configurations. Motivated by this, in the present
study, we have studied the Einstein ring structure for the lensed response of
the complex scalar field as a probe wave propagating in the charged Rastall
AdS BH in the framework of AdS/CFT correspondence. In general, real quantum
materials are engineered at a finite chemical potential. So, for a better
understanding of this, we need to consider the bulk electromagnetic field and
hence, the BH should be charged. Figure 2 (left panel) displays that the
temperature gradually decays with respect to the inverse of the horizon
$v_{h}$. On the other hand, it gradually increases with the increasing of the
RG parameter $\lambda$ see the right panel of Fig. 2. These results imply that
the Hawking temperature closely depends on the parameter $\lambda$ as well as
the BH horizon $v_{h}$. This distinct feature of the horizon temperature may
be used as a tool to differentiate the charged Rastall BH solutions from other
studies R75 ; R76 ; R77 .

The nature of the absolute amplitude of the total response function for
different values of the involved parameters is illustrated in Figs. 3 to 5,
which shows the corresponding behavior of the wave oscillations according to
the variation of state parameters. The left plot of Fig. 3 shows that when
$\lambda$ has smaller values, the corresponding amplitude increases and vice
versa. The right plot of Fig. 3 depicts that when the wave source $\omega$ has
smaller values the time period of the scalar wave increases, and it gradually
decreases with the increasing of $\omega$. The effects of bulk electromagnetic
field $e$ and finite chemical potential $\mu$ on the amplitude of the total
response function are presented in Fig. 4. This figure indicates that when
both $e$ and $\mu$ approach smaller values, the corresponding amplitude of the
response function gradually decreases. In addition, we also plot the profile
of the total response function for the different values of the horizon
temperature $T$ in Fig. 5, which shows that when $T=0.458$, the amplitude has
the maximum value, and it sharply decreases with the increasing of the
temperature $T$. For a better understanding of the considering BH space-time,
we further construct the virtual optical system in a three-dimensional flat
space with a thin convex lens and a hemispherical screen as shown in Fig. 6.
We copy the response function to the virtual optical system as the incident
wave on the lens and build its image on the screen. This physical assumption
provides a more realistic scenario to understand the interesting features of
the resulting Einstein ring images on the screen in the presence of the bulk
electromagnetic field. Based on this setup, we illustrate the optical
appearance of the resulting Einstein ring for different values of parameter
$\lambda$ and the observational angle $\theta_{\text{obs}}$ in Fig. 7.

We observe that when $\theta_{\text{obs}}=0^{o}$, the image of the AdS BH
appears as a bright ring and there is a series of concentric stripes, which
corresponds to the diffraction of the total response function. We observe that
this brightest ring changes into light arcs or a dim light spot when
$\theta_{\text{obs}}$ approaches $90^{o}$. From top to bottom, as the value of
parameter $\lambda$ grows, such as when $\lambda=0.16$ and
$\theta_{\text{obs}}=0^{o}$ (see Fig. 7 (m)), one can see that the series of
concentric strip patterns are more obvious and the radius of the brightest
ring gradually moves towards the centre of the screen. Moreover, when
$\theta_{\text{obs}}$ approaches $90^{o}$ (see Fig. 7 (p)), there are two dim
light spots in the screen, lying away from the boundary. To analyze the glance
as well as the radius of the Einstein ring image, we further plot the
brightness profiles of the lensed response on the screen under the numerous
values of the RG parameter $\lambda$, as depicted in Fig. 8. It is observed
that as the parameter $\lambda$ grows, the radius of the Einstein ring is
gradually shifted towards the center of the screen. However, the corresponding
brightness of the resulting Einstein ring is slightly changed with respect to
the variation of parameter $\lambda$. These results are also consistent with
Fig. 7, when $\theta_{\text{obs}}=0^{o}$. In Fig. 9, we analyzed the influence
of the scalar wave $\omega$ on the obtained images of AdS BH for the fixed
value of the convex lens such as $\eta=0.05$ and $d=0.6$. Here, one can find
that as the values of $\omega$ decrease from left to right, the corresponding
bands of the resulting rings are more obvious. Moreover, the gap between the
bands also increases with decreasing $\omega$. The corresponding brightness
profiles of Fig. 9 are also plotted in Fig. 10 under the same set of
parameters. These profiles show that with decreasing $\omega$, the gap between
the curves and brightness of the rings, both increased.

We further study the effect of horizon temperature $T$ on Einstein ring images
as presented in Fig. 11. Our results showed that when $T$ has smaller values
such as $T=0.0249$, the resulting ring lies close to the center. As the values
of temperature $T$ grow, the corresponding rings are gradually shifted
outwards, which means that the radius of corresponding rings is increased
appreciably. This effect can also be seen in Fig. 12, where the positions of
the peak curves are significantly changed and shifted towards the boundary.
Next, we observed the Einstein ring images when $\theta_{\text{obs}}=0^{o}$ at
different chemical potentials $\mu$, displayed in Fig. 13, where we observed
that the variation of the image with respect to chemical potential $\mu$. When
$\mu=0.05$ and $0.35$, the Einstein ring does not change. However, when the
value of $\mu$ increases to $\mu=0.65$, the corresponding ring is slightly
shifted inwards and this effect can be seen clearly when $\mu=0.95$, the
resulting ring is closer to the center and its optical appearance is a very
bright light spot in the center of the screen. This phenomenon is also shown
in Fig. 14, where at smaller values of $\mu$, the positions of the peak curves
are relatively the same. But at the large values of $\mu$, the positions of
these curves are sharply shifted inwards, see Fig. 14 (c) and 14 (d).
Nevertheless, from both Figs. 13 and 14, as the values of $\mu$ are grew, the
corresponding ring’s radius is also decreased.

All the derived results are based on the holographic framework. So, to verify
the correctness of these results, we obtained the radius of the photon ring
and the corresponding ingoing angle through the geometric optics technique. In
Figs. 17 and 18, we plotted the radius of BH horizon $r_{h}$, location of the
photon orbit $r_{c}$ and the radius of the Einstein ring (see black curve) for
different values of Rastall parameter $\lambda$. However, in both figures, the
discrete red points denote the Einstein ring radius, which is obtained through
wave optics. From Fig 17 (a), one can see that when $\lambda=0.01$, the
Einstein ring radius is appreciably increased at the initial values of the
horizon $r_{h}$ and when $r_{h}\sim 0.3$, the black curve is changed into a
straight line. As the value of $\lambda$ further increased, we observed that
the Einstein ring radius is slightly increased at very smaller values of the
horizon $r_{h}$ and kept unchanged throughout the horizon $r_{h}$. In all
cases, we noticed that the discrete red points always lie on the black curve
or its surroundings, indicating that our results are consistent with geometric
optics. Further, from Fig. 18, we see that the Einstein ring radius gradually
decreases with the increasing values of $\mu$. When $\lambda=0.01$, one can
find that the discrete red points almost lie in the surrounding of the black
curve. But as the values of $\lambda$ increase, these red points are shifted
away or lie exactly on the black curve at a value of $\mu$. For example, in
Fig.18 (c), at the smaller values of $\mu$, the red points are gradually
shifted away from the black curve and lie exactly on it when $\mu\sim 0.75$.
And consequently, as expected, the Einstein ring radius obtained by wave
optics is consistent with that of geometric optics in all cases.

By summarizing, the current analysis of the holographic Einstein ring of
charged AdS BH in the presence of a bulk electromagnetic field has offered
significant and insightful observations regarding the behavior of these types
of systems. Further, the holographic images can be used as an effective tool
to distinguish different types of BH solutions for the fixed wave source and
optical system. In this regard, it is also very interesting to further
investigate such a holographic framework in other MTG as well. Finally, we
thank EHT collaboration for publishing the first image of a BH and beautifully
explaining the physical quantities of the BH space-time structure, prompting
us to further investigate the associated phenomenological consequences more
deeply.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grants Nos. 11675140, 11705005, 12375043), 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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