11institutetext:  Université Côte d’Azur, CNRS, LJAD, France
11email<EMAIL_ADDRESS>
# Project proposal: A modular reinforcement learning based automated theorem
prover ††thanks: This work has been supported by the French government,
through the 3IA Côte d’Azur Investments in the Future project managed by the
National Research Agency (ANR) with the reference number ANR-19-P3IA-0002

Boris Shminke 11

###### Abstract

We propose to build a reinforcement learning prover of independent components:
a deductive system (an environment), the proof state representation (how an
agent sees the environment), and an agent training algorithm. To that purpose,
we contribute an additional Vampire-based environment to gym-saturation
package of OpenAI Gym environments for saturation provers. We demonstrate a
prototype of using gym-saturation together with a popular reinforcement
learning framework (Ray RLlib). Finally, we discuss our plans for completing
this work in progress to a competitive automated theorem prover.

##  0.1 Introduction and related work

Reinforcement learning (RL) is applied widely in the automated reasoning
domain. There are RL-related (including iterating supervised learning
algorithms without applying recent RL advances) projects for interactive
theorem provers (ITPs) (e.g. HOList [2] for HOL Light [8], ASTactic [33] for
Coq [31], or TacticZero [32] for HOL4 [27]) as well as for automated theorem
provers (ATPs) (e.g. Deepire [28] for Vampire [16], ENIGMA [25] for E [12], or
rlCoP [13] for leanCoP [18]). Despite the variety of solutions and ideas, we
are not aware of cases of significant code reuse between such projects.

We envision a prover capable of learning from its experience and composed of
pluggable building blocks. We hope that such architecture could promote faster
experimentation and easier flow of ideas between different projects for
everyone’s progress. For an RL-based prover, we identify at least three types
of modules. They are a deductive system (an environment), a proof state
representation (how an agent sees it), and an agent training algorithm.

When choosing whether to learn to guide an ITP or an ATP, we prefer the latter
since ATPs can be relatively easy compared as black boxes [30] in contrast to
RL guided ITPs, which often come with their distinctive benchmarks.

Among ATPs, one can consider saturation provers less suitable for the RL
(e.g., see design considerations from [22]), but several existing projects
(like ENIGMA, Deepire or TRAIL [5]) show encouraging results. Keeping that in
mind, we decided to concentrate on guiding clause selection in the saturation
algorithm by RL.

Inspired by HOList, CoqGym (from ASTactic) and lean-gym [20], we have created
gym-saturation [26] — an OpenAI Gym [4] environment for training RL agents to
prove theorems in clausal normal form (CNF) of the Thousands of Problems for
Theorem Provers (TPTP) library [29] language.

##  0.2 Recent work in progress

Contemporary RL training algorithms are notorious for the number of details
that can differ from one implementation to another [9]. To eliminate the risk
of abandoning an RL algorithm as unsuitable for guiding an ATP only because of
flaws in our implementation of it, we plan to rely on existing RL frameworks
containing tested implementations of well-known baselines. As a starting
point, we have chosen Ray RLlib [17] as a library claiming both deep learning
(DL) framework independence and extendability. Similar solutions like
Tensorflow Agents [7] or Catalyst.RL [15] tend to support only one DL
framework, which we wanted to avoid for greater generality.

In contrast to CoqGym and others, gym-saturation is not only a ‘gym’ in some
general sense, but it implements the standard OpenAI Gym API. It makes it
easier to integrate with libraries like Ray RLlib. We
contribute111https://github.com/inpefess/basic-rl-prover a prototype of such
integration. Even together with some domain related patches, the prototype
remains a lightweight collection of wrappers around standard RLlib classes,
taking only around 300 lines of Python code.

Since we postulated interchangeability of modules, we added a Vampire-based
environment to gym-saturation (see the project
page222https://pypi.org/project/gym-saturation/ for more details) in addition
to the already existing naïve implementation of a saturation loop. Despite a
different backend, one can plug a new environment into the prover prototype
without additional edits of RL related code.

### Similar systems for connection tableaux

There exists a FLoP (Finding Longer Proofs) project [34] which implements a
ProofEnv OpenAI Gym environment for a connection tableaux calculus, which can
guide two different provers (leanCoP and its OCaml reimplementation fCoP
[14]). FLoP shares many architectural features with our work, and we plan to
test its approaches in saturation provers setting.

##  0.3 Prototype implementation details

Since this research is still in an early stage, we don’t report any conclusive
results of its performance, only describing the architecture. A prototype
prover has two main parts: gym-saturation as an environment and a patched DQN
[10] implementation from Ray RLlib training an agent. An episode starts with
the environment reset. On environment reset, a random TPTP problem from a
training subset is loaded, transformed to the CNF, and becomes a proof state.
After an agent makes an action (selects a clause), the episode can stop for
three reasons: a given clause is empty (refutation proof found, the reward is
1.0; in other cases, it’s 0.0), we reach the step limit (a soft timeout), we
reach the maximal number of clauses in the proof state (a soft memory limit).
Only episodes with a positive final reward go to the memory buffer. Before
storing in the buffer, the reward is spread evenly between the clauses from
the proof (others remain zero). A memory buffer can contain the same proof for
the same problem twice or different proofs (maybe of different lengths) for
one problem. A training batch can contain steps from different episodes (and
thus different initial environment states). We sample the memory buffer with
higher weights for more recent episodes.

##  0.4 Future plans and discussion

In the prototype, we represent each clause in a proof state only by its size
and order number, applying a logistic regression as a Q-value function. We
will need an elaborate feature extraction procedure to complete this
oversimplified model to a competitive ATP. We plan to use graph neural
networks similar to those used for lazyCoP [23] and then compare and combine
them with the graph representation of clause lineage pioneered by Deepire. We
also plan to test training algorithm interchangeability by using IMPALA [6]
and Ape-X [11] in addition to DQN.

A finished project will have to address many different problems. Here we list
several obvious ones.

### Delayed reward

One of the well-known peculiarities of an ATP is the fact that a reward can be
assigned only after proof is found, which can take a large number of steps in
an RL episode. To make an agent learn to discern good steps, one has to spread
the final reward to all the steps in a finished trajectory. A typical solution
is to post-process a trajectory by assigning positive advantage values only to
the steps encountered in a proof, and negative (or zero) values to all the
rest. Here one can argue in favour of both higher values for longer proofs
(since the ability to produce longer proofs is desirable) and higher values
for shorter ones (since more concise proofs for simpler problems are
preferable to verbose ones which in turn could help to find longer proofs
otherwise unreachable because of time and memory constraints). A contrarian
approach is to assign positive advantage values for all the steps in a
trajectory on which proof was found, and non-positive to all the steps from
trajectories finished because of the resource limitations. Such an approach
works well, for example, in the Atari Pong game, where it’s practically
impossible to judge which action led to a goal.

### Sparse positive reward

Another well-known problem of applying RL to ATPs is related to the fact that
even sub-human performance still seems out of reach. The majority of proof
attempts finish without proof found. Discarding failed episodes seems too
wasteful, although obvious as a first attempt. An opposite solution (assigning
non-positive advantage values to all failed episodes) makes the training
dataset too imbalanced. One possible solution to this is to use replay buffers
and sample from them balanced train batches. This explains why we decided not
to neglect DQN despite its known limitations when compared to on-policy
algorithms like PPO [24].

### Multiple proofs

Many problems have multiple possible proofs, equivalent in some sense or not.
An agent will have to decide which proofs are preferable to replicate. Again,
replay buffers can be used for that. Ranking proofs can be based on their
length or other important properties (reuse of previously proved lemmata,
using only a selected subset of deduction rules or tactics etc)

### High environment’s inhomogeneity

Some problems are inherently harder than others and can belong to areas of
mathematics not connected in a given formalization. Curriculum learning [3] or
at least limiting the training scope to a reasonable subset of the TPTP
library will be needed.

### State representation

Usually, contemporary RL algorithms expect the observed state to have a form
of a vector. Representing logic formulae as such is an active domain of
research. We plan to try both logic-specific approaches like [21] and general
abstract syntax tree encoding models like code2vec [1] or ast2vec [19].

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# Holographic Einstein Rings of Black Holes in Scalar-Tensor-Vector Gravity

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  M. Israr Aslam<EMAIL_ADDRESS>Department
of Mathematics, COMSATS University Islamabad, Lahore-Campus, Lahore $54000$
Pakistan.  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

With the help of AdS/CFT correspondence, we analyze the holographic Einstein
images via the response function of the complex scalar field as a probe wave
on the AdS Schwarzschild scalar-tensor-vector gravity (STVG) black hole (BH).
We find that the amplitude of the response function $|\langle O\rangle|$
decreases with the increasing values of the coupling parameter $\alpha$, while
it increases with the decreasing values of temperature $T$. The frequency
$\omega$ of the wave source also plays a significant role in wave periods, as
we increase the values of $\omega$, we find a decrease in periods of waves,
which means that the total response function closely depends on the wave
source. Further, we investigate the optical appearance of the holographic
images of the BH in bulk. We found that the holographic ring always appears
with the concentric stripe surrounded when the observer is located at the
north pole, and an extremely bright ring appears when the observer is at the
position of the photon sphere of the BH. With the change of the observational
angle, this ring will change into a luminosity-deformed ring or a bright light
spot. The corresponding brightness profiles show that the luminosity of the
ring decreases, and the shadow radius increases with increasing values of
$\alpha$. The relation between temperature $T$ and the inverse of the horizon
$h_{e}$ is discussed, which shows the smaller values at the beginning of the
horizon $h_{e}$, and then increases as the horizon radius increases. This
effect can be used to distinguish the STVG BH solution from other BH
solutions. Moreover, these significant features are also reflected in the
Einstein ring and the corresponding brightness profiles. In addition, we
compare the results obtained by wave optics and geometric optics, which are
aligned well, implying that the holographic scheme adopted in this paper is
valid.

Keywords: Scalar-tensor-vector Gravity; Shadows; AdS/CFT Correspondence.

##  I Introduction

During the last decades, significant progress has been made in dealing with
the strongly coupled systems owing to the Anti-de-Sitter (AdS)/conformal field
theory (CFT) correspondence sv1 ; sv2 ; sv3 , which provides a concrete
relation between strongly coupled quantum field theories and dynamics of a
classical gravitational theory in the bulk of gravitational space-time sv4 .
This theory relates gravity in a ($d+1$)-dimensional AdS space-time to a
strongly coupled $d$-dimensional quantum field theory living on its boundary.
In this perspective, AdS/CFT correspondence is successfully applied in
different eras such as the relation of the strong coupling dynamics of quantum
chromodynamics and the electro-weak theories, physics of black holes (BHs) and
quantum gravity, relativistic hydrodynamics, the study of condensed matter
physics, superfluidity and superconductivity sv5 ; sv6 ; sv7 ; sv8 ; sv9 ;
sv10 ; sv11 ; sv12 ; sv13 ; sv14 ; sv15 . Further, quantum information uses
AdS/CFT as a powerful tool and provides us with an interesting result on
multi-body systems, like holographic entanglement entropy sv16 , mutual
information sv17 , entanglement of purification sv18  and holographic
complexity sv19 ; sv20  etc.

According to AdS/CFT correspondence, we know that quantum field theories or
quantum materials possess their gravity dual. Nevertheless, no one knows about
the reality of this material, especially which has its gravitational dual in
our universe. However, the characteristics of such a material make it possible
to analyze with classical or even quantum gravity in tabletop experiments.
Hence, it is necessary to investigate the nature of such material and whether
it has a gravitational dual. Recently, based on AdS/CFT correspondence, the
authors in sv21  found sources on the boundary that generate one wave packet
drawing a null geodesic inside the bulk. They analyzed the behavior of such a
wave packet inside the bulk and built a technique that is helpful in
identifying the holographic materials. In sv22 , authors proposed a method to
create a star orbiting in an asymptotically AdS space-time and then they
investigated the angular position of the star with the help of lensed response
function in the framework of AdS/CFT correspondence. Therefore, the AdS/CFT
correspondence strongly supports a better understanding of various physical
topics.

On the other hand, we have witnessed the beginning of a new era in BH
gravitational physics triggered by the leap in the quantity, quality, and
variety of observational data from different probes. A BH is a particular
astrophysical object of our Universe that possesses a powerful gravitational
field, resulting in a significantly curved space-time. A BH is a solution of
Einstein’s field equations, describing the regions of space-time geometry that
have undergone gravitational collapse. In this scenario, in 2019, the Event
Horizon Telescope (EHT) provided the first strong evidence of the existence of
BHs in nature and released the first ultra-high angular resolution image of
the super-massive BH M$87$, which put a new spirit to further analyze the BH
physics sv23 ; sv24 . Later, in $2021$, EHT measured the polarization of M$87$
when the magnetic fields and plasma properties are taken into account, which
provides a piece of significant information about the launching of energetic
jets from its core sv25 ; sv26 . As a milestone in the history of BH, these
findings are scientifically important because these images reveal the
structural information about BHs such as the accretion process of various
matter, radiation mechanism, and some other relevant astrophysical
consequences as well as provide strong evidence for general relativity (GR).

A prominent feature of this image is that a dark interior is surrounded by a
bright ring-shaped accretion disk, in which the central dark region and the
bright ring are so-called the shadow and the photon ring of the BH,
respectively. The light trajectories emitted from the accretion material are
deflected due to the strong gravitational field near the BH, which thus makes
it possible to analyze the geometry of BH under different accretion flows. For
a distant observer, the image of the BH shadow is projected onto the local sky
of the observer due to gravitationally lensed, which is observed by the strong
gravitational field near the BH. Hence, the study of shadow and gravitational
lensing of BH are feasible methods to investigate the interesting features of
the gravitational field sv27 ; sv28 . Since, due to different interesting
scenarios of matter accretions around the BHs, the study of BH shadows and
their optical appearance has reached a peak position among the scientific
community. Gralla et al. sv29  analyzed the shadow and photon ring of
Schwarzschild BH and found that the total luminosity of BH shadow depends on
the location and form of accretion material around the BH. By establishing
different accretion flow models, the study of BHs shadows has become a subject
of great interest, for instance, see Refs. sv30 ; sv31 ; sv32 ; sv33 ; sv34 ;
sv35 ; sv36 .

Although the properties of BH shadow and its relevant dynamics have been
extensively studied in the literature, which gives us concrete information
about BH physics, there are still many mysteries to uncover through some more
realistic models. In this perspective, Hashimoto et al. sv37 ; sv38  proposed
a method to construct the holographic images of the BH in the bulk, when the
scalar wave emitted by the source at the AdS boundary, enters the bulk field
and propagates in the bulk space-time. Particularly, they observed the
Einstein ring on the holographic images and the size of the ring is consistent
with the size of the BH photon sphere, which is calculated through geometrical
optics. Liu et al. sv39  investigate the Einstein ring structure for the
lensed response of the complex scalar field as a probe wave on the charged AdS
BH in the framework of AdS/CFT correspondence. They observed that such an
Einstein ring emerged at high frequencies and could be well captured by the
photon sphere, away from the BH horizon, in the geometric optics
approximation, and this distinct feature supports the existence of gravity
dual. Further, in the framework of AdS/CFT correspondence, the authors in sv40
; sv41 ; sv42  also investigated some distinct features of the Einstein ring
in modified theories of gravity. The photon ring of the asymptotically AdS BH,
dual to a superconductor, is observed on a two-dimensional sphere sv43 , where
the influence of the charge scalar condensate on the image is investigated and
concluded that the bulk images depicted the discontinuous change in the size
of the photon ring.

Einstein’s theory of GR is one of the most successful and well-established
gravitational theories in modern physics, but there are many theoretical and
observational shortcomings attached to it, such as the current behavior of our
Universe sv44 , rotation curves of galaxies sv45  and some tension in
cosmological data sv46 . In GR, we often encounter the presence of
singularities and a lack of self-consistent theory of quantum gravity. In
addition, from the observational point of view, GR does not address the
galactic, extra-galactic, and cosmic dynamics without dark matter and dark
energy sv47 ; sv48 ; sv49 . In order to resolve the problem of exotic matter,
the research community divided the GR into two groups, either dark matter
exists or alternatively modified Einstein’s field equations. Therefore,
several modified theories have been developed as the generalization of GR to
resolve such problems sv50 ; sv51 . In sv52 , authors calculated the velocity
of galaxies in clusters and found that the gravitational mass is greater than
the luminous matter. In this scenario, we require modified theories, where the
gravitational part is non-minimally coupled with the dynamical vector fields
and hence, the dark matter/energy problem is resolved through understandable
dynamical vector fields.

In literature, some promising modified gravitational theories have been
proposed from time to time, such as the Gauss-Bonnet theory sv53 , $f(R)$
theory (in which $R$ is the Ricci scalar) sv54  and its extended versions with
minimal/non-minimal interactions among various fields like $f(R,T)$ (where $T$
is the trace of energy-momentum tensor) sv55 , scalar-tensor theories and its
generalized versions sv56  and teleparallel theory of gravity sv57  etc. One
of the well-known modified theories of gravity in the literature is the so-
called STVG theory proposed by Moffat sv58 , in which the principle of action
is used and calculated through the metric tensor, three scalar fields, and a
massive vector field. In sv59 , Moffat gave the BH solution in the context of
STVG. This theory explains the cosmic phenomena without introducing the dark
matter sv60 , successfully fits the rotation curves of galaxies sv61 , dynamic
of galactic clusters sv62 , and the cosmic microwave background (CMB) acoustic
power spectrum sv63 . In sv64 , authors investigated the quasinormal modes of
the generalized ABG (Ayón-Beato-García) BHs in STVG gravity. In the framework
of the STVG model, the thermodynamics phase transition of AdS Schwarzschild BH
is investigated in sv65 . Li et al. sv66  studied the thermal fluctuation,
deflection angle, and greybody factor of the high-dimensional Schwarzschild BH
in STVG. Cai and Miao sv67  calculated the high-dimensional static spherically
symmetric Schwarzschild BH in STVG (which is a high-dimensional modification
of STVG) and studied the quasinormal modes of a massless scalar field and BH
shadow. Recently, the shadow of regular BH and its energy emission rate has
been analyzed in STVG theory sv68 .

In the context of AdS/CFT correspondence, in the present study, we analyze the
holographic Einstein images of an AdS Schwarzschild STVG BH solutions in the
bulk from a particular response function on one side of the AdS boundary, in
which the response function is produced by the source, which lies far away
from the response function on the other side of the AdS boundary. We employ an
oscillatory Gaussian wave source $\mathcal{J}_{\mathcal{O}}$, on one side of
the AdS boundary, a bulk scalar wave generated by the source, is injected into
the bulk from the AdS boundary. The scalar wave propagates inside BH space-
time and reaches the other side of the AdS boundary, corresponding response
function will be generated such as $\langle O\rangle$, which gives the
information about the bulk structure of the BH space-time sv37 ; sv38 . We
depicted this setup in Fig. 1.

Figure 1: The schematic diagram of the working principle for imaging a dual
BH.

Considering a special optical method, we calculate a formula, which converts
the extracted response function $\langle O\rangle$, to the image of the dual
BH, 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
and a probe massless bulk scalar field in the space-time.

The layout of our present study is settled as follows. In section II, we
define the summary of the AdS Schwarzschild STVG BH metric, and the
holographic setup for the given response function is defined in section III.
In section IV, we introduce the optical image formation in wave optics and
discuss the null geodesics for the AdS Schwarzschild STVG BH on the basis of
geometrical optics. We also provide a comparison of the Einstein radius of the
photon sphere, which is obtained from the geometric optics with that of the
holographic ring image constructed from the response function in the wave
optics in the same section. The last section contains the outlook of our work.

##  II Basic Formulation of AdS Schwarzschild STVG BH

Starting from the general action of STVG theory sv58 , which is given below

$\mathcal{I}=\mathcal{I}_{Gr}+\mathcal{I}_{\Phi}+\mathcal{I}_{S}+\mathcal{I}_{M},$
(1)

where

$\displaystyle\mathcal{I}_{Gr}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int
d^{4}{x\sqrt{-g}}\bigg{[}\frac{1}{G}\bigg{(}R+2\Lambda\bigg{)}\bigg{]},$ (2)
$\displaystyle\mathcal{I}_{\Phi}$ $\displaystyle=$ $\displaystyle-\int
d^{4}{x\sqrt{-g}}\bigg{[}\mathcal{T}\bigg{(}\frac{1}{4}B^{\xi\zeta}B_{\xi\zeta}+V(\Phi)\bigg{)}\bigg{]},$
(3) $\displaystyle\mathcal{I}_{S}$ $\displaystyle=$ $\displaystyle\int
d^{4}{x\sqrt{-g}}\bigg{[}\frac{1}{G^{3}}\bigg{(}\frac{1}{2}g^{\xi\zeta}\nabla_{\xi}G\nabla_{\zeta}G-V(G)\bigg{)}\bigg{]}+\int
d^{4}{x\sqrt{-g}}\bigg{[}\frac{1}{G}\bigg{(}\frac{1}{2}g^{\xi\zeta}\nabla_{\xi}\mathcal{T}\nabla_{\zeta}\mathcal{T}-V(\mathcal{T})\bigg{)}\bigg{]}$
$\displaystyle+$ $\displaystyle\int
d^{4}{x\sqrt{-g}}\bigg{[}\frac{1}{\xi^{2}G}\bigg{(}\frac{1}{2}g^{\xi\zeta}\nabla_{\xi}\xi\nabla_{\zeta}\xi-V(\xi)\bigg{)}\bigg{]}.$

in which $\mathcal{I}_{Gr}$ is the gravity part, which is defined by the
Einstein Hilbert action with additional cosmological parameter $\Lambda$,
$\mathcal{I}_{\Phi}$ is the modified massive vector field of the action,
$\mathcal{I}_{S}$ represents the scalar part of the action and
$\mathcal{I}_{M}$ is the matter action. Further, $g$ denotes the determinant
of the metric tensor, $R$ is the Ricci scalar,
$V(\Phi)=\frac{1}{2}\xi^{2}\Phi^{\xi}\Phi_{\xi}$ is the self-interacting
potential, $\Phi_{\xi}$ refers the massive vector field with mass parameter
$\xi$ and $B_{\xi\zeta}=\partial_{\xi}\Phi_{\zeta}-\partial_{\zeta}\Phi_{\xi}$
is the anti-symmetric linear part and $\mathcal{T}$ is a dimensionless scalar
field with potential $V(\mathcal{T})$. $G(x)$ and $\xi(x)$ are the scalar
fields and $V(G)$ and $V(\xi)$ are the corresponding potentials, respectively.
$\nabla_{\xi}$ is the covariant derivative for a metric tensor $g_{\xi\zeta}$.
Since, the influence of the vector field $\Phi_{\xi}$ with mass $\xi$ becomes
prominent just at kiloparsec distances from the gravitational sources, it can
be neglected when solving the field equations for compact objects like BHs
sv65 ; sv68 .

Further, at the slowly varying regime, one can consider Newton’s gravity
coupling parameter as $G=G_{N}(1+\alpha)$, in which $G_{N}$ is Newton’s
gravitational constant and $\alpha$ is a dimensionless parameter, which comes
from the alternative participation of the above defined action of the scalar
part $G(x)$ at the slowly varying regime. In addition, GR can be recovered
when $\alpha=0$, which is usually known as the gravitational charge, so we can
regard deviation of the STVG theory with respect to GR given by parameter
$\alpha$. Recently, the authors in sv69  analyzed the influence of the
coupling parameter $\alpha$ on the stabilization of orbits of magnetized
particles moving around the Schwarzschild BH in STVG theory. The thermodynamic
behavior of the Schwarzschild BH in the framework of STVG theory is discussed
in sv70 , where BH heat capacity depicted the change of the sign at critical
mass with the presence of Hawking radiation.

Recently, the BH solutions with a non-vanishing cosmological constant
$\Lambda$ reached a peak position among the researchers due to two interesting
features such as they play a significant role in the phenomenology of the
AdS/CFT correspondence sv1 ; sv3 ; sv71  which associates the cosmological
constant with the rank of the gauge group and the observational data shows
that the universe may have a small positive value of $\Lambda$ sv72 . In fact,
many authors bring our understanding to investigating the interesting features
of AdS BHs in the context of different modified theories. In sv73 , the author
analyzed that the cosmological parameter raises the efficiency of the Penrose
process in the AdS BH for the case of $\Lambda=0$.

The line element of the AdS Schwarzschild BH in STVG theory is given as sv65

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$ (5)

where

$f(r)=1-\frac{2(1+\alpha)M}{r}+\frac{\alpha(1+\alpha)M^{2}}{r^{2}}+\frac{r^{2}}{l^{2}},$
(6)

in which $G_{N}=1$, $M$ is the mass of the BH, $l$ represents the radius of
the AdS space, which is associated with the cosmological parameter as
$l^{2}=-\frac{3}{\Lambda}$ for $\Lambda<0$ and
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. Since negative $\Lambda$
in the bulk is generally related to the degrees of freedom of the dual CFT on
the boundary. For simplicity, we hereafter set $l=1$. Next, we will analyze
the Einstein ring structure of the above metric in the context of AdS/CFT.

##  III The Holographic Framework of AdS Schwarzschild STVG BH

Associated with the above metric, we define the holographic mechanism of AdS
Schwarzschild STVG BH in the framework of AdS/CFT correspondence. For this, we
set $h=1/r$ and $f(r)=h^{-2}f(h)$. In this perspective, we define the metric
(5) in a new coordinate as following

$ds^{2}=\frac{1}{h^{2}}[-f(h)dt^{2}+\frac{dh^{2}}{f(h)}+d\Omega^{2}].$ (7)

For a massless scalar field, we define the Klein-Gordon equation, which is
helpful in determining the dynamics, which are defined as sv38

$\square\Psi=0.$ (8)

In order to solve the above equation, we define the ongoing Eddington
coordinates as sv39

$\displaystyle v=t+h_{\star}=t-\int\frac{1}{f(h)}dh,$ (9)

and then the metric function can be further expressed as

$ds^{2}=\frac{1}{h^{2}}[-f(h)dv^{2}-2dhdv+d\Omega^{2}].$ (10)

The asymptotic behavior of the scalar field close to the AdS boundary becomes

$\Psi(v,h,\theta,\phi)=\mathcal{J}_{\mathcal{O}}(v,\theta,\phi)+h\partial_{v}\mathcal{J}_{\mathcal{O}}(v,\theta,\phi)+\frac{1}{2}h^{2}D^{2}\mathcal{J}_{\mathcal{O}}(v,\theta,\phi)+\langle
O\rangle h^{3}+\mathcal{O}(h^{4}),$ (11)

in which $D^{2}$ represents the scalar Laplacian on unit $\mathcal{S}^{2}$.
From the AdS/CFT point of view, $\mathcal{J}_{\mathcal{O}}$ and $\langle
O\rangle$ is interpreted as the external source and the corresponding expected
response function in dual CFT, respectively sv74 . We consider that an axis-
symmetric and monochromatic oscillating Gaussian wave packet source, which is
located at the south pole of the AdS boundary, i.e., $\theta=\pi$. In this
regard, we have

$\displaystyle\mathcal{J}_{\mathcal{O}}(v,\theta)=e^{-i\omega
v}\exp\big{[}\frac{-(\pi-\theta)^{2}}{2\rho^{2}}\big{]}\frac{1}{2\pi\rho^{2}}=e^{-i\omega
v}\sum_{l=0}^{\infty}Y_{l0}(\theta)N_{lo},$ (12)

in which $\rho$, $Y_{l0}(\theta)$, and $N_{lo}$ interpreted the width of the
Gaussian wave source, the spherical harmonics function and its coefficient,
respectively. Due to the smallest value of the Gaussian tail, we only consider
the case of $\rho<<\pi$. Further, the value of $N_{l0}$ can be calculated as

$N_{l0}=(-1)^{l}\bigg{(}\frac{(l+1/2)}{2\pi}\bigg{)}^{\frac{1}{2}}\exp\bigg{[}-\frac{(l+1/2)^{2}\rho^{2}}{2}\bigg{]}.$
(13)

One can further decomposed the scalar field $\Psi(v,h,\theta,\phi)$ as

$\Psi(v,h,\theta,\phi)=e^{-i\omega
v}\sum_{l=0}^{\infty}\sum_{n=-l}^{l}Y_{l0}H_{l}(h)N_{l0},$ (14)

and the corresponding response function can be written as

$\langle O\rangle=e^{-i\omega v}\sum_{l=0}^{\infty}Y_{l0}\langle
O\rangle_{l}N_{lo}(\theta).$ (15)

From Eq. (14), we obtain $H_{l}$, which satisfies the equation of motion

$h^{2}f(h)H^{\prime\prime}_{l}+[h^{2}f^{\prime}(h)-2hf(h)+2i\omega
h^{2}]H^{\prime}_{l}+[-2i\omega h-l(l+1)h^{2}]H_{l}=0,$ (16)

and its asymptotic behavior close to the AdS boundary can be written as

$\lim\limits_{h\to 0}H_{l}=1-i\omega h+\frac{h^{2}}{2}(-l(1+l))+\langle
O\rangle_{l}h^{3}+\mathcal{O}(h^{4}).$ (17)

Clearly, Eq. (16) has two boundary conditions for function $H_{l}$, i.e.,
horizon boundary condition for which $f(h)$ in Eq. (16) vanishes, and the AdS
boundary condition where $H_{l}(0)=1$. Our main task is to solve Eq. (16)
according to these boundary conditions and obtain the function $H_{l}$, with
the help of the pseudo-spectral method sv37 ; sv38 . In this case, we can
extract $\langle O\rangle_{l}$ based on Eq. (17) . Using the extracted
$\langle O\rangle_{l}$ and Eq. (15), one can obtain the value of the total
response function as we depicted the profiles of the total response function
in Figs. 2 to 4. In these profiles the physical interference pattern arises
from the scalar wave propagates in the bulk field and is diffracted by the BH.

Figure 2: The amplitude of $|\langle O\rangle|$ for different values of
$\alpha$ with $h_{e}=1$ and $\omega=80$.

From Fig. 2, clearly one can see that the amplitude of the response function
shows the maximum values of the propagating wave for $\alpha=0.01$ and then
decreases with the increasing values of $\alpha$. Hence, the coupling
parameter of STVG theory significantly changes the behavior of the propagating
waves in the bulk. We plotted Fig. 3 for different values of $\omega$, which
is interpreted as the period of the oscillations of scalar waves, produced by
the Gaussian source. Particularly, the period of the scalar wave is maximum
when $\omega=30$, and it gradually decreases, as we increase the values of
$\omega$. Further, Fig. 4 shows the amplitude of the response function for
different values of Hawking temperature $T$, which is associated with the
event horizon for a fixed $\alpha$. For instance, when $T=0.351$, then
$h_{e}=0.8$, the amplitude of the total response function reaches the peak
position. And when $T=0.568$, which corresponds to $h_{e}=0.4$, the profile of
the amplitude does not significantly oscillate and shows a narrow ramp in the
middle, which is hard to see, as compared to the previous one. Hence, the
different values of the event horizon significantly change the behavior of the
scalar waves in the bulk field, implying the temperature of the BH plays an
important role in the variation of propagating waves. From the above
discussion, we concluded that the optical appearance of the total response
function entirely depends on the Gaussian source and the space-time geometry
of the considering framework. In the next section, we further consider a
special optical system, which can be converted to such a response function in
the observed images, reflecting the space-time geometry more obviously.

Figure 3: The amplitude period of $|\langle O\rangle|$ for different values of
$\omega$ with $\alpha=0.1$ and $h_{e}=1$. Figure 4: The amplitude of $|\langle
O\rangle|$ for different values of $T$ with $\alpha=0.1$ and $\omega=80$. From
top to bottom, the values of $T$ corresponds to $h_{e}=0.4,~{}0.6$ and $0.8$,
respectively.

##  IV The Optical System For Einstein Rings Formation

After analysis of the physical interpretation of the response function, we
will use it to directly interpret the image of the BH on the screen. We define
a special optical system on the AdS boundary, where an observer looks up into
the AdS bulk. The mechanism of our constructed optical system, which shows the
image of the AdS BH from the response function, is depicted in Fig. 5. We
suppose a particular observational area at
($\theta,~{}\varphi)=(\theta_{obs},~{}0$) on the AdS boundary, where the
observation region is shown in a yellow circle in the left side of Fig. 5.
Rotating the original spherical coordinate system ($\theta,~{}\varphi$) to a
new one ($\theta^{\prime},~{}\varphi^{\prime}$) as

$\sin\theta^{\prime}\cos\varphi^{\prime}+i\cos\theta^{\prime}=e^{i\theta_{obs}}(\sin\theta\cos\varphi+i\cos\theta),$
(18)

($\theta^{\prime}=0,~{}\varphi^{\prime}=0$), which corresponds to the central
point of the observational region. In addition, we define cartesian
coordinates ($x,y,z$) with
$(x,y)=(\theta^{\prime}\cos\varphi^{\prime},~{}\theta^{\prime}\sin\varphi^{\prime})$
on the AdS boundary. To construct an optical framework virtually, in the
middle, we set the convex lens on the ($x,y$) plane. The focal length and
radius of the convex lens are denoted by $f$ and $d$, respectively. Moreover,
we adjust a spherical screen at $(x,y,z)=(x_{sr},y_{sr},z_{sr})$, satisfying
$x^{2}_{sr}+y^{2}_{sr}+z^{2}_{sr}=f^{2}$ sv37 ; sv38 . The role of the convex
lens is considered a converter between the plane and spherical waves. Imagine
that an incident wave coming from the left side reaches the position of the
convex lens, this wave will be converted to a transmitted wave at the focus,
which will be observed on the screen, as shown in the right side of Fig. 5.

Figure 5: The observer’s setup to see the image formed on the screen.

(a) $\alpha=0.1,~{}\theta_{obs}=0^{o}$

(b) $\alpha=0.1,~{}\theta_{obs}=30^{o}$

(c) $\alpha=0.1,~{}\theta_{obs}=60^{o}$

(d) $\alpha=0.1,~{}\theta_{obs}=90^{o}$

(e) $\alpha=0.5,~{}\theta_{obs}=0^{o}$

(f) $\alpha=0.5,~{}\theta_{obs}=30^{o}$

(g) $\alpha=0.5,~{}\theta_{obs}=60^{o}$

(h) $\alpha=0.5,~{}\theta_{obs}=90^{o}$

(i) $\alpha=0.9,~{}\theta_{obs}=0^{o}$

(j) $\alpha=0.9,~{}\theta_{obs}=30^{o}$

(k) $\alpha=0.9,~{}\theta_{obs}=60^{o}$

(l) $\alpha=0.9,~{}\theta_{obs}=90^{o}$

(m) $\alpha=1.3,~{}\theta_{obs}=0^{o}$

(n) $\alpha=1.3,~{}\theta_{obs}=30^{o}$

(o) $\alpha=1.3,~{}\theta_{obs}=60^{o}$

(p) $\alpha=1.3,~{}\theta_{obs}=90^{o}$

Figure 6: Observational appearance of the lensed response on the screen at
various observation angles for different values of $\alpha$ with $h_{e}=1.2$
and $\omega=80$.

At the point of observation, the response function is considered as a wave
$\Xi(\hat{x})$, which will be transformed as the transmitted wave
$\Xi_{Tr}(\hat{x})$ through the convex lens having a frequency $\omega$.
Mathematically, this transmitted wave can be written as

$\Xi_{Tr}(\hat{x})=e^{-i\omega\frac{|\hat{x}|^{2}}{2f}}\Xi(\hat{x}),$ (19)

where $\hat{x}=(x,y,0)$ is the coordinate position on the AdS boundary. In
this perspective, the optical appearance of the wave function on the screen is
defined as

$\displaystyle\Xi_{sr}(\hat{x}_{sr})=\int_{|\hat{x}|\leq
d}d^{2}x\Xi_{Tr}(\hat{x})e^{i\omega\mathbb{D}}\propto\int_{|\hat{x}|\leq
d}d^{2}x\Xi(\hat{x})e^{-i\frac{\omega}{f}\hat{x}.\hat{x}_{sr}}=\int
d^{2}x\Xi(\hat{x})\Pi(\hat{x})e^{-i\frac{\omega}{f}\hat{x}.\hat{x}_{sr}},$
(20)

in which $\hat{x}_{sr}$ represents the cartesian-like coordinates on the
screen, $\mathbb{D}$ is the distance between $\hat{x}$ and $\hat{x}_{sr}$ and
$\Pi(\hat{x})$ is the window function, which is defined as

$\Pi(\hat{x})=\begin{cases}\text{$1,$ \quad$0\leq|\hat{x}|\leq d$},\\\
\text{$0$,\quad~{}$|\hat{x}|>d$}.\end{cases}$ (21)

With the help of Eq. (20), we can observe the image on the screen, which is
obtained through the Fourier transformation of the incident wave sv75 ; sv76 ;
sv77 . We depicted different profiles of the dual BH on the screen when the
observer is located at different positions of the AdS boundary. For instance,
we plotted the holographic Einstein images for various values of parameter
$\alpha$ and observational angles for a fixed value of $h_{e}=1$ and
$\omega=80$, as shown in Fig. 6. When $\alpha=0.01$ and $\theta=0^{o}$,
meaning that the observer is located at the north pole of the AdS boundary,
here the observer will see a clear ring with a series of concentric striped
patterns as shown in the left column of Fig. 6 (top row). These patterns are
caused by a diffraction effect with imaging and are related to the lens radius
$d$ and the frequency $\omega$. As we increase the values of $\alpha$ from top
to bottom, we observed that the rings lie far away from the center in all
cases, however the luminosity of the ring and shadow radius changes according
to the values of $\alpha$, as we shown in Fig. 7. When $\alpha=0.1$ (see the
left panel of Fig. 7), one can see that the brightness curves reached the peak
position at the two endpoints on the abscissa such as $-0.97$ and $0.97$.
Here, we observe the strongest brightness at these points. This effect changes
when $\alpha=0.5$, where the brightness curves change the positions at the
points $-0.98$ and $0.98$, it means $\alpha$ has a minor influence on the
shadow radius and the brightness of the ring sharply decreases (see the second
panel from the left side of Fig. 7). Further, in the third panel of Fig. 7,
when $\alpha=0.9$, there is no change in the shadow radius but we found that
the brightness of the ring gradually decreases. Further, when $\alpha$ grows
like $\alpha=1.3$, we see that the position of the brightness curves is
changed at the points on the abscissa $-0.99$ and $0.99$ (see the right panel
of Fig. 7).

In this case, we note that the shadow’s radius is slightly increased, and the
corresponding brightness of the ring gets smaller values. Hence, according to
the above discussion, we found that increases of $\alpha$ lead to a change in
the shadow’s radius and luminosity of the ring implying that the parameter
$\alpha$ is sensitive enough to analyze the luminosities and the shadow’s
radius of the dual BH image. Next, we fix $\theta_{obs}=30^{o}$ as the values
of the parameter $\alpha$ increase, the bright ring transformed into a
luminosity deformed ring and now the shape of the strict axis-symmetric ring
disappears, see the second column from the left side of Fig. 6. Further, it is
noted that, when $\alpha$ has smaller values, the right side of the ring is
brighter while the left side of the ring is darker. And particularly, when
$\alpha=0.5$, the left side of the ring also depicted a little brightness but
less than the right side of the ring. And from top to bottom, we see that the
overall brightness of the ring decreases, as the values of the parameter
$\alpha$ increase. Now, the concentration further goes towards the position of
the angle at $\theta_{obs}=60^{o}$, see the third column from the left side of
Fig. 6.

(a) $\alpha=0.1$

(b) $\alpha=0.5$

(c) $\alpha=0.9$

(d) $\alpha=1.3$

Figure 7: The brightness of the lensed response on the screen for various
$\alpha$ with $h_{e}=1.2$ and $\omega=80$.

In this case, when $\alpha$ has smaller values, we see that a just bright
light arcs on the right side of the screen, and when $\alpha=0.5$, a dimmer
light ring also appears on the left side of the screen. From top to bottom, as
the values of $\alpha$ increase, we see that there are only dimmer light arcs
appearing on the left side of the screen. When we set the observation angle at
$\theta_{obs}=90^{o}$, a pair of bright light spots appeared in the middle of
the left and right side of the screen, as shown in the right column of Fig. 6
(top row). And when $\alpha=0.5$, there is only one bright light spot appears
on the left side of the screen. As the parameter $\alpha$ increases, the
bright light spot does not change the luminosity and lies far away from the
center in all cases. These results are also consistent with sv41 ; sv42 .

Now, we are going to analyze the effect of wave source on the characteristics
of the holographic Einstein image, which is observed at the position of the
north pole with $\alpha=0.1$ and $h_{e}=1.2$, as shown in Fig. 8. The
corresponding profiles, which show the brightness of the lensed response on
the screen for the same parameters, are shown in Fig. 9. To see the influence
of the wave source, we fix $\rho=0.02$ and $d=0.6$ for the convex lens, as the
value of the frequency increases, the resulting ring becomes sharper. From
this perspective, we concluded that the Einstein image that is captured
through the geometric optics approximation gives a good optical appearance in
the high frequency limit. This effect can also be seen in Fig. 9, where the
peak curve of brightness is getting steeper with the increasing values of the
$\omega$.

(a) $\omega=70$

(b) $\omega=50$

(c) $\omega=30$

(d) $\omega=10$

Figure 8: The profiles of the lensed response for various $\omega$ at the
observational angle $\theta=0^{o}$ with $\alpha=0.1$ and $h_{e}=1.2$.

(a) $\omega=70$

(b) $\omega=50$

(c) $\omega=30$

(d) $\omega=10$

Figure 9: The brightness of the lensed response for various $\omega$ with
$\alpha=0.1$ and $h_{e}=1.2$. Figure 10: The relation between temperature $T$
and inverse of event horizon $h_{e}$ with $\alpha=0.1$.

We further investigate the influence of the horizon temperature $T$ on the
images of the lensed response of the above holographic Einstein image, which
is observed for the fixed values of the parameters $\alpha=0.1$ and the
frequency $\omega=80$, as shown in Figs. 11 and 12 corresponding to small
values of the horizon $h_{e}$, respectively. For a better understanding of the
physical meaning of the horizon temperature, we plot the relation of
temperature $T$ and the inverse of the horizon $h_{e}$ as shown in Fig. 10
with $\alpha=0.1$. It is found that the influence of temperature on shadow is
different at small and large event horizons. It can be seen clearly that at
the beginning the horizon temperature $T$ has the smaller value and shows the
lowest value at point ($T=0.2706$, $h_{e}=1.6$). After that, it sharply
increases with the increasing value of the horizon $h_{e}$. This feature may
be used as a method to distinguish the STVG BH solution from other BH
solutions sv39 ; sv40 ; sv42 . When the horizon $h_{e}=1.6$, we have
$T=0.2706$, and similarly we have $T=0.2725,~{}0.3103,~{}0.3831$ corresponding
to $h_{e}=1.5,1,0.7$, respectively. One can see that the horizon temperature
increases with the decreasing value of the horizon $h_{e}$. In all cases, one
can see that a series of axis-symmetric concentric strict rings have appeared
on the screen as shown in Fig. 11. Here, we observed that the luminosity of
the rings does not change and lies far away from the center in all cases. This
phenomenon can also be seen in Fig. 12, where the brightness peak of the
lensed response is far away from the center as the temperature $T$ increases
for the same values of the parameters as defined in Fig. 11. Figure 12 (see
the left panel) corresponds to $T=0.2706$, illustrating that the brightness
curves obtain the peak position at points $-0.96$ and $0.96$ on the abscissa
and here the brightness of the ring has the maximum value.

When the temperature grows such as $T=0.2725$ (see the second panel from the
left side of Fig. 12), the shadow’s radius shows a slight increase, however,
the brightness of the ring decreases significantly. Similarly, as temperature
increases such as $T=0.3103$ (see the third panel from the left side of Fig.
12), we observe a further increase in the shadow’s radius, but the brightness
of the ring continues to decrease. When the temperature further increases and
reaches $T=0.3831$ (see the right panel of Fig. 12), the shadow’s radius
obtains the larger values and the brightness of the ring is still going to get
smaller values. This means that increasing the horizon temperature leads to a
slight increase in the shadow’s radius and a decrease in the brightness of the
ring. From these results, we can summarize that the smaller values of horizon
$h_{e}$ are also sensitive to analyzing the optical appearance and the
corresponding shadow’s radius of the Einstein ring. Further, we also observe
that in the middle of peak curves, there is also a relatively smaller curve of
brightness, which corresponds to a very dim light spot in the center of the
screen, which is hard to see in Fig. 11.

(a) $T=0.2706$

(b) $T=0.2725$

(c) $T=0.3103$

(d) $T=0.3831$

Figure 11: The profiles of the lensed response on the screen at the fixed
value of the observational angle $\theta=0^{o}$ for the small $h_{e}$ with
$\alpha=0.1$ and $\omega=80$.

We further study the effect of the large values of the horizon $h_{e}$ on the
image of the dual BH as shown in Fig. 13. When the horizon $h_{e}=3$, we have
$T=0.3113$ (see the left panel of Fig. 13), we observed that a series of axis-
symmetric concentric rings appeared on the screen, and one of them is a
particularly bright ring, lies far away from the center. When the horizon
grows to $h_{e}=5$ corresponding to $T=0.4354$ (see the second panel from the
left side), we see smaller axis-symmetric concentric rings appearing in the
image and the bright ring goes closer to the center. When we further increase
the value of the horizon such as $h_{e}=7$ which corresponds to $T=0.5773$
(see the third panel from the left side), we observe that the bright ring
comes closer to the center and ultimately the size of the ring becomes
smaller. When the horizon $h_{e}=9$ at $T=0.7252$ (see the right panel), the
brightest ring gradually moves inward. We also plot the brightness curves of
the lensed response in Fig. 14 corresponding to the sub figures of Fig. 13
under the same values of parameters. Particularly when $T=0.3113$, we see that
the peaks of brightness curves lie far away from the center, and when the
temperature gradually increases such as $T=0.4354$, the peaks of the
brightness curves become closer to the center. When the temperature grows
$T=0.5773$, the peak of the brightness curves further moves towards the
center. And when the temperature rises to $T=0.7252$, the brightness curves
gradually move towards the center. Further, it is noted that in Fig. 14,
between the peaks of the brightness curves, there is also a relatively smaller
curve of brightness, which corresponds to the dim light spot in the center of
the screen as shown in Fig. 13. These figures show that when the value of the
temperature is smaller, the position of the brightest ring lies far away from
the center, and at higher values, it gradually moves inward.

(a) $T=0.2706$

(b) $T=0.2725$

(c) $T=0.3103$

(d) $T=0.3831$

Figure 12: The brightness of the lensed response on the screen for the small
$h_{e}$ with $\alpha=0.1$ and $\omega=80$.

(a) $T=0.3113$

(b) $T=0.4354$

(c) $T=0.5773$

(d) $T=0.7252$

Figure 13: The profiles of the lensed response on the screen at the fixed
value of the observational angle $\theta=0^{o}$ for the large $h_{e}$ with
$\alpha=0.1$ and $\omega=80$.

(a) $T=0.3113$

(b) $T=0.4354$

(c) $T=0.5773$

(d) $T=0.7252$

Figure 14: The brightness of the lensed response on the screen for the large
$h_{e}$ with $\alpha=0.1$ and $\omega=80$.

In order to analyze the light deflection of AdS Schwarzschild STVG BH, we need
to find how the light rays move in this space-time. Particularly, we
investigate the optical geometry of the brightest ring in the image, which is
located at the position of the photon sphere. In the spherically symmetric
space-time, an orbit of geodesics lies in a plane passing through the center
of the BH. Therefore, without loss of generality, we can always rotate an
orbital plane of the null geodesic to coincide with the equatorial plane.
Hence, the Lagrangian of this system is defined as sv33 ; sv42

$\displaystyle\mathcal{L}=\frac{1}{2}g_{\kappa\lambda}\dot{x}^{\kappa}\dot{x}^{\lambda}=\frac{1}{2}\big{(}-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}+r^{2}(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2})\big{)},$
(22)

where $\dot{x}^{\kappa}$ is the four velocity of the light ray and “.” is the
derivative with respect to the affine parameter $s$. As we consider an
equatorial plane, we impose the initial conditions as $\theta=\pi/2$ and
$\dot{\theta}=0$. Since the metric coefficients do not depend explicitly on
time $t$ and azimuthal angle $\phi$, we have two conserved quantities, such as
energy $E$ and angular momentum $L$ of the photon, defined as

$\displaystyle E=f(r)\dot{t},\quad L=r^{2}\dot{\phi}.$ (23)

Based on the null geodesics, we have
$g_{\kappa\lambda}\dot{x}^{\kappa}\dot{x}^{\lambda}=0$, and using Eq. (23), we
can write

$\dot{r}^{2}=E^{2}-V(r)L^{2},$ (24)

where $V(r)=f(r)/r^{2}$ is an effective potential. We further define the
impact parameter $b=L/E$, which is the vertical distance between the geodesic
line and the parallel line passing through the origin. In this space-time,
there exists one null geodesic on the equatorial plane, having a circular
shape. This is essentially the photon sphere, projected on the equatorial
plane, yielding a ring-like circular shape, known as a photon circular orbit.
Now, the effective potential $V(r)$ is used to represent the radial geodesic,
which is defined as follows:

$\dot{r}^{2}+V(r)=\frac{1}{b^{2}}.$ (25)

At the photon sphere, i.e., ($r=r_{ph}$), the motions of the photons satisfy
the conditions $\dot{r}=0$ and $\ddot{r}=0$, which also means that

$\displaystyle V(r_{ph})=\frac{1}{b^{2}_{ph}},\quad V^{\prime}(r_{ph})=0,$
(26)

which corresponds to the maximum position of the effective potential $V(r)$.
We plot the profile of the effective potential $V(r)$ in Fig. 15 for a fixed
value of $\alpha$, as an example.

Figure 15: The profile of the effective potential $V(r)$ for $\alpha=1.5$ with
$M=1$. The dashed line indicates the radii of the photon sphere $r_{ph}$. The
region $1$, $2$ and $3$ corresponds to $V(r)<1/b^{2}_{ph}$,
$V(r)=1/b^{2}_{ph}$ (green line) and $V(r)>1/b^{2}_{ph}$, respectively.

We see that the effective potential vanishes on the event horizon. Then, it
increases and reaches the peak position at the photon sphere $r_{ph}$ and
subsequently moves down with respect to the radius $r$. In general, when a
photon moves in the radially inward direction, it shows different behavior of
motion, and we can classify it into three different regions. In region $1$, if
the photon starts its motion at $r>r_{ph}$, the photon will encounter the
potential barrier and be reflected in the outward direction. Further, when
$b<b_{ph}$ (correspond to region $3$), the photon does not face any obstacles
and hence, it will continue moving in the inward direction and finally fall
into the BH singularity. Particularly, in the region $2$, namely $b=b_{ph}$,
the photons start their motion at the position of the photon sphere, it will
rotate around the BH in a state of constant rotation. Hence, the closer the
value of the impact parameter is to $b_{ph}$, the more cycles of photons
revolve around the BH.

Further, we can naturally define the ingoing angle $\theta_{in}$ of the photon
with the normal vector of boundary $n^{\tau}\equiv\partial/\partial r^{\tau}$
is given as follows sv38

$\displaystyle\cos\theta_{in}=\frac{g_{\zeta\xi}v^{\zeta}n^{\xi}}{|v||n|}\bigg{|}_{r=\infty}=\sqrt{\frac{\dot{r}^{2}/f(r)}{\dot{r}^{2}/f(r)+L^{2}/r^{2}}}\bigg{|}_{r=\infty},$
(27)

or equivalently

$\displaystyle\sin\theta^{2}_{in}=1-\cos\theta^{2}_{in}=\frac{L^{2}V(r)}{\dot{r}^{2}+L^{2}V(r)}\bigg{|}_{r=\infty}=\frac{L^{2}}{E^{2}}.$
(28)

Figure 16: A schematic diagram of the dominant contribution to the final
response function comes from the incident photon rotating once around the BH.

Hence, the ingoing angle of photon orbit from the boundary satisfies:

$\sin\theta_{in}=\frac{L}{E},$ (29)

which is shown in Fig. 16. Here, the light ray approaches the photon sphere
and revolves around the BH many times, since the angular velocity is non-zero.
We further focused on the main contribution to the final response function,
which originated from the special angular momentum as $L_{p}$, starting from
the south pole, moves around the BH once and reaches the north pole, which can
be determined by the following conditions as

$\displaystyle\dot{r}=0,\quad\frac{dV}{dr}=0.$ (30)

When an observer on the AdS boundary looks up into the AdS bulk, the resulting
angle $\theta_{in}$ provides the angular distance of the image of the incident
ray from the zenith. If both ends of the geodesics and the center of the BH
are in alignment, the observer recognizes a picture of the ring having a
radius corresponding to the incident angle $\theta_{in}$ because of axis-
symmetry sv38 . In addition, as shown in Fig. 17, one can obtain the angle of
the Einstein ring, which is formed on the screen, having ring radius
$\theta_{R}$ as given below

$\sin\theta_{R}=\frac{r_{R}}{f}.$ (31)

Further, when the angular momentum is sufficiently large such as
$\sin\theta_{in}=\sin\theta_{R}$, then we have the following relation as sv38

$\frac{L_{p}}{E}=\frac{r_{R}}{f}.$ (32)

Since, both the angle of the incident of the photon and the angle of the
photon ring describes the angle at which the observer can see the photon ring,
and hence these should be essentially equal, which will be verified
numerically, in the next. We depicted the Einstein ring radius and the
Einstein radius of the photon orbit for different values of $\alpha$ and
$\omega$ in Figs. 18 and 19, respectively.

Figure 17: The relation between $\theta_{R}$ and $r_{R}$.

In both Figs. the blue curve depicts the photon ring position, which is the
prediction of the geodesics, providing the angle of incidence of the photon to
the AdS boundary. Whereas the red points are the Einstein ring radius,
constructed from the response function in wave optics, where the angular
momentum can be considered as some wave packets, realized by the superposition
of the spherical harmonics. From Fig. 18, we observe that the Einstein ring
radius decreases with increasing temperature and the discrete red points
always lie on the blue line or in its vicinity. Further, Fig. 19 (left panel)
shows that when $\omega=80$ at the smaller values of the temperature, the
discrete red points nearly lie to the blue line, but they gradually move away
from it with increasing values of the temperature. But as we increase $\omega$
to $\omega=120$ (see the right panel of Fig. 19) at the smaller values of the
temperature, the discrete red points lie nearly below to blue line but with
increasing values of temperature, they crossed it and lie on the blue line or
its vicinity. Further, both panels of Fig. 19 illustrate that the Einstein
ring radius decreases with increasing the temperature for both values of
$\omega$. Hence, one can observe that the accuracy of high and low-temperature
fitting is different with different frequencies, implying that the variation
of $\omega$ also significantly affects the results of both wave and geometric
optics. In all cases, as expected, the position of the photon rings given by
the geometric optics always lies on the blue line or in its vicinity with that
of wave optics.

(a) $\alpha=0.1$

(b) $\alpha=0.5$

(c) $\alpha=0.93$

(d) $\alpha=1.3$

Figure 18: Comparison of Einstein ring radius (discrete red points) in the
unit of $f$ with the Einstein radius of the photon orbit (blue curve) for
different values of $\alpha$ with a fixed $\omega=80$.

(a) $\omega=80$

(b) $\omega=120$

Figure 19: Comparison of Einstein ring radius (discrete red points) in the
unit of $f$ with the Einstein radius of the photon orbit (blue curve) for
different values of $\omega$ with a fixed $\alpha=0.5$.

##  V Conclusions

Gravitational lensing by BHs is one of the most peculiar astrophysical tools
for analyzing the strong field properties of gravity and provides us the
information about the significant properties of distant stars which are too
dim to be observed. It can help us to identify exotic objects and hence, it is
fruitful to investigate the modified theories of gravity. Suppose that there
is a light source behind the gravitational body. When the light source, the
gravitational body, and the observer are in alignment, the observer will
observe a bright ring-like shape of the light source, which is so-called the
Einstein Ring. When the gravitational body is a BH, then some light rays are
strongly deflected and revolve around the BH many times especially infinite
times on the photon sphere. In the present study, we investigated the
holographic images of a BH in the bulk from a given response function on the
AdS boundary. Considering the oscillating Gaussian source
$\mathcal{J}_{\mathcal{O}}$ with frequency $\omega$ produced at a point on the
boundary, we derive the local response function in the context of AdS/CFT
correspondence, which is presented on the other side of the AdS boundary. When
the scalar wave passes through BH, it always generates a diffraction pattern
in the bulk geometry of the BH space-time, which is analyzed through the local
response function. Our results show that the absolute amplitude of the total
response function significantly varies with the variation of coupling
parameter $\alpha$, frequency $\omega$, and the positions of the horizon
$h_{e}$.

As the value of $\alpha$ increases the resulting diffraction pattern decreases
and this effect is vice versa when we change the values of $\omega$ and
positions of the horizons. Although we analyzed the diffraction pattern of the
response function, it does not directly produce concrete information on BH.
With the help of an optical system, which consists of a convex lens and
spherical screen, we obtained the images of the BH from the response function
based on the wave optics method. We depicted the resulting Einstein ring image
on the screen for different values of $\alpha$ and observational angles in
Fig. 6. When the position of the observer changes, the resulting Einstein
image changes into a luminosity deformed ring, light arcs, or a bright light
spot. In addition, we also show the effect of $\alpha$ on the brightness
curves of the lensed response and it seems that the peak of the curves always
lies far away from the center in all cases as shown in Fig. 7. Here one can
see that, when $\alpha$ has smaller values, the luminosity of the ring shows
the maximum value, and it gradually decreases with the increasing values of
$\alpha$. Further, in all cases, it is noted that the shadow radius increases
with respect to increasing values of $\alpha$, and hence, $\alpha$ is
sensitive to observing the holographic Einstein image of the STVG BH.
Similarly, Figs. 8 and 9 are showing Einstein ring image and corresponding
brightness profiles for different values of $\omega$. In Fig. 8, it seems that
the width of the Einstein ring becomes larger for lower values of $\omega$ and
becomes sharper for large values. This effect can also be seen in Fig. 9,
where the peak curves of the brightness show sharper wave oscillations with
increasing values of $\omega$. In general, the optical appearance of the
Einstein ring and the brightness profiles are closely related to the BH-
involved parameters, the optical system, and the wave source.

We also show the optical appearance of the horizon temperature with respect to
the inverse of horizon $h_{e}$ with a fixed value of parameter $\alpha=0.1$ as
shown in Fig. 10. It is found that at the beginning of the horizon $h_{e}$,
the temperature shows smaller values, and it increases with the increasing
values of the horizon radius. This particular behavior of the horizon
temperature may be used as a tool to distinguish the STVG BH solutions from
previous studies sv39 ; sv40 ; sv42 . Further, the influence of the
temperature with respect to smaller and larger values of the horizon $h_{e}$
is also depicted on the profiles of the holographic images and the brightness
curves of the lensed response, as shown in Figs. 11-14. It is observed that in
the case of smaller values of the horizon $h_{e}$, there exists a series of
concentric striped patterns in which one of them is the brightest ring, lies
far away from the center, as shown in Fig. 11. This effect can also be
observed in the brightness profiles (see Fig. 12), where the luminosity of the
ring obtains smaller values as we increase the horizon temperature and the
peak curves always stay close to the boundary. In addition, we found that at
the smaller values of the horizon temperature, the shadow radius has minimum
values, but it increases with the increasing values of the temperature. On the
other hand, for large values of the horizon $h_{e}$, we found that the
brightest ring gradually moves towards the center with increasing values of
temperature. The proper visualization of this effect is depicted in Fig. 13,
and the corresponding brightness curves of the lensed response are shown in
Fig. 14. One can see that when $T=0.3113$, the luminosity of the ring and
shadow radius show the maximum values, and the increasing values of the
horizon temperature lead to a decrease in the luminosity of the ring and
shadow radius (see Fig. 14).

Based on the geometric optics, we also analyzed the Einstein ring radius
corresponding to the position of the photon orbit, which is calculated from
the geodesic analysis with the Einstein radius of the image constructed from
the response function in the wave optics as shown in Figs. 18 and 19 for
different values of parameter $\alpha$ and $\omega$, respectively. Figure 18
indicates that the discrete red points always lie on the blue line at the
smaller values of the horizon temperature and then gradually move away from it
with increasing values of temperature in all cases. Whereas from Fig. 19, it
is noted that the discrete red points always lie above and below or closely to
the blue line at the smaller or larger values of the horizon temperature.
Hence, the change in the positions of the photon orbit and the Einstein radius
obtained from the wave optics are observed with respect to different values of
parameter $\alpha$ and $\omega$ in both figures. As expected, in all cases,
the Einstein ring radius obtained by our wave optics fits well with that of
geometric optics. Finally, we concluded that the holographic Einstein ring
provides us with immense information about the geometric structure of space-
time and helps us to analyze different gravity models comprehensively.
Moreover, holographic images can be used as an effective tool to distinguish
different types of BHs for fixed wave source and optical system. We hope these
observations may provide a more intuitive understanding of the Einstein rings
and their related consequences, for the tabletop experiments in the future.

## 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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