# Influence of plasma on the optical appearance of spinning black hole in
Kalb-Ramond gravity and its Existence around M87* and Sgr A*

Muhammad Ali Raza<EMAIL_ADDRESS>Department of Mathematics, COMSATS
University Islamabad, Lahore Campus, Lahore, Pakistan  M. Zubair
mzubairkk<EMAIL_ADDRESS>Department of Mathematics,
COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan  Eiman Maqsood
<EMAIL_ADDRESS>Department of Mathematics, COMSATS University
Islamabad, Lahore Campus, Lahore, Pakistan

###### Abstract

The visible universe is filled with different types of plasma media in the
form of stars, nebulas and other forms of excited gases. These matter fields
have a high influence on the gravity and are likely to be present around the
black holes due to the immense gravitational attraction. Since a plasma medium
affects the speed of light, therefore we investigated the null geodesics and
various optical features around the rotating black hole in Kalb-Ramond gravity
immersed in plasma medium. Various plasma distributions are considered to
develop a comparative study for their influence on unstable null circular
orbits, shadows and evaporation rate of the black hole in the presence of a
plasma medium. Moreover, the shadow results are also compared with Event
Horizon Telescope data for M78* and Sgr A* in order to estimate the parametric
bounds for which the rotating black hole in Kalb-Ramond gravity is considered
either M87* or Sgr A* under the different values of plasma parameters. From
this analysis, we also found the distribution of plasma that has a significant
impact on the above mentioned features and is most likely to be present around
M87* and Sgr A*.
Keywords: Black hole, Kalb-Ramond field, plasma, shadow, M87*, Sgr A*.

##  I Introduction

According to Newton’s concept, the gravity being a force of attraction is
instantaneous and eventually becomes weaker as the separation between the two
objects increases. This concept did not support the cosmic catastrophe
problem, that is, the Earth would leave its orbit immediately as the Sun
vanishes. This implies that a signal traveled to Earth from the Sun faster
than the cosmic speed. To answer such complicated mysteries of the universe,
Albert Einstein laid the foundation of a new concept of gravity, in 1915,
based on the concept of spacetime curvature known as General Relativity (GR) 1
; 2 ; 3 ; 4 . It describes how the force of gravity actually works. Soon after
this discovery, Schwarzschild 5  proposed the pioneering solution of the
Einstein field equations in the most ideal scenario resulting into a black
hole (BH) known as Schwarzschild BH. Later, various other BH solutions were
presented based on different characteristics and properties such as Reissner-
Nordström 6 ; 7 , Kerr 8  and Kerr-Newman 9 ; 10  BHs being the most
fundamental solutions.

Besides the fundamental background of the theory, another group of scientists
was looking to unify gravity with the other subatomic fundamental forces of
nature. In this context, Kaluza 11  and Klein 12 ; 13  proposed Kaluza-Klein
theory that describes the unification of the electromagnetic and gravitational
theories based on 5D structure with quantum interpretation. Over the decades,
several attempt were made by the scientists in order to construct a theory
based on the principles of GR and quantum mechanics. Several theories have
been proposed, however the technological deficiencies restrict us to verify
some of these theories. However, we are still capable enough to use the
advancement of the technology to a certain extent. In this regard, GR has been
tested observationally at high resolutions and obtained great precision 14 ;
15 . For the theoretical description of the results, it was required to
construct the modified and alternative gravitational theories. A modified
theory can be constructed by modifying the Einstein-Hilbert action that can be
accomplished by introducing the Kalb-Ramond (KR) field as a possible example
16 . Since, the hybridization of a superstring and a bosonic string generates
a closed string that gives rise to heterotic string theory. Hence, the KR
field, being inherently a quantum field, exhibits a strong connection to the
excitation of closed strings within the framework of heterotic string theory
17 . The KR field is described by an antisymmetric tensor of rank two whose
vacuum expectation value (VEV) remains non-zero. Spontaneous Lorentz symmetry
breaking (LSB) manifests as a consequence of a nonminimal coupling between
gravity and a non-zero VEV 18 . Khodadi 19  considered a BH with spin in
Einstein-bumblebee gravity for which he studied the effect of the spontaneous
LSB on superradiance scattering and instability. He found the weakening of
superradiance scattering for scalar wave with low frequencies and the positive
values of LSB parameter. However, for the negative values, the scattering
becomes stronger. Moreover, Khodadi 20  also investigated that LSB parameter
gives rise to a BH with fast rotation that is distinct from Kerr BH. For this
particular BH, the investigation focused on analyzing the energy extraction
arising from magnetic reconnection occurring within the ergoregion. He found
when the BH is surrounded by a plasma region with a weak magnetization, the
energy extraction due to magnetic reconnection increases for negative values
of LSB parameter. Khodadi et al. 21  explored the rate of energy deposition
for gamma-rays induced by the annihilation of neutrino pairs for a slowly
rotating BH, with modifications due to spontaneous LSB, specifically
considering the plane passing through the equator. It was found that the
energy deposition rate is increased for positive values of LSB parameter. The
cosmological aspects of LSB has also been investigated, see Refs. 22 ; 23 .
Some properties are deduced for the KR field such as a tensor of third rank
which is antisymmetric and is proposed as the source of spacetime torsion 24 ,
intrinsic angular momentum and different massive structures in the universe
arising from the topological defects 25 . The KR field has also been
considered to study its gravitational properties and its impact on particles
26 ; 27 ; 28 ; 29 ; 30 . Recently, Lessa et al. 29  developed a metric
described as power law hairy BH in KR gravity that encompasses some
interesting properties and reduces to fundamental BHs, that is, Schwarzschild
and Reissner-Nordström BHs under various restrictions of the BH parameters.
Later, Kumar et al. 30  converted this static hairy BH into its rotating
counterpart and investigated its shadow and strong lensing. The coupling of
gravity with KR field may be useful to obtain some desired and feasible
quantum field theoretic results in curved spacetime, quantum geometrodynamics
and quantum gravity.

It remains a fundamental question that what do these BHs appear to a human
eye. Such a visual image of a BH is actually not the image of the BH itself,
instead it is a 2D silhouette known as shadow of the BH. The photon sphere is
made up of photon orbits that are circular in which the photons are captured
in the stable and unstable orbits. In the unstable orbits, the light particles
cannot reside for a long time and thus scatter away or fall into the horizon
generating a bright circular ring or a dark 2D image, respectively 31 . Some
pioneering and early shadow studies are given in Refs. 32 ; 33 ; 34 ; 35 ; 36
. Later, various BH solutions were considered to calculate their shadows, see
37 ; 38 ; 39 ; 40 ; 41 ; 42 ; 43 ; 44 ; 45 ; 46 ; 47 ; 48 ; 49 . Sharif and
Iftikhar 50  considered a noncommutative rotating BH comprising a charge. They
found that the shadow deviates from a circular shape with decrease in the
noncommutative charge. Lee et al. 51  considered the anisotropic matter
surrounding a rotating BH for which the shadows have been calculated. It was
found that the shadow observables are highly influenced by the anisotropy of
the matter. Amarilla and Eiroa 52  investigated the shadows of a braneworld BH
in the Randall-Sundrum model. Their study revealed that the shadow undergoes
significant distortion due to the influence of angular momentum and terms
associated with tidal charge.

To probe the physical reality of BHs, the collaborative efforts of the Event
Horizon Telescope (EHT) have yielded valuable observational data. This dataset
encompasses shadow images capturing M87* 53 ; 54 ; 55 ; 56 ; 57 ; 58  at the
galactic center of Messier 87 and Sgr A* 59  at the galactic center of the
Milky Way, providing essential insights into the existence and characteristics
of BHs. Utilization of the EHT observational data, scientists have undertaken
studies to compare theoretical BH shadow sizes with those observed for M87*
and Sgr A*. This analysis enables us to determine those values of the BH
parameters for which the shadow size lies in agreement with that of M87* and
Sgr A*. Consequently, the theoretically considered BH may be regarded as M87*
or Sgr A*. For details, we refer to 60 ; 61 ; 62 ; 63 ; 64 ; 65 .

A plasma is the fourth fundamental state of matter that comprises charged
particles in the form of electrons or ions. The largest amount of ordinary
matter in the visible universe is plasma that mostly exists in stars,
intracluster and intergalactic media. Therefore, a BH may also be surrounded
by a material media such as a plasma which is observationally proposed in the
Ref. 66 . Therefore, in the context of shadows and null geodesics around a BH,
if the BH is surrounded by a plasma medium, it may affect the motion of light
particles as compared to that in the non-plasma medium. It is quite certain
that a material media such as plasma lowers the speed of light due to which
the photons do not travel along the null geodesics and as a result the shadows
get affected. The speed of light in a plasma is dependent on its electron
frequency and hence the index of refraction. The study of light propagation in
plasma having no pressure but is magnetized, is governed by the Hamiltonian
formalism that was derived precisely by Breuer and Ehlers 67 ; 68 . Later,
Perlick 69  derived the Hamiltonian formulation for a rather simpler case of a
plasma without pressure and magnetization. Perlick et al. 70  also explored
the shadow of static and spherically symmetric BHs immersed in a plasma
medium. Perlick and Tsupko 71  applied the Hamiltonian and Hamilton-Jacobi
formulations to determine the geodesic equations for studying the shadow of
Kerr BH in plasma. They considered several cases of plasma distributions and
proposed a separating function in $r$ and $\theta$. Recently, various BH
solutions have been considered to investigate their shadows in the presence of
plasma, see Refs. 72 ; 73 ; 74 ; 75 ; 76 ; 77 .

Since, the composition of a plasma is mostly excited gaseous matter and
charged particles such as electrons. Therefore, the quantum mechanical
principles may apply in understanding various phenomena and the interaction of
these particles may also give rise to quantum aspects at the subatomic regime.
For instance, an electron being an elementary particle is an element in the
standard model of physics, obeys various principles of quantum field theory.
Therefore, it is quite certain that a plasma would also follow such quantum
theoretic principles. Since, due to quantum nature of KR field, a BH generated
in KR gravity may give rise to certain quantum effects in the nearby curved
spacetime. Therefore, such a BH when surrounded by a plasma is more likely to
give rise to quantum aspects in its vicinity and it is quite possible to give
rise to new physics at the quantum and subatomic level. Therefore, with this
motivation, we consider the rotating BH in KR gravity that is surrounded by
plasma described by different distribution functions and investigate the
optical features and the impact of plasma on the photon motion.

The manuscript is divided into the sections as: The following section comprise
the action of the theory and BH metrics. We also discussed the horizon
structure briefly. The third section is related to the null geodesics and
effective potential. This section is further divided into three subsections
corresponding to the plasma distribution functions. Moreover, the first
subsection has further two subcases. The fourth section is related to the
shadows and is also divided into further subsections corresponding to the
cases discussed in the third section. Similarly, corresponding to the plasma
cases, the distortion and the rate of emitted energy is discussed in fifth
section. The comparison of shadow sizes with observational EHT data is given
in sixth section and finally in seventh section, we conclude the manuscript.
Note that throughout the manuscript, we consider $G=c=M=1$, where $G$, $c$ and
$M$ are Newton’s constant, speed of light and mass of BH, respectively.

##  II The Black Holes in Kalb-Ramond Gravity

The mathematical formulation representing the nonminimal coupling between a
self-interacting KR field and gravity is succinctly expressed by the action
described as 18 ; 29

$\displaystyle S=\int\sqrt{-g}d^{4}x\bigg{[}\frac{R}{16\pi
G}-\frac{1}{12}H_{\alpha\mu\nu}H^{\alpha\mu\nu}-V\big{(}B_{\mu\nu}B^{\mu\nu}\pm
b_{\mu\nu}b^{\mu\nu}\big{)}+\frac{1}{16\pi
G}\big{(}\xi_{2}B^{\mu\lambda}B^{\nu}_{\lambda}R_{\mu\nu}+\xi_{3}B_{\mu\nu}B^{\mu\nu}R\big{)}\bigg{]},$
(1)

where the symbol $R$ denotes the Ricci scalar and $g=det(g_{\mu\nu})$. The
second rank tensor $B_{\mu\nu}$ is anti-symmetric and defines the KR field
that is responsible for the LSB vacuum expectation value (VEV) $b_{\mu\nu}\neq
0$, where this VEV has some relation with the potential $V$ 16 . With this
VEV, we can obtain two spacelike vectors and a timelike vector by decomposing
the tensor field $B_{\mu\nu}$ in the same way as we decompose the Maxwell’s
linear field tensor $F_{\mu\nu}$ 18 . The tensor field
$H_{\alpha\beta\delta}=\partial_{\alpha[}B_{\beta\delta]}$ being
antisymmetric, is analogical to the Maxwell’s linear field tensor
$F_{\mu\nu}$, while the KR field tensor $B_{\mu\nu}$ has an analogy with the
vector potential. Therefore, we can write the KR action in analogy with the
electrodynamics. The symbols $\xi_{i}$ are the nonminimal coupling constants,
where $i=2,3$. Lessa et al. 29  considered the vacuum KR field as
$B_{\mu\nu}B^{\mu\nu}=b_{\mu\nu}b^{\mu\nu}$ to study how the KR VEV affects
the gravitational field. In a flat spacetime, the LSB VEV follows
$\partial_{\sigma}b_{\mu\nu}=0$ and
$j^{2}=\eta^{\alpha\beta}\eta^{\mu\nu}b_{\alpha\mu}b_{\beta\nu}$. Furthermore,
if the VEV is constant, then the strength of KR field vanishes 18 . In this
regard, we can consider the KR VEV $b_{\mu\nu}$ as a constant whose
Hamiltonian vanishes. Under this analogy in the curved spacetime background,
we can also consider the VEV as a constant, that is,
$\nabla_{\sigma}b_{\mu\nu}=0$ such that the KR field strength and Hamiltonian
also vanish. To derive a spacetime metric describing a BH in KR gravity, the
norm of VEV $b_{\mu\nu}$ must be a constant whose Hamiltonian vanishes. The
gravitational field equations are modified to be

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa T_{\mu\nu}^{\xi_{2}},$
(2)

where $R_{\mu\nu}$ and $T_{\mu\nu}^{\xi_{2}}$ are the Ricci and the energy-
momentum tensors, respectively. The line element for a 4D static and
spherically symmetric metric is given as

$\displaystyle
ds^{2}=-\chi_{t}(r)dt^{2}+\chi_{r}(r)^{-1}dr^{2}+r^{2}d\Omega_{2}^{2}$ (3)

with $d\Omega_{2}^{2}$ defining the 2-sphere. The KR VEV ansatz are

$\displaystyle b_{2}=-\epsilon(r)dt\wedge dr,$ (4)

where $\epsilon=-b_{tr}$. As said earlier that the norm of the ansatz of KR
VEV is $b^{2}$ which is a constant, then for the metric (3) we obtain a
pseudo-electric field

$\displaystyle\epsilon(r)=|b|\sqrt{\frac{\chi_{t}(r)}{2\chi_{r}(r)}}$ (5)

that is static and is in radial direction. Therefore, the values of the metric
functions come out to be

$\chi_{t}(r)=\chi_{r}(r)=1-\frac{R_{sc}}{r}+\gamma r^{-\frac{2}{\lambda}},$
(6)

where $R_{sc}=2M$ is the Schwarzschild radius. Hence, the metric (3) becomes

$\displaystyle ds^{2}=-\bigg{(}1-\frac{R_{sc}}{r}+\gamma
r^{-\frac{2}{\lambda}}\bigg{)}dt^{2}+\bigg{(}1-\frac{R_{sc}}{r}+\gamma
r^{-\frac{2}{\lambda}}\bigg{)}^{-\frac{1}{2}}dr^{2}+r^{2}d\Omega_{2}^{2}$ (7)

that describes BH in KR gravity. Here, $\gamma$ and $\lambda$ are considered
to be the corresponding spontaneous LSB parameters. The parameter $\lambda$ is
defined in terms of nonminimal coupling parameter and VEV as
$\lambda=\xi_{2}|b|^{2}$, where $b^{2}=b_{\mu\nu}b^{\mu\nu}$. Whereas, the
parameter $\gamma$ is a constant of integration. The parameters $M$ and
$\gamma$ has dimensions of $[length]$ and $[length]^{\frac{2}{\lambda}}$,
respectively. The metric (7) can also be termed as power-law hairy BH
solution. For a fixed $\gamma$ and taking either $|b|^{2}=0$ or $\xi_{2}=0$,
we get $\lambda\rightarrow 0$ and thus the BH solution (7) becomes
Schwarzschild solution. Therefore, for $\lambda\in(0,\infty)$ the BH in KR
gravity deviates from the Schwarzschild BH. Since the gravitational field is
not significantly affected by the LSB, so the necessity arises for the
coupling constant to possess a small value. However, it is anticipated that
the LSB occurs at Planck scale. Therefore, the VEV $b^{2}$ that gives rise to
the LSB coupling, might also occur at Planck scale. In this context, some
systems are generated by the spontaneous LSB that are described by small value
of coupling constant but a large value of VEV. It can also be noted that the
metric (7) reduces to Schwarzschild de-Sitter BH corresponding to $\lambda=-1$
defining an asymptotically non-flat BH. However, it reduces to asymptotically
flat Reissner-Nordström BH for $\lambda=1$. Therefore, BH metric (7) is
asymptotically non-flat metric for $\lambda\leq 0$ and hence, we consider only
positive values of $\lambda$.

The rotating BHs are defined by a spin parameter and encompasses various
interesting features, especially, the particle motion and optical features.
Since, the BHs located at the center of galaxies are usually spinning BHs.
Therefore, it is more convenient and feasible to consider a spinning BH for a
rigorous comparative analysis for the shadows of the theoretical and
supermassive BHs. The Newman-Janis algorithm is a concise and useful method to
convert a static BH metric into its rotating counterpart. Therefore, by
incorporating this algorithm, the BH metric (7) was transformed into the
rotating metric by Kumar et al. 30  that is given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{\Delta(r)-a^{2}\sin^{2}{\theta}}{\zeta^{2}}\bigg{)}dt^{2}+\frac{\zeta^{2}}{\Delta(r)}dr^{2}+\zeta^{2}d\theta^{2}+\frac{\sin^{2}\theta}{\zeta^{2}}\bigg{(}\mathcal{U}(r;a)^{2}-\Delta(r)a^{2}\sin^{2}\theta\bigg{)}d\phi^{2}$
(8)
$\displaystyle+\frac{2a\sin^{2}\theta}{\zeta^{2}}\bigg{(}\Delta(r)-\mathcal{U}(r;a)\bigg{)}dtd\phi,$

with the metric functions defined as

$\displaystyle\Delta(r)$ $\displaystyle=$ $\displaystyle
a^{2}+r^{2}\chi_{t}(r)=\mathcal{U}(r;a)-R_{sc}r+\gamma
r^{\frac{2(\lambda-1)}{\lambda}},$ (9) $\displaystyle\zeta^{2}$
$\displaystyle=$ $\displaystyle r^{2}+a^{2}\cos^{2}\theta,$ (10)

with $a$ as spin parameter and $\mathcal{U}(r;a)=r^{2}+a^{2}$. By supposing
$\lambda=0$ and $\lambda=1$ in (8), we can recover the Kerr and Kerr-Newman
BHs, respectively. In addition to this, the static metric (7) is recovered by
supposing $a=0$ in the metric (8). Since the metric coefficients in (8) are
independent of the coordinates $t$ and $\phi$. Therefore, the Killing vectors
$(\partial_{t})^{\mu}$ and $(\partial_{\phi})^{\mu}$ exist that define the
rotational and time symmetries. We focus on the study of the behavior of
various plasma distributions surrounding the rotating BH in KR gravity. We
also find out that what distribution of plasma is more influential on the
optical features of the BH (8). It will also enable us to find that the BH (8)
surrounded by a particular plasma medium is whether M87* or Sgr A*. Hence, we
draw a comparison between the angular radius of the shadows of the BH (8) with
M87* and Sgr A*.

The outer horizon, also known as event horizon defines the size of the BH that
has been plotted along with Cauchy horizon for with respect to the spin for
different values of $\lambda$ and $\gamma$ in Fig. 1. To study the horizon
structure, we need to determine the roots $r_{h}$ by solving the metric
function $\Delta(r)\big{|}_{r_{h}}=0$. For each curve in the figure, it is
evident that the Cauchy horizon keeps on increasing and the event horizon
keeps on decreasing with respect to $a$, until the extremal value of $a$ is
achieved where the metric function $\delta$ has a double root and both
horizons become identical. Moreover, the extreme value of $a$ decreases as
$\lambda$ and $\gamma$ increase in both plots. The event and Cauchy horizons
in the left plot decrease for the fixed value of $a$ as $\lambda$ increases
except for $\lambda=0$. For $\lambda=0$, the Kerr BH that shows the largest
extremal value of $a$ and has the largest difference of horizons. For
$\lambda\neq 0$, it deviates from the Kerr and approaches the Kerr-Newman BH
for $\lambda=1$. For a particular value of $a$ as $\gamma$ is elevated in the
right plot, it can be seen that the value of event horizon drops but the
Cauchy horizon is elevated.

Figure 1: Plots illustrating the variation of horizons $r_{h}$ vs $a$ for
various values of $\lambda$ and $\gamma$.

##  III Null Geodesics

There is a critical region in the neighborhood of the BH where the light rays
are trapped that emerge out of a bright source. This region is known as photon
sphere that comprise the unstable circular null orbits. Some photons in these
orbits fall inside the event horizon, whereas the remaining photons scatter
away to the infinity. As a result of this phenomenon, the shadow of the BH is
formed as a 2D image. We know that a photon in the unstable circular null
orbit may revolve around the BH for more than one time before leaving the
orbit. Whereas, a photon resides forever in the stable circular null orbit.
The effective potential describes these circular null orbits. Since, it is
quite certain that the astrophysical BHs are mostly immersed in material
media, especially a plasma medium. Hence, our study delves into the
characteristics of null geodesics and the effective potential governing the
motion of photons around a rotating black hole within the framework of KR
gravity. This investigation is conducted in the presence of a plasma medium,
and various distribution functions are considered to describe its properties.
Hence, the properties of particle orbits around the BH are governed by the
effective potential that can be determined by incorporating the Hamilton-
Jacobi 78  and Hamiltonian formulations. Since, we encounter four unknown
variables for which we need four constants of motion in order to solve the
system of equations completely. Therefore, we obtain two constants by using
the Hamiltonian formulation, namely, the angular momentum $L$ along $\phi$
direction and energy $E$ of the particle. The particle’s mass being the third
constant, we employ the Hamilton-Jacobi method that generates a constant
termed as Carter constant 78  which is treated as the fourth constant. This
formalism also ensures the separability of terms as a function of $r$ and
$\theta$ coordinates. We divide this analysis into different subsections based
on various plasma distributions.

###  III.1 Case I

We begin with a general case in which the frequency of electrons in plasma is
assumed a function of the $r$ and $\theta$ 31 ; 71 ; 72 . The Hamiltonian that
describes the photon motion in a plasma medium without pressure and
magnetization is given as

$\mathcal{H}=\frac{1}{2}\Big{[}g^{\mu\nu}p_{\mu}p_{\nu}+\omega_{p}(r,\theta)^{2}\Big{]},$
(11)

where, $p_{\mu}$ are the generalized momenta. The plasma electron frequency
$\omega_{p}(r,\theta)$ is related to the refractive index $n(r,\theta)$ as

$n(r,\theta)=\sqrt{1-\frac{\omega_{p}(r,\theta)^{2}}{\omega(r,\theta)^{2}}},$
(12)

where, $\omega(r,\theta)$ is the frequency of photon measured by a static
observer outside the event horizon. The frequency of electrons in plasma is
written in terms of the number density of electrons as

$\omega_{p}(r,\theta)=\frac{4\pi e^{2}}{m_{e}}N_{e}(r,\theta),$ (13)

where, $e$, $m_{e}$ and $N_{e}$ are the charge, mass and number density of the
electron, respectively. The Hamiltonian corresponds to the Hamilton’s
equations

$\dot{x}^{\mu}=\partial_{\tau}x^{\mu}=\partial_{p_{\mu}}\mathcal{H},\qquad\dot{p}_{\mu}=\partial_{\tau}p_{\mu}=-\partial_{x^{\mu}}\mathcal{H}$
(14)

with $\tau$ being an affine parameter. Since, the plasma medium is dense and
dispersive, therefore, the motion of photons is affected by the frequencies of
electrons in the plasma and photons in a light ray. Therefore, the frequency
of the propagating photon must be greater than the frequency of the electrons
in plasma, that is, $\omega(r,\theta)^{2}\geq\omega_{p}(r,\theta)^{2}$. Since
the observer is static with the four-velocity in comoving coordinates being
$U^{\mu}(r,\theta)=(-g_{tt}(r,\theta))^{-1/2}$ and the photon frequency is
$\omega(r,\theta)=-p_{\mu}U^{\mu}(r,\theta)=-p_{t}U^{t}(r,\theta)$. The
Planck’s relation relates the energy and the angular frequency as
$E=\hbar\omega_{0}$ which gives $p_{t}=-\omega_{0}$ by setting the units for
$\hbar=1$. Therefore, we obtain

$\omega(r,\theta)=\omega_{0}\big{(}-g_{tt}(r,\theta)\big{)}^{-\frac{1}{2}}.$
(15)

Using the Hamiltonian approach, we obtain the following two equations

$\displaystyle\zeta^{2}\dot{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta(r)}\Big{[}\Big{(}\mathcal{U}(r;a)^{2}-a^{2}\Delta(r)\sin^{2}\theta\Big{)}E+a\big{(}\Delta(r)-\mathcal{U}(r;a)\big{)}L\Big{]},$
(16) $\displaystyle\zeta^{2}\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta(r)}\Big{[}\big{(}\Delta(r)\csc^{2}\theta-a^{2}\big{)}L-a\big{(}\Delta(r)-\mathcal{U}(r;a)\big{)}E\Big{]}.$
(17)

Here, we obtained the two constants; the energy $E$ and angular momentum $L$,
where the third constant is taken as $\mathcal{H}=0$. Therefore, by virtue of
the expression $p_{\mu}=\partial_{x^{\mu}}S_{J}$, the Hamilton-Jacobi equation
can be expressed in the form

$g^{\mu\nu}\partial_{x^{\mu}}\mathcal{S_{J}}\partial_{x^{\nu}}\mathcal{S_{J}}=0$
(18)

with Jacobi action

$\mathcal{S_{J}}=-Et+L\phi+\mathcal{A}_{r}(r)+\mathcal{A}_{\theta}(\theta)$
(19)

with $\mathcal{A}_{r}(r)$ and $\mathcal{A}_{\theta}(\theta)$ as unknown
arbitrary functions. We obtain as a result of solving the Hamilton-Jacobi
equation,

$\Delta(r)\big{(}\partial_{r}\mathcal{A}_{r}(r)\big{)}^{2}-\frac{\Big{(}\mathcal{U}(r;a)E-aL\Big{)}^{2}}{\Delta(r)}+(L-aE)^{2}+\big{(}\partial_{\theta}\mathcal{A}_{\theta}(\theta)\big{)}^{2}+\big{(}L^{2}\csc^{2}\theta-a^{2}E^{2}\big{)}\cos^{2}\theta+\zeta^{2}\omega_{p}^{2}(r,\theta)=0.$
(20)

Since, the last term on the left hand side of Eq. (20) is mixed in $r$ and
$\theta$ coordinates, so we cannot separate this equation. For this, we assume
31 ; 71 ; 72

$\omega_{p}(r,\theta)=\frac{\sqrt{f_{r}(r)+f_{\theta}(\theta)}}{\zeta},$ (21)

where, $f_{r}(r)$ and $f_{\theta}(\theta)$ are arbitrary functions that
ensures the separability of the Eq. (20). Therefore, the separation gives the
following equations:

$\displaystyle\partial_{r}\mathcal{A}_{r}(r)$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{\Big{(}\mathcal{U}(r;a)E-aL\Big{)}^{2}}{\Delta(r)^{2}}-\frac{f_{r}(r)+(aE-L)^{2}+\mathcal{Z}}{\Delta(r)}\bigg{]}^{\frac{1}{2}},$
(22) $\displaystyle\partial_{\theta}\mathcal{A}_{\theta}(\theta)$
$\displaystyle=$
$\displaystyle\sqrt{\mathcal{Z}-\big{(}L^{2}\csc^{2}\theta-a^{2}E^{2}\big{)}\cos^{2}\theta-
f_{\theta}(\theta)},$ (23)

where, $\mathcal{Z}$ is the Carter constant 78  that is considered as the
fourth constant of motion. Simplifying further, the null geodesic equations
for the photon motion can be written as

$\displaystyle\zeta^{2}\dot{t}$ $\displaystyle=$ $\displaystyle
a\big{(}L-aE\sin^{2}{\theta}\big{)}+\frac{\mathcal{U}(r;a)}{\Delta(r)}\big{(}\mathcal{U}(r;a)E-aL\big{)},$
(24) $\displaystyle\zeta^{2}\dot{r}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\mathcal{R}(r)},$ (25)
$\displaystyle\zeta^{2}\dot{\theta}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\Theta(\theta)},$ (26)
$\displaystyle\zeta^{2}\dot{\phi}$ $\displaystyle=$
$\displaystyle\big{(}L\csc^{2}\theta-
aE\big{)}-\frac{a}{\Delta(r)}\big{(}aL-\mathcal{U}(r;a)E\big{)},$ (27)

where

$\displaystyle\mathcal{R}(r)$ $\displaystyle=$
$\displaystyle\big{(}aL-\mathcal{U}(r;a)E\big{)}^{2}-\Delta(r)\big{(}f_{r}(r)+\mathcal{Z}+(aE-L)^{2}\big{)},$
(28) $\displaystyle\Theta(\theta)$ $\displaystyle=$
$\displaystyle\mathcal{Z}+a^{2}E^{2}\cos^{2}\theta-L^{2}\cot^{2}\theta-
f_{\theta}(\theta).$ (29)

The constants of motion are renamed as $L=E\Gamma$ and
$\mathcal{Z}=E^{2}\Sigma$ and thus

$\displaystyle\mathcal{R}(r)$ $\displaystyle=$
$\displaystyle\big{(}\mathcal{U}(r;a)-a\Gamma\big{)}^{2}-\Delta(r)\big{(}\Sigma+(\Gamma-a)^{2}+f_{r}(r)\big{)}.$
(30)

The radial equation is related to the effective potential $V_{eff}(r)$ as
$\dot{r}^{2}+2V_{eff}(r)=0$. Then, the effective potential at the equator
$\Big{(}\theta=\frac{\pi}{2}\Big{)}$ becomes

$\displaystyle
V_{eff}(r)=-\frac{\mathcal{R}(r)}{2r^{4}}=-\frac{\big{(}\mathcal{U}(r;a)-a\Gamma\big{)}^{2}-\Delta(r)\big{(}\Sigma+(\Gamma-a)^{2}+f_{r}(r)\big{)}}{2r^{4}}.$
(31)

Our major aim is to determine the size and shape of the shadows that are
dependent upon the unstable circular light orbits in the neighborhood of the
BH. Since these orbits form a sphere, therefore, the photons revolving in
these orbits around the BH must stay at the surface of Euclidean sphere
described by the equation $r=A_{0}$, where $A_{0}$ is a constant. The circular
null orbits are described by $\dot{r}=0=\ddot{r}$ that implies
$V_{eff}(r_{p})=0=\partial_{r}V_{eff}(r_{p})$, where $r_{p}$ is the radius of
photon sphere. These conditions are connected to the function $\mathcal{R}(r)$
as $\mathcal{R}(r_{p})=0=\partial_{r}\mathcal{R}(r_{p})$. The unstable orbits
are described by the local maxima in effective potential curve and is
mathematically given by the condition $\partial^{2}_{r}V_{eff}(r_{p})<0$.
Next, we consider two subcases in which we assume different values of the
functions $f_{r}(r)$ and $f_{\theta}(\theta)$ to study the size of photon
sphere.

####  III.1.1 Case Ia

In this case, we consider 71

$\displaystyle f_{r}(r)=\omega_{c}^{2}\sqrt{M^{3}r},\qquad
f_{\theta}(\theta)=0,$ (32)

where, the constant $\omega_{c}$ has the same dimensions as the frequency. By
using the values of functions given in Eq. (32) into (31), we plotted the
effective potential $V_{eff}(r)$ vs $r$ in the upper panel of Fig. 2. In each
plot, we varied the values of $\omega_{c}$ for each curve with fixed $\gamma$,
$\lambda$ and $a$. The peaks observed in the effective potential curves
signify the positions of unstable circular orbits for light trajectories. By
increasing the value of $\omega_{c}$, there is no effect on the size of the BH
as $\omega_{c}$ is not a BH parameter. However, the size of the unstable
circular null orbits increased by a small amount. This is because the function
$f_{r}(r)$ appears in the relation (31) linearly such that
$f_{r}(r)\propto\omega_{c}^{2}$ and $f_{r}(r)\propto r$. Eventually, the Eqs.
(13) and (21) show that $N_{e}\propto\omega_{c}^{2}$ and $N_{e}\propto r$.
Therefore, the increase in $\omega_{c}$ causes increase in $N_{e}$ and thus
strengthening the plasma medium. Moreover, as $N_{e}\propto r$, so the number
of electrons are further increased raising the mass of the plasma field
because electrons are massive particles. This causes an extra increase in
gravity due to plasma by a small amount. Therefore, with increase in
$\omega_{c}$ causing the increase in gravity, the photons are captured
slightly away from the BH, raising the size of unstable null orbits.

####  III.1.2 Case Ib

In this case, we consider 71

$\displaystyle f_{r}(r)=0,\qquad
f_{\theta}(\theta)=\omega_{c}^{2}M^{2}\big{(}1+2\sin^{2}\theta\big{)}.$ (33)

Because of the symmetry on the equatorial plane, the effective potential does
not depend on $\theta$ and hence $f_{\theta}(\theta)$. Therefore, for this
case, the unstable circular null orbits and photon sphere behave as if there
is no plasma medium. This behavior of photon sphere is quite usual and has
been studied in detail in Ref. 79 .

###  III.2 Case II

Now, we consider the case in which the refractive index $n$ is a function of
the coordinate $r$ only and possesses no dependence on $\theta$ 73 .
Therefore, we can write $n=n(r)$ and thus the plasma frequency becomes

$\omega_{p}^{2}(r)=\big{(}n(r)^{2}-1\big{)}g^{tt}p_{t}^{2}.$ (34)

The Hamiltonian (11) for this case becomes

$\mathcal{H}=\frac{1}{2}\Big{[}g^{\mu\nu}p_{\mu}p_{\nu}+\big{(}n(r)^{2}-1\big{)}g^{tt}p_{t}^{2}\Big{]}.$
(35)

Solving for the $t$ and $\phi$ coordinates, we obtain

$\displaystyle\zeta^{2}\dot{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta(r)}\Big{[}n(r)^{2}\Big{(}\mathcal{U}(r;a)^{2}-a^{2}\Delta(r)\sin^{2}\theta\Big{)}E+a\big{(}\Delta(r)-\mathcal{U}(r;a)\big{)}L\Big{]},$
(36) $\displaystyle\zeta^{2}\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta(r)}\Big{[}\big{(}\Delta(r)\csc^{2}\theta-a^{2}\big{)}L-a\big{(}\Delta(r)-\mathcal{U}(r;a)\big{)}E\Big{]}.$
(37)

In the same way as in the case III.1, $E$ and $L$ are the two constants of
motion with $\mathcal{H}=0$ as the third constant of motion. Therefore, for
the fourth constant, we solve the Hamilton-Jacobi equation (18) with Jacobi
function (19) that comes out to be

$\displaystyle\Delta(r)\big{(}\partial_{r}\mathcal{A}_{r}(r)\big{)}^{2}-\frac{\Big{(}\mathcal{U}(r;a)E-aL\Big{)}^{2}}{\Delta(r)}+(L-aE)^{2}+\big{(}\partial_{\theta}\mathcal{A}_{\theta}(\theta)\big{)}^{2}+\big{(}L^{2}\csc^{2}\theta-a^{2}E^{2}\big{)}\cos^{2}\theta$
$\displaystyle-\frac{\big{(}n(r)^{2}-1\big{)}E^{2}\mathcal{U}(r;a)^{2}}{\Delta(r)}+\big{(}n(r)^{2}-1\big{)}a^{2}E^{2}\sin^{2}\theta=0.$
(38)

Since, the last term on the left hand side of Eq. (38) comprise
$n(r)^{2}a^{2}E^{2}\sin^{2}\theta$ which is mixed in $r$ and $\theta$
coordinates and is generally non-separable equation. For this, we assume that
the orientation of the system at the equatorial plane is slightly perturbed by
shifting off the plane by a fractionally small angle $\psi$ such that
$\theta=\frac{\pi}{2}+\psi$. It is known that the unstable circular orbits of
light are the part of photon sphere and can be found at any plane. Therefore,
a distant observer located at equatorial plane can observe the photon orbits
off the equatorial plane. The approximation $\theta=\frac{\pi}{2}+\psi$ is
valid because $\psi$ is very small and we may obtain the desirable results for
BH shadow. Hence, we place the observer at the equatorial plane. However, the
null geodesics will be modified under this assumption. By taking
$\theta=\frac{\pi}{2}+\psi$ in Eq. (38), we obtain

$\displaystyle(L-aE)^{2}-\frac{\Big{(}\mathcal{U}(r;a)E-aL\Big{)}^{2}}{\Delta(r)}+\Delta(r)\big{(}\partial_{r}\mathcal{A}_{r}(r)\big{)}^{2}+\big{(}\partial_{\psi}\mathcal{A}_{\psi}(\psi)\big{)}^{2}+\big{(}n(r)^{2}-1\big{)}a^{2}E^{2}$
$\displaystyle-\frac{\big{(}n(r)^{2}-1\big{)}E^{2}\mathcal{U}(r;a)^{2}}{\Delta(r)}=0,$
(39)

which is separated into:

$\displaystyle\Delta(r)\big{(}\partial_{r}\mathcal{A}_{r}(r)\big{)}^{2}$
$\displaystyle=$
$\displaystyle\frac{\Big{(}\mathcal{U}(r;a)E-aL\Big{)}^{2}}{\Delta(r)}-(L-aE)^{2}-\big{(}n(r)^{2}-1\big{)}a^{2}E^{2}+\frac{\big{(}n(r)^{2}-1\big{)}E^{2}\mathcal{U}(r;a)^{2}}{\Delta(r)}-\mathcal{Z},$
(40) $\displaystyle\big{(}\partial_{\psi}\mathcal{A}_{\psi}(\psi)\big{)}^{2}$
$\displaystyle=$ $\displaystyle\mathcal{Z},$ (41)

where, the Carter constant $\mathcal{Z}$ is considered as the fourth constant
of motion. Further simplifying, the null geodesic equations governing photon
motion can be expressed as

$\displaystyle r^{2}\dot{t}$ $\displaystyle=$ $\displaystyle
a\big{(}L-n(r)^{2}aE\cos^{2}\psi\big{)}+\frac{\mathcal{U}(r;a)}{\Delta(r)}\big{(}n(r)^{2}\mathcal{U}(r;a)E-aL\big{)},$
(42) $\displaystyle r^{2}\dot{r}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\mathcal{R}(r)},$ (43) $\displaystyle r^{2}\dot{\psi}$
$\displaystyle=$ $\displaystyle\pm\sqrt{\Theta(\psi)},$ (44) $\displaystyle
r^{2}\dot{\phi}$ $\displaystyle=$ $\displaystyle\big{(}L\sec^{2}\psi-
aE\big{)}-\frac{a}{\Delta(r)}\big{(}aL-\mathcal{U}(r;a)E\big{)},$ (45)

where

$\displaystyle\mathcal{R}(r)$ $\displaystyle=$
$\displaystyle\big{(}n(r)^{2}-1\big{)}\mathcal{U}(r;a)^{2}E^{2}+\big{(}\mathcal{U}(r;a)E-aL\big{)}^{2}-\Delta(r)\big{(}\mathcal{Z}+(L-aE)^{2}+\big{(}n(r)^{2}-1\big{)}a^{2}E^{2}\big{)},$
(46) $\displaystyle\Theta(\psi)$ $\displaystyle=$ $\displaystyle\mathcal{Z}.$
(47)

In terms of $\Gamma$ and $\Sigma$, the function $\mathcal{R}(r)$ in Eq. (46)
becomes

$\displaystyle\mathcal{R}(r)=\big{(}n(r)^{2}-1\big{)}\mathcal{U}(r;a)^{2}+\big{(}\mathcal{U}(r;a)-a\Gamma\big{)}^{2}-\Delta(r)\big{(}\Sigma+(\Gamma-a)^{2}+\big{(}n(r)^{2}-1\big{)}a^{2}\big{)}.$
(48)

From the radial equation, the effective potential $V_{eff}(r)$ near the
equatorial plane becomes

$\displaystyle
V_{eff}(r)=-\frac{\mathcal{R}(r)}{2r^{4}}=-\frac{\big{(}n(r)^{2}-1\big{)}\mathcal{U}(r;a)^{2}+\big{(}\mathcal{U}(r;a)-a\Gamma\big{)}^{2}-\Delta(r)\big{(}\Sigma+(\Gamma-a)^{2}+\big{(}n(r)^{2}-1\big{)}a^{2}\big{)}}{2r^{4}}.$
(49)

We suppose the function

$\displaystyle n(r)^{2}=1-\frac{k}{r},$ (50)

where $k$ is a constant that describes the strength of plasma around the BH.
Assuming this value of $n(r)$ in Eq. (49), we plot the effective potential
$V_{eff}(r)$ vs $r$ in the middle panel of Fig. 2. In both plots, we varied
the values of $k$ for each curve and kept $\gamma$, $\lambda$ and $a$ fixed.
By increasing the value of $k$, there is no effect on the size of the BH as it
defines the plasma medium only. However, the size of the unstable circular
null orbits decreased by a small amount, in converse to the upper panel. This
is because the function $n(r)^{2}-1=-\frac{k}{r}$ appears in the relation (49)
linearly such that $N_{e}\propto k$ and $N_{e}\propto\frac{1}{r}$. Therefore,
the increase in $k$ causes an increase in $N_{e}$ but the inverse relation
with $r$ affects the strength of the plasma medium. As a result, the increase
in mass of the plasma medium is affected by $N_{e}\propto\frac{1}{r}$, unlike
the case III.1.1. Thus, the gravity does not increase significantly due to
plasma itself. Hence, with the increase in $k$, the photons are captured
slightly closer to the BH, reducing the size of unstable null orbits.

###  III.3 Case III

In this case, we consider that the refractive index $n$ is a constant 73 .
Therefore, we may write

$\displaystyle n^{2}=1-k,$ (51)

where $k$ is the same parameter as in the case III.2. Since, the last term on
the left hand side of Eq. (38) comprise $n(r)^{2}a^{2}E^{2}\sin^{2}\theta$
which is mixed in $r$ and $\theta$ coordinates corresponding to the case
III.2. However, in this case, this term is only function of $\theta$ because
$n\neq n(r)$. Therefore, the Eq. (38) is separable. We obtain

$\displaystyle\Delta(r)\big{(}\partial_{r}\mathcal{A}_{r}(r)\big{)}^{2}$
$\displaystyle=$
$\displaystyle\frac{\Big{(}\mathcal{U}(r;a)E-aL\Big{)}^{2}}{\Delta(r)}-(L-aE)^{2}+\frac{\big{(}n(r)^{2}-1\big{)}E^{2}\mathcal{U}(r;a)^{2}}{\Delta(r)}-\mathcal{Z},$
(52)
$\displaystyle\big{(}\partial_{\theta}\mathcal{A}_{\theta}(\theta)\big{)}^{2}$
$\displaystyle=$
$\displaystyle\mathcal{Z}+a^{2}E^{2}\cos^{2}\theta-L^{2}\cot^{2}\theta-\big{(}n^{2}-1\big{)}a^{2}E^{2}\sin^{2}\theta.$
(53)

Therefore, following Eqs. (36) and (37), and simplifying further, the Eqs.
(52) and (53), the null geodesic equations for the photon motion can be
written as

$\displaystyle\zeta^{2}\dot{t}$ $\displaystyle=$ $\displaystyle
a\big{(}L-n^{2}aE\sin^{2}\theta\big{)}+\frac{\mathcal{U}(r;a)}{\Delta(r)}\big{(}n^{2}\mathcal{U}(r;a)E-aL\big{)},$
(54) $\displaystyle\zeta^{2}\dot{r}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\mathcal{R}(r)},$ (55)
$\displaystyle\zeta^{2}\dot{\theta}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\Theta(\theta)},$ (56)
$\displaystyle\zeta^{2}\dot{\phi}$ $\displaystyle=$
$\displaystyle\big{(}L\csc^{2}\theta-
aE\big{)}-\frac{a}{\Delta(r)}\big{(}aL-\mathcal{U}(r;a)E\big{)},$ (57)

where

$\displaystyle\mathcal{R}(r)$ $\displaystyle=$
$\displaystyle\big{(}n^{2}-1\big{)}\mathcal{U}(r;a)^{2}E^{2}+\big{(}\mathcal{U}(r;a)E-aL\big{)}^{2}-\Delta(r)\big{(}\mathcal{Z}+(L-aE)^{2}\big{)},$
(58) $\displaystyle\Theta(\theta)$ $\displaystyle=$
$\displaystyle\mathcal{Z}-\big{(}n^{2}-1\big{)}a^{2}E^{2}\sin^{2}\theta+a^{2}E^{2}\cos^{2}\theta-L^{2}\cot^{2}\theta.$
(59)

In terms of $\Gamma$ and $\Sigma$, the function $\mathcal{R}(r)$ in Eq. (58)
becomes

Figure 2: The effective potential $V_{eff}$ is sketched versus $r$ to analyze
the variation for different values of $a$, $\lambda$ and $\gamma$. The top,
middle and bottom panels correspond to the case III.1.1, III.2 and III.3,
respectively.

$\displaystyle\mathcal{R}(r)=\big{(}n^{2}-1\big{)}\mathcal{U}(r;a)^{2}+\big{(}\mathcal{U}(r;a)-a\Gamma\big{)}^{2}-\Delta(r)\big{(}\Sigma+(\Gamma-a)^{2}\big{)}.$
(60)

From the radial equation, the effective potential $V_{eff}(r)$ near the
equatorial plane becomes

$\displaystyle
V_{eff}(r)=-\frac{\mathcal{R}(r)}{2r^{4}}=-\frac{\big{(}n^{2}-1\big{)}\mathcal{U}(r;a)^{2}+\big{(}\mathcal{U}(r;a)-a\Gamma\big{)}^{2}-\Delta(r)\big{(}\Sigma+(\Gamma-a)^{2}\big{)}}{2r^{4}}.$
(61)

By using the given value of $n$ in Eq. (51) into (61), we plot the effective
potential $V_{eff}(r)$ vs $r$ in the lower panel of Fig. 2. In both plots, we
fixed the values of $\gamma$, $\lambda$ and $a$ as in the upper and middle
panels and varied $k$ for each curve. By increasing the value of $k$, there is
no effect on the size of the BH as it defines the plasma medium only. However,
the size of the unstable circular null orbits decreased by a small amount, as
in the case of middle panel. This is because the function $n^{2}-1=-k$ appears
in the relation (61) linearly such that $N_{e}\propto k$. Therefore, the
increase in $k$ causes an increase in $N_{e}$ but there is no relation with
$r$. As a result, the increase in mass of the plasma medium is not enhanced by
the relation $N_{e}\propto r$ or diminished by the relation
$N_{e}\propto\frac{1}{r}$. Hence, with the increase in $k$, the photons are
captured slightly closer to the BH by a very small amount, reducing the size
of unstable null orbits.

##  IV Shadows

This section comprise the optical images of the BH (8) immersed in plasma
medium corresponding to the cases discussed in the section III. Since, out of
all circular null orbits, the unstable orbits contain the photons that form
the shadow of the BH. The BH (8) constructed by Kumar et al. 30  was further
analyzed in the same paper for which they calculated the shadows as viewed
from the equator in the absence of plasma. However, in Ref. 79 , we studied
the shadows as viewed by an observer away from the equator varying different
BH parameters in non-plasma medium for the same BH (8) in this work. Hence, we
only calculate the shadows by varying the plasma parameters, keeping the other
BH parameters fixed. The 2D shadow image of a spatially 3D BH is projected in
terms of coordinates $\alpha$ and $\beta$ as

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\lim\limits_{r_{0}\rightarrow\infty}\bigg{(}r_{0}^{2}\sin\theta_{0}\bigg{[}\frac{d\phi}{dr}\bigg{]}_{\theta\rightarrow\theta_{0}}\bigg{)},$
(62) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\lim\limits_{r_{0}\rightarrow\infty}\bigg{(}r_{0}^{2}\bigg{[}\frac{d\theta}{dr}\bigg{]}_{\theta\rightarrow\theta_{0}}\bigg{)},$
(63)

where, the location of the observer is $(r_{0}\rightarrow\infty,\theta_{0})$.
The mathematical formulation for calculating the shadows is derived by
considering the circular photon orbits conditions for different cases of
plasma distributions.

###  IV.1 Case I

Using Eq. (30) in the conditions
$\mathcal{R}(r_{p})=0=\partial_{r}\mathcal{R}(r_{p})$, we get the values of
$\Gamma$ and $\Sigma$ given by

$\displaystyle\Gamma(r_{p})$ $\displaystyle=$
$\displaystyle\frac{\mathcal{U}(r;a)\Delta^{\prime}(r)-\Delta(r)\big{[}2r+\sqrt{4r^{2}-f_{r}^{\prime}(r)\Delta^{\prime}(r)}\big{]}}{a\Delta^{\prime}(r)}\bigg{|}_{r=r_{p}},$
(64) $\displaystyle\Sigma(r_{p})$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{1}{a^{2}\Delta^{\prime}(r)^{2}}\bigg{(}8r^{2}\Delta(r)\big{[}a^{2}-\Delta(r)\big{]}-\big{[}r^{4}+a^{2}f_{r}(r)\big{]}\Delta^{\prime}(r)^{2}$
(65)
$\displaystyle+\big{[}4r^{3}-\big{(}a^{2}-\Delta(r)\big{)}f_{r}^{\prime}(r)\big{]}\Delta(r)\Delta^{\prime}(r)+2r\Delta(r)\big{[}2\big{(}a^{2}-\Delta(r)\big{)}+r\Delta^{\prime}(r)\big{]}\sqrt{4r^{2}-f_{r}^{\prime}(r)\Delta^{\prime}(r)}\bigg{)}\bigg{]}\bigg{|}_{r=r_{p}}.$

Now, we consider two different combinations of $f_{r}(r)$ and
$f_{\theta}(\theta)$ as in Eqs. (32) and (33) to obtain the exact shadow
equations.

####  IV.1.1 Case Ia

We simplify the differentials $\frac{d\phi}{dr}$ and $\frac{d\theta}{dr}$ in
Eqs. (62) and (63) by using Eqs. (25)-(29) and (32). Then, by applying the
limits, we get

$\displaystyle\alpha(r_{p})$ $\displaystyle=$
$\displaystyle-\Gamma(r_{p})\csc\theta_{0},$ (66) $\displaystyle\beta(r_{p})$
$\displaystyle=$
$\displaystyle\pm\sqrt{\Sigma(r_{p})+a^{2}\cos^{2}\theta_{0}-\Gamma(r_{p})^{2}\cot^{2}\theta_{0}}.$
(67)

The null sphere around the rotating BH has a finite width with a variable
radial distance $r_{p}\in\big{[}r_{p,min},r_{p,max}\big{]}$, where $r_{p,min}$
and $r_{p,max}$ denote the minimum and maximum radii of the null sphere.
Therefore, $r_{p}$ behaves as a parameter in calculating shadows. The values
$r_{p,min}$ and $r_{p,max}$ are the real and positive roots of the equation
$\beta(r_{p})=0$. We consider the observer’s location
$\big{(}\infty,\frac{\pi}{2}\big{)}$, the celestial coordinates reduce to

$\displaystyle\alpha(r_{p})$ $\displaystyle=$ $\displaystyle-\Gamma(r_{p}),$
(68) $\displaystyle\beta(r_{p})$ $\displaystyle=$
$\displaystyle\pm\sqrt{\Sigma(r_{p})}.$ (69)

The shadows have been computed and graphed, holding values of $a$, $\lambda$
and $\gamma$ constant, while varying $\omega_{c}$ for each individual curve in
Fig. 3. Clearly, the shadow size decreases with increases in $\omega_{c}$.
Apparently, there is no difference in both plots. However, with the change in
$\gamma$ and $\lambda$, there is a small change in sizes of the shadows.
Moreover, we observe an obvious horizontal shift in the shadow due to the
increase in $a$. The increase in $\omega_{c}$ causes increase in $f_{r}(r)$
and thus $\omega_{p}^{2}$. Therefore, with increase in $\omega_{p}^{2}$, the
light slows down in plasma due to which the spiral paths formed in the domain
of outer communication become smaller. This causes the shadows to shrink.

####  IV.1.2 Case Ib

We evaluate the differentials $\frac{d\phi}{dr}$ and $\frac{d\theta}{dr}$ in
Eqs. (62) and (63) by using Eqs. (25)-(29) and (33). Then, by applying the
limits, we get

$\displaystyle\alpha(r_{p})$ $\displaystyle=$
$\displaystyle-\Gamma(r_{p})\csc\theta_{0},$ (70) $\displaystyle\beta(r_{p})$
$\displaystyle=$
$\displaystyle\pm\sqrt{\Sigma(r_{p})+a^{2}\cos^{2}\theta_{0}-\Gamma(r_{p})^{2}\cot^{2}\theta_{0}-\omega_{c}^{2}M^{2}\big{(}1+2\sin^{2}\theta_{0}\big{)}}.$
(71)

For the observer to be located at $\big{(}\infty,\frac{\pi}{2}\big{)}$, the
celestial coordinates become

$\displaystyle\alpha(r_{p})$ $\displaystyle=$ $\displaystyle-\Gamma(r_{p}),$
(72) $\displaystyle\beta(r_{p})$ $\displaystyle=$
$\displaystyle\pm\sqrt{\Sigma(r_{p})-3\omega_{c}^{2}M^{2}}.$ (73)

By varying the value of $\omega_{c}$ for each curve and fixing $a$, $\lambda$
and $\gamma$, we plotted the shadows in the Fig. 4. Clearly, the shadow
shrinks by increasing the value of $\omega_{c}$. The variation in shrinking of
shadow is higher than that in the previous subcase. There is a very small
difference in shadows in both plots with the change in $\gamma$ and $\lambda$.
Moreover, the increase in $a$ causes the shadow to shift towards the right. As
in the previous subcase, the shadows shrink due to the strengthening of the
plasma medium that affects the speed of light in plasma. As a result, the
spirals outside the photon sphere become smaller.

###  IV.2 Case II

Using Eq. (48) in the conditions
$\mathcal{R}(r_{p})=0=\partial_{r}\mathcal{R}(r_{p})$, we get

$\displaystyle\Gamma(r_{p})$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{1}{a\Delta^{\prime}(r)}\bigg{(}\mathcal{U}(r;a)\Delta^{\prime}(r)-2r\Delta(r)-\mathcal{T}\bigg{)}\bigg{]}\bigg{|}_{r=r_{p}},$
(74) $\displaystyle\Sigma(r_{p})$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{1}{a^{2}\Delta^{\prime}(r)^{2}}\bigg{(}8r^{2}\big{[}a^{2}-\Delta(r)\big{]}\Delta(r)+2\Delta^{\prime}(r)\Big{(}2r\Big{[}a^{2}\mathcal{U}(r;a)\big{(}n(r)^{2}-1\big{)}$
(75)
$\displaystyle+\big{[}a^{2}+2r^{2}-\mathcal{U}(r;a)n(r)^{2}\big{]}\Delta(r)\Big{]}+n(r)\big{(}a^{2}-\Delta(r)\big{)}n^{\prime}(r)\Big{[}\mathcal{U}(r;a)^{2}-a^{2}\Delta(r)\Big{]}\Big{)}$
$\displaystyle+r^{2}\Delta^{\prime}(r)^{2}\Big{(}\Big{(}2a^{2}+r^{2}\Big{)}n(r)^{2}-2\mathcal{U}(r;a)\Big{)}+2r\mathcal{T}\big{[}2\big{(}a^{2}-\Delta(r)\big{)}+r\Delta^{\prime}(r)\big{]}\bigg{)}\bigg{]}\bigg{|}_{r=r_{p}},$

where,

$\displaystyle\mathcal{T}$ $\displaystyle=$
$\displaystyle\Big{[}4r^{2}\Delta(r)^{2}+2\Delta(r)\Delta^{\prime}(r)\Big{(}2r\mathcal{U}(r;a)\big{(}n(r)^{2}-1\big{)}$
(76)
$\displaystyle+n(r)n^{\prime}(r)\big{[}\mathcal{U}(r;a)^{2}-a^{2}\Delta(r)\big{]}\Big{)}-\Delta^{\prime}(r)^{2}\mathcal{U}(r;a)^{2}\big{(}n(r)^{2}-1\big{)}\Big{]}^{\frac{1}{2}}.$

Evaluating the differentials $\frac{d\phi}{dr}$ and $\frac{d\theta}{dr}$ in
Eqs. (62) and (63) by using Eqs. (43)-(47) and inserting the value of the
function $n(r)^{2}=1-\frac{k}{r}$ and solving the limits, we get

$\displaystyle\alpha(r_{p})$ $\displaystyle=$ $\displaystyle-\Gamma(r_{p}),$
(77) $\displaystyle\beta(r_{p})$ $\displaystyle=$
$\displaystyle\pm\sqrt{\Sigma(r_{p})}.$ (78)

This case directly corresponds to the null geodesics in a near equatorial
plane observed by an observer located at the equatorial plane. In the Fig. 5,
we have plotted the shadows by varying the parameter $k$ for each curve and
kept $a$, $\lambda$ and $\gamma$ fixed. According to the figure, the shadows
shrink with increases in $k$. In this case, a small variation in $k$ shows an
enormous variation in the shadow size. Whereas, we had to consider a larger
variation in $\omega_{c}$ for a visible change in the shadow size in the
previous cases given in Figs. 3 and 4. Therefore, the parameter $k$ is more
sensitive than $\omega_{c}$. For higher values of $a$, $\gamma$ and $\lambda$,
the flatness in the shadow increases with increase in $k$.

Figure 3: Plots showing the behavior of shadows for different values of
$\omega_{c}$ and fixed $a$, $\lambda$ and $\gamma$ corresponding to the case
IV.1.1.

Figure 4: The behavior of shadows for different values of $\omega_{c}$ and
fixed $a$, $\lambda$ and $\gamma$ corresponding to the case IV.1.2.

Figure 5: Plots illustrating the variation of shadows for different values of
$k$ and fixed $a$, $\lambda$ and $\gamma$ corresponding to the case IV.2.

Figure 6: Shadow plots for various values of $k$ and fixed $a$, $\lambda$ and
$\gamma$ corresponding to the case IV.3.

###  IV.3 Case III

Using Eq. (60) in the conditions
$\mathcal{R}(r_{p})=0=\partial_{r}\mathcal{R}(r_{p})$, we get

$\displaystyle\Gamma(r_{p})$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{1}{a\Delta^{\prime}(r)}\bigg{(}\mathcal{U}(r;a)\Delta^{\prime}(r)-2r\Delta(r)-\mathcal{P}\bigg{)}\bigg{]}\bigg{|}_{r=r_{p}},$
(79) $\displaystyle\Sigma(r_{p})$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{1}{a^{2}\Delta^{\prime}(r)^{2}}\bigg{(}r^{4}\Delta^{\prime}(r)^{2}\big{[}n^{2}-2\big{]}+4r\Delta(r)\Big{(}2r\big{[}a^{2}-\Delta(r)\big{]}-\Delta^{\prime}(r)\big{[}a^{2}\big{(}n^{2}-1\big{)}+r^{2}\big{(}n^{2}-2\big{)}\big{]}\Big{)}$
(80)
$\displaystyle+2r\Big{(}2\big{[}a^{2}-\Delta(r)\big{]}+r\Delta^{\prime}(r)\Big{)}\mathcal{P}+a^{2}\big{(}n^{2}-1\big{)}\Delta^{\prime}(r)\Big{[}2r^{2}\big{(}2r+\Delta^{\prime}(r)\big{)}+a^{2}\big{(}4r+\Delta^{\prime}(r)\big{)}\Big{]}\bigg{)}\bigg{]}\bigg{|}_{r=r_{p}},$

where,

$\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle\sqrt{4r^{2}\Delta(r)^{2}+\Delta^{\prime}(r)\big{(}n^{2}-1\big{)}\mathcal{U}(r;a)\big{[}4r\Delta(r)-\mathcal{U}(r;a)\Delta^{\prime}(r)\big{]}}.$
(81)

The differentials $\frac{d\phi}{dr}$ and $\frac{d\theta}{dr}$ in Eqs. (62) and
(63) are simplified by using Eqs. (55)-(59) and (51). Then solving the limits,
we get

$\displaystyle\alpha(r_{p})$ $\displaystyle=$
$\displaystyle-\frac{\Gamma(r_{p})\csc\theta_{0}}{n},$ (82)
$\displaystyle\beta(r_{p})$ $\displaystyle=$
$\displaystyle\pm\frac{\sqrt{\Sigma(r_{p})+a^{2}\big{(}1-n^{2}\sin^{2}\theta_{0}\big{)}-\Gamma(r_{p})^{2}\cot^{2}\theta_{0}}}{n}.$
(83)

Whereas, for the equatorial observer,

$\displaystyle\alpha(r_{p})$ $\displaystyle=$
$\displaystyle-\frac{\Gamma(r_{p})}{n},$ (84) $\displaystyle\beta(r_{p})$
$\displaystyle=$
$\displaystyle\pm\frac{\sqrt{\Sigma(r_{p})-a^{2}\big{(}n^{2}-1\big{)}}}{n}.$
(85)

We have plotted the shadows in the Fig. 6 by varying $k$ representing each
curve and fixing the values of $a$, $\lambda$ and $\gamma$. With smaller
values assigned to the BH parameters, an increase in $k$ results in minimal
changes in the size of the shadow. Whereas, for higher values of BH
parameters, the shape and size of the shadow show a significant variation. The
parameter $k$ behaves differently in this case as compared to the one in the
case IV.2. As for the higher values of $a$, $\gamma$ and $\lambda$, the shape
of the shadows changes significantly with increase in $k$.

##  V Distortion and Energy Emission Rate

The rotating BH shadow results carry some useful information apart from its
shape and size. We apply these shadow results in determining the deviation in
the shape of shadow from a circular shadow image of the static BH, termed as
distortion. This is because the rotating BH shadow is flattened on one side.
We also apply the shadow results in estimating the energy emission rate that
describes evaporation rate of the BH.

The shadow radius 80 ; 81  is given as

$R_{sh}=\frac{D_{c}^{2}+\beta_{t}^{2}}{2|D_{c}|},$ (86)

where, $D_{c}=\alpha_{t}-\alpha_{r}$. It is required in order to define the
distortion. We consider that the points $(\alpha_{t},\beta_{t})$,
$(\alpha_{b},\beta_{b})$ and $(\alpha_{r},0)$ of the shadow coincides with the
imaginary circle. The indices $b$, $r$ and $t$ correspond to the bottom, right
and top most points on the shadow, respectively. One can see the Fig. 9 in 81
for a detailed view of the image describing these points. The relation (86)
does not apply in determining the radius of static BH shadows. The distortion
is mathematically expressed as

$\delta_{s}=\frac{|\tilde{D}_{c}|}{R_{sh}},$ (87)

where, $\tilde{D}_{c}=\bar{\alpha}_{l}-\alpha_{l}$. The point $(\alpha_{l},0)$
lies on the shadow and $(\bar{\alpha}_{l},0)$ lies on the hypothetical circle.
The index $l$ represents the points on the shadow on the left side of the
vertical axis, whereas the index $\bar{l}$ represents the points on the
imaginary circle on the left side of the vertical axis.

Classically, an object or a particle cannot escape out, once it falls in a BH.
However, quantum mechanically, a BH emits energy in the form of radiation. The
quantum fluctuations in a BH creates or annihilates particles, generating an
enormous pressure caused by the rise in density, resulting in a substantial
quantity of energy. Hence, the particles that possess positive energy, escape
out of the BH due to a process known as quantum tunneling. Hence, the BH
evaporates as the energy is released carried by the particles. It is measured
as an absorption process called absorption cross section which is a measure of
the probability. The BH shadow can be useful in determining the absorption
cross section at high energies. The geometric area of the shadow is
approximately equal to $\sigma_{lim}\approx\pi R_{sh}^{2}$. The energy
emission rate is written as 82 ; 83 ; 84

$\mathcal{E}_{\omega
t}:=\frac{d^{2}\mathcal{E}(\omega)}{dtd\omega}=\frac{2\pi^{2}\omega^{3}\sigma_{lim}}{e^{\frac{\omega}{T_{H}}}-1}\approx\frac{\pi^{3}\omega^{3}}{2\big{(}e^{\frac{\omega}{T_{H}}}-1\big{)}}\frac{\big{(}(\alpha_{t}-\alpha_{r})^{2}+\beta_{t}^{2}\big{)}^{2}}{|\alpha_{t}-\alpha_{r}|^{2}}.$
(88)

In above Eq. (88), the parameter $\omega$ denotes the photon’s angular
frequency and $T_{H}=\frac{\kappa}{2\pi}$ defines the Hawking temperature. The
relation

$\kappa(r_{h})=\lim\limits_{\theta=0,r\rightarrow
r_{h}}\frac{\partial_{r}\sqrt{g_{tt}}}{\sqrt{g_{rr}}}$ (89)

determines the surface gravity at the event horizon of a rotating BH 85 . For
the metric (8) the surface gravity reduces to

$\kappa(r_{h})=\frac{\Delta^{\prime}(r_{h})}{2(r_{h}^{2}+a^{2})}$ (90)

for the rotating metric. By using Eqs. (87) and (88), we plot the behavior of
distortion and BH evaporation rate for the plasma cases as discussed in the
previous sections. It is believed that the KR field is a quantum field that
has some obvious connections with String Theory. In this regard, such a
quantum field may have a great impact on a BH in KR gravity with various
underlying quantum effects. We can possibly encounter such quantum effects in
studying the energy emission rate that corresponds to the BH evaporation rate.
It has already been mentioned that the quantum tunneling process enables the
particles carrying energy to escape away from a BH and we are intended to
measure this release of energy in this work. Inside a BH, due to sufficiently
high pressure and density, many quantum processes take place during the
creation and annihilation of particles. This is one reason to believe that a
BH in KR gravity is highly preferable to study the quantum aspects of
spacetime and BHs. We also know that a plasma medium is a super hot ionized
gaseous matter made up of particles that are ions. Due to high temperature of
the medium and therefore the pressure, there may arise various quantum
processes because of electrons splitting away from the nuclei. In this work,
we are studying the BH in KR gravity that is immersed in plasma medium.
Therefore, it is most likely that the quantum processes take place in the
vicinity of the BH and it might fascinate the people working in quantum
gravity and quantum field theory in curved spacetimes.

###  V.1 Case I

This case corresponds to the plasma distribution with frequency given by Eq.
(21). For the functions $f_{r}(r)$ and $f_{\theta}(\theta)$, we further divide
this case into two subsections.

####  V.1.1 Case Ia

For the case when $f_{r}(r)$ and $f_{\theta}(\theta)$ are given by the Eq.
(32), the upper right plot in the Fig. 7 depicts the variation in the
distortion. The smaller values of the BH parameters generate a constant
behavior of the distortion as the plasma parameter $\omega_{c}$ grows larger.
When we increase the values of the BH parameters, the distortion decreases as
the value of $\omega_{c}$ increases. Moreover, for the higher values of the BH
parameters, the distortion is significantly higher for a fixed value of
$\omega_{c}$. The energy emission rate for this case is plotted in the top
panel of Fig. 8. It is found that for the smaller values of the BH parameters,
the energy emission rate and hence the BH evaporation rate decreases with
increase in $\omega_{c}$. Similar behavior of the BH evaporation rate has been
observed when the values of the BH parameters are increased. However, for the
fixed values of $\omega_{c}$, the BH evaporation slows down with increase in
the BH parameters.

Figure 7: The distortion behavior for different values of plasma parameters
and fixed $a$, $\lambda$ and $\gamma$.

Figure 8: The graphical variation in the energy emission rate for different
values of plasma parameters and fixed $a$, $\lambda$ and $\gamma$.

####  V.1.2 Case Ib

Considering the values of $f_{r}(r)$ and $f_{\theta}(\theta)$ as given by the
Eq. (33), we plotted the distortion in the Fig. 7 as indicated by the
corresponding label. We found that the smaller values of the BH parameters
give the a constant behavior of the distortion as we increase $\omega_{c}$.
When we increase the values of the BH parameters, the distortion decreases as
the value of $\omega_{c}$ increases. Moreover, for the higher values of the BH
parameters, the distortion is significantly higher for a fixed value of
$\omega_{c}$. This shows that there is no significant change in the distortion
behavior as compared to that in the case V.1.1. The energy emission rate for
this case is plotted in the second panel of Fig. 8. We found that for the
smaller values of the BH parameters, the BH evaporation rate decreases with
increase in $\omega_{c}$. By increasing the values of the BH parameters, a
similar kind of behavior of the BH evaporation rate has been observed.
However, for the fixed values of $\omega_{c}$, the BH evaporation slows down
with increase in the BH parameters. It also shows that in terms of BH
evaporation, $\omega_{c}$ is more sensitive in this case as compared to the
case V.1.1.

###  V.2 Case II

This case corresponds to the plasma distribution defined in terms of
refractive index as given in Eq. (50). We plotted the distortion in the lower
left plot in Fig. 7 as indicated by the corresponding label. We can see that
the distortion increases at a very slow rate with respect to increase in the
value of $k$ and for the smaller values of the BH parameters. When we
increased the values of the BH parameters, the distortion increases rapidly as
the value of $k$ increases. Moreover, for a fixed value of $k$, the distortion
is significantly higher for the higher values of the BH parameters. This shows
a converse behavior of distortion as compared to the previous two subcases
V.1.1 and V.1.2. The energy emission rate for this case is plotted in the
third panel of Fig. 8. The plots show that for the smaller values of the BH
parameters, the BH evaporation slows down with increase in $k$. By increasing
the values of the BH parameters, the variation in the BH evaporation rate is
same as for the smaller values of the BH parameters. However, for the fixed
values of $k$, the BH evaporation is significantly delayed with increase in
the BH parameters.

###  V.3 Case III

For this case, the plasma distribution is defined as the constant value of
refractive index given as in Eq. (51). We plotted the distortion in the Fig. 7
as indicated by the corresponding label. We can see that for when BH
parameters are kept smaller, there is a slow elevation in distortion with
increase in $k$. When the values of the BH parameters are elevated, a rapid
growth in the distortion is observed as the value of $k$ grows larger.
Moreover, when the value of $k$ is kept fixed, the distortion is significantly
higher for the higher values of the BH parameters. This shows an identical
behavior of distortion as compared to the case V.2. The energy emission rate
for this case is plotted in the bottom panel of Fig. 8. As the BH parameters
take on smaller values in the left plot, the rate of BH evaporation remains
constant with the ascending values of $k$. Elevating the BH parameters leads
to a minor fluctuation in the rate of BH evaporation. However, this variation
shows that with increase in the value of $k$, the peak rises and the BH
evaporation rate increases. This is an entirely different behavior of the BH
evaporation rate among all of the discussed cases.

##  VI Constraints on Black Hole Parameters

Considering the BH in plasma medium, we will establish the bounds on the BH
parameters $\gamma$, $a$ and $\lambda$ in this section by utilizing the data
acquired through the observations at EHT for M87* and Sgr A*. Therefore, we
establish a comparative analysis of the angular radii of the shadows of BH
defined by the metric(8) in plasma with M87* and Sgr A*. The BH parameters are
said to be constrained when the BH shadow falls within 1-$\sigma$ uncertainty.
Corresponding to these limits on the parameters, the BH (8) immersed in plasma
is considered as M87* or Sgr A*. This work exclusively focuses on rotating BH
because supermassive BHs, by nature, exhibit rotational characteristics. It
ensures a feasible and rigorous comparative study. A coordinate-independent
formalism, vaguely known as Kumar-Ghosh method 86 ; 87 , is employed in which
the shadow area $A_{sh}$ will be used that is defined as

$\displaystyle A_{Sh}$ $\displaystyle=$ $\displaystyle
2\int_{r_{-}}^{r_{+}}dx^{r}\beta(r)\partial_{r}\alpha(r).$ (91)

The values $r_{+}$ and $r_{-}$ represent the size of the retrograde and
prograde stable circular orbits as measured from the origin, respectively. Let
us consider, the BH and the observer are separated by the linear distance $d$,
then the diameter of the BH shadow is measured by using the relation for arc
length as follows 88 ; 89

$\displaystyle\theta_{d}=\frac{2R_{A_{sh}}}{d},$ (92)

where $R_{A_{sh}}=\sqrt{\frac{A_{sh}}{\pi}}$ is the shadow radius. By
employing the relations (91) and (92), the angular diameter of the BH shadow
is expressed as a function involving the BH parameters and $\theta_{0}$. For
the comparison of the shadows, we need to know the distance $d$ of Earth from
M87* and Sgr A*, the mass $M$ and the shadow size $\theta_{d}$ of M87* and Sgr
A*. For M87*, we get 53 ; 57 ; 58

$\displaystyle d=16.8Mpc,\qquad M=6.5\times
10^{9}M_{\odot},\qquad\theta_{d}=42\pm 3\mu as,$ (93)

where $M_{\odot}$ denotes the solar mass. For simplicity, we have disregarded
uncertainties in the measurements of mass and distance. For Sgr A*, we get 59
; 90

$\displaystyle d=8kpc,\qquad M=4\times
10^{6}M_{\odot},\qquad\theta_{d}=48.7\pm 7\mu as.$ (94)

Ultimately, the angular diameter of the shadow of the BH (8) can be
calculated. This calculated value can then be compared with the angular
diameters of Sgr A* and M87* to assess constraints on the BH parameters. As we
know that the largest amount of matter in the visible universe is plasma in
the form of stars, nebulas and auroras etc. Therefore, due to strong gravity
near a BH, it is quite certain that a plasma medium exists around a BH. Such a
plasma medium is significantly crucial and influential on physics, especially
the particle motion around the BH and various quantum aspects. Under the said
comparative study, we mainly estimate the influence of plasma on the
parametric bounds. From this analysis, we can deduce that what distribution of
plasma and under what conditions it is likely to be present around the BH, if
the BH (8) immersed in plasma is one of M87* and Sgr A*. Therefore, the
results obtained under this analysis might be useful for people working in
observational astronomy and astroparticle physics, especially those who are
interested in BH imaging. Note that the EHT observations were conducted at the
inclination angles at $17^{\circ}$ for M87* and $<50^{\circ}$ for Sgr A*.
Therefore, we perform the calculations for the shadows by considering the
inclination angles of $17^{\circ}$ for M87* and $30^{\circ}$ for Sgr A*.
Moreover, the black solid curves correspond to the values $\theta_{d}=42\mu
as$ for M87* and $\theta_{d}=48.7\mu as$ for Sgr A*. Whereas, the dashed and
dotted curves represent the boundaries of 1-$\sigma$ and 2-$\sigma$
uncertainty intervals.

###  VI.1 Case I

Considering the plasma distribution with frequency given by Eq. (21) and
further assuming the two subcases for the functions $f_{r}(r)$ and
$f_{\theta}(\theta)$, we determined the angular diameters of the BH shadows in
terms of the parametric spaces.

####  VI.1.1 Case Ia

The density plots depicted in the Fig. 9 illustrate the angular diameter
across parametric spaces for M87*. These plots correspond to the specific case
outlined by the Eq. (32). The top panel corresponds to the parametric space
$a$-$\gamma$ with $\lambda=0.6$ and two different values of $\omega_{c}$. In
the left plot, we can find that for the given values of $\lambda$ and
$\omega_{c}$, the limits of $a$ and $\gamma$ are shown by the dashed curve in
the parametric space. For the specified parametric values below the dashed
curve, the size of the shadow falls within the 1-$\sigma$ error level.
However, in the right plot, it becomes evident that elevating the value of
$\omega_{c}$ results in the dotted curve delineating the boundaries for both
$a$ and $\gamma$. This indicates that the shadow size is confined inside the
2-$\sigma$ uncertainty level. Therefore, the BH (8) immersed in plasma medium
can be considered as M87* for the smaller value of $\omega_{c}$ and the
corresponding parametric values under the dashed curve. The panel at the
middle illustrates the parametric space $a$-$\lambda$ with $\gamma=0.4$ and
two different values of $\omega_{c}$. In the left plot, the parametric space
illustrates the constraints on $a$ and $\lambda$ for a given value of $\gamma$
and $\omega_{c}$, represented by the dashed and dotted curves. The size of the
shadow falls within the 1-$\sigma$ error level for all values of the
parameters situated below the dashed curve. Meanwhile, for parametric values
below the dotted curve, the size of the shadow is within the 2-$\sigma$
uncertainty interval. The right plot shows that by increasing the value of
$\omega_{c}$, the dotted curve determines the limits on $a$ and $\lambda$
which means that the shadow size is bounded within 2-$\sigma$ error level.
Therefore, the BH (8) in plasma background can be considered as M87* for the
smaller value of $\omega_{c}$ and the corresponding parametric values bounded
by the dashed curve. The lower panel corresponds to the parametric space
$\gamma$-$\lambda$ with $a=0.5$ and two different values of $\omega_{c}$. In
both plots, we find that for the given value of $a$ and $\omega_{c}$, the
limits of $\gamma$ and $\lambda$ are shown by the dotted curves in the
parametric space. The size of the shadow falls within the 2-$\sigma$
confidence level for all specified values of the parameters located below the
dotted curves. Therefore, we exclude the consideration of the BH (8) in plasma
background as M87* for both values of $\omega_{c}$ and the corresponding
parametric values.

In the graphical representation denoted by Fig. 10, the plots illustrate the
angular diameter across various parametric spaces, specifically pertaining to
Sgr A*, subjected to the case specified by Eq. (32). The top panel corresponds
to parametric space $a$-$\gamma$ with $\lambda=0.6$ and two different values
of $\omega_{c}$. In both plots, we can find that for the given values of
$\lambda$ and $\omega_{c}$, the angular diameter falls inside 1-$\sigma$
uncertainty level for all values of $a$ and $\gamma$ for which the density
plot exists. Therefore, the BH (8) immersed in plasma medium can be considered
as Sgr A* for both values of $\omega_{c}$ and all of the corresponding
parametric values for which the density plot exists. The middle panel
corresponds to parametric space $a$-$\lambda$ with $\gamma=0.4$ and two
different values of $\omega_{c}$. We found that for both values of
$\omega_{c}$, the angular diameter falls inside 1-$\sigma$ level of error for
each value of $a$ and $\lambda$ for which the density plot exists. Therefore,
the BH (8) in plasma background can be considered as Sgr A* for both values of
$\omega_{c}$ and all of the corresponding parametric values for which the
density plot exists as in the upper panel. The lower panel corresponds to the
parametric space $\gamma$-$\lambda$ with $a=0.5$ and two different values of
$\omega_{c}$. For the given value of $a$ and $\omega_{c}$, both plots show
that the angular diameter lies in 1-$\sigma$ level of uncertainty for every
value of $\gamma$ and $\lambda$ for which the density plot exists. Therefore,
the BH (8) in plasma background can be considered as Sgr A* for both values of
$\omega_{c}$ and all of the corresponding parametric values for which the
density plot exists.

Figure 9: The density plot depicting the variation of angular size of the
shadow of M87* in terms of parametric spaces for various values of plasma
parameters for the case described by Eq. (32).

Figure 10: Plots showing the variation of angular size of the shadow of Sgr A*
in terms of parametric spaces for various values of plasma parameters for the
case described by Eq. (32).

Figure 11: The variation of angular size of the shadow of M87* in terms of
parametric spaces for various values of plasma parameters for the case
described by Eq. (33).

Figure 12: The behavior of angular size of the shadow of Sgr A* in terms of
parametric spaces for various values of plasma parameters for the case
described by Eq. (33).

Figure 13: Plots showing the behavior of angular size of the shadow of M87* in
terms of parametric spaces for various values of plasma parameters for the
case described by Eq. (51).

Figure 14: The density plots showing the angular size of the shadow of Sgr A*
in terms of parametric spaces for various values of plasma parameters for the
case described by Eq. (51).

####  VI.1.2 Case Ib

The parametric spaces in Fig. 11 correspond to the plots for angular diameter
of M87* for the case described by the Eq. (33). The top panel corresponds to
parametric space $a$-$\gamma$ with $\lambda=0.6$ and two different values of
$\omega_{c}$. The left plot is associated with a lower $\omega_{c}$ value and
given the $\lambda$ value, it demonstrates that for all parametric values
situated below the dashed curve, the shadow size falls within the 1-$\sigma$
error interval. Nevertheless, in the right plot, it becomes apparent that with
an increase in the $\omega_{c}$ value, the parametric values confined by the
dotted curve indicate that the angular diameter of the shadow falls within the
2-$\sigma$ level. Consequently, we consider the BH (8) immersed in plasma
medium as M87* for only the smaller value of $\omega_{c}$ together with the
corresponding parametric values under the dashed curve. It is also obvious
that the interval of $a$ and $\gamma$ bounded by the dashed curve is larger
for this case as compared to that in the Fig. 9. The panel at the middle of
the figure is associated to the parametric space $a$-$\lambda$ with
$\gamma=0.4$ and two different values of $\omega_{c}$. The left plot shows
that for the given values of $\gamma$ and $\omega_{c}$, the intervals of $a$
and $\lambda$ are bounded by the dashed and dotted curves in the parametric
space. On the other hand, in the right plot, elevating the value of $\omega$
reveals that solely the dotted curve delineates the limits on both $a$ and
$\lambda$, ensuring that the shadow size resides within the 2-$\sigma$
interval. In this way, the BH (8) in plasma background is considered as M87*
only for the smaller value of $\omega_{c}$ corresponding to parametric values
under the dashed curve. It is also observed that for this case, the interval
of $a$ and $\lambda$ bounded by the dashed curve is larger as compared to that
in the Fig. 9. The lower panel corresponds to parametric space
$\gamma$-$\lambda$ with $a=0.5$ and two different values of $\omega_{c}$.
Corresponding to the given values of $a$ and $\omega_{c}$ in the left plot,
the limits of $\gamma$ and $\lambda$ are shown by the dashed and dotted curves
in the parametric space. However, in the right plot, it can be seen that by
increasing the value of $\omega_{c}$, the bounds on $\gamma$ and $\lambda$
lies only within 2-$\sigma$ interval. Therefore, we consider the BH (8) in
plasma background as M87* for the smaller value of $\omega_{c}$ corresponding
to the parametric values under the dashed curve. For the corresponding case in
Fig. 9, we did not find any bound on the BH parameters.

Corresponding to the Eq. (33), the parametric spaces in Fig. 12 show the
angular diameter for Sgr A*. The parametric space $a$-$\gamma$ with
$\lambda=0.6$ and two different values of $\omega_{c}$ describe the top panel.
In both plots, for the given values of $\lambda$ and $\omega_{c}$, the angular
diameter falls inside 1-$\sigma$ level of error for every value of $a$ and
$\gamma$ for which the density plot exists. Thus, the BH (8) immersed in
plasma medium can be considered as Sgr A* for both values of $\omega_{c}$ and
all of the corresponding parametric values for which the density plots exist.
The middle panel corresponds to parametric space $a$-$\lambda$ with
$\gamma=0.4$ and two different values of $\omega_{c}$ for which the angular
diameter resides inside the error interval 1-$\sigma$ for all values of $a$
and $\lambda$ for which the density plots exist. Likewise, in the lower panel
in which the parametric space $\gamma$-$\lambda$ with $a=0.5$ and two
different values of $\omega_{c}$, the angular diameter is confined inside
error interval 1-$\sigma$ for every value of $\gamma$ and $\lambda$ for which
the density plots exist. Therefore, the BH (8) immersed in plasma can be
considered as Sgr A* for both values of $\omega_{c}$ and all of the
corresponding parametric values for which the density plots exist in middle
and lower panels.

###  VI.2 Case II

To construct a comparison considering the plasma distribution with refractive
index given by Eq. (51), we determined the angular diameters of the BH shadows
in terms of the parametric spaces as in the previous case.

The parametric spaces in Fig. 13 show the angular diameter for M87* and the
bounds on the parameters. The top panel corresponds to the parametric space
$a$-$\gamma$ with $\lambda=0.6$ and two different values of $k$. It is found
that for the given values of $\lambda$ and $k$ in both plots, the limits of
$a$ and $\gamma$ are shown by the dashed curve in the parametric space
corresponding to which the size of shadow lies within the 1-$\sigma$ level of
error. Therefore, for all these parametric values, the BH (8) immersed in
plasma is considered to be M87*. The panel lying at the middle refers to the
parametric space $a$-$\lambda$ with $\gamma=0.4$ and two different values of
$k$. In both plots, we can find that for the given value of $\gamma$ and $k$,
the limits of $a$ and $\lambda$ are shown by the dashed and dotted curves in
the parametric spaces. The 1-$\sigma$ error level encompasses the shadow size
for all parametric values positioned beneath the dashed curves. Therefore, the
BH (8) in plasma background can be considered as M87* for both values of $k$
and the corresponding parametric values under the dashed curve. The lower
panel corresponds to parametric space $\gamma$-$\lambda$ with $a=0.5$ and two
different values of $k$. The plots in this panel exhibit an exactly identical
behavior as compared to the middle panel. Therefore, the spinning BH (8) in
plasma background can be considered as M87* for both values of $k$ and the
corresponding parametric values under the dashed curve. There is no
significant variation in the plots with the change in the value of $k$ in all
panels. Moreover, the intervals of the BH parameters bounded by the dashed
curves in this case are larger than those in the all previous cases for M87*.

As a final case, we discuss the density plots in Fig. 14 that delineates the
angular diameter through the parametric spaces for Sgr A*. The top, middle and
lower panels corresponds to parametric spaces $a$-$\gamma$, $a$-$\lambda$ and
$\lambda$-$\gamma$ with fixed values of the third BH parameter, respectively
and two different values of $k$ in each column. In each plot, we can find that
the angular diameter falls inside 1-$\sigma$ error level for all respective
parametric values. Therefore, we determine the bounds on the parameters of the
BH and for both values of $k$ from which we can deduce that the BH (8)
immersed in plasma medium can be regarded as the analog of Sgr A*.

##  VII Conclusion

In this work, we mainly emphasized on the impact of different descriptions of
plasma fields on the light trajectories and optical images of the BH (8). A
meticulous discussion is presented on the effect of various plasma parameters
affect describing the corresponding plasma medium. We summarize as:

  * •

The BH (8) reduces to Kerr and Kerr-Newman BHs for $\lambda=0$ and
$\lambda=1$, respectively. The event horizon goes down by elevating both
$\lambda$ and $\gamma$ with consistent value of the other parameter. With this
increase in parameters, the extremal value of $a$ also decreases.

  * •

The unstable null orbits are not affected by $\theta$ coordinate because the
relation (31) depends only on $r$. For the case III.1.1, the size of unstable
null orbits is elevated with raise in $\omega_{c}$. Whereas in the following
two cases, the size of unstable orbits of light decrease with elevation in
$k$. This shows an entirely different behavior of the plasma media on the
unstable orbits.

  * •

The two subcases in IV.1 reveal that the shadow shrinks for both subcases with
a rise in $\omega_{c}$. However, for the subcase IV.1.2, the shrinking rate of
the shadow is much more than the other subcase. This shows that $\omega_{c}$
is more sensitive for the case IV.1.2 than for the case IV.1.1. In the cases
IV.2 and IV.3, the parameter $k$ has quite unusual behavior. For the case
IV.2, the shadow shrinks significantly with increase in $k$. Whereas, in the
case IV.3, the shadow size remains consistent, instead the distortion is
highly increased for the larger values of the BH parameters.

  * •

The distortion is diminished with increase in $\omega_{c}$ for the cases V.1.1
and V.1.2. Whereas, it increases rapidly with increase in $k$ for the cases
V.2 and V.3. Therefore we may infer that the parameter $k$ for both respective
cases supports the deviation of shadows from circularity than $\omega_{c}$.

  * •

In analogy with the shadows, the BH evaporation slows down with increase in
$\omega_{c}$ for both respective cases. However, for the second case, the
parameter $\omega_{c}$ is more sensitive. Moreover, for the other cases
described by the parameter $k$, the first of the two cases shows that the BH
evaporation decelerates with elevating value of $k$. However, for the other
case, there is almost no BH evaporation.

  * •

A rigorous analysis shows that the BH (8) is more probable to be considered as
Sgr A* rather than M87* because the parametric bounds for Sgr A* are obtained
with larger intervals than for M87*. Moreover, both smaller and larger values
of plasma parameters conform to the findings by the EHT collaboration for Sgr
A*. However, for many cases, the larger value of plasma parameter did not
support the EHT data for M87*. We also propose that if there exists a plasma
medium around the BH described by the metric (8) that is considered as Sgr A*,
then this plasma medium is described the Eq. (32). It is because for this
case, the angular diameters of the shadow are more convergent in 1-$\sigma$
interval.

We conclude that all of the plasma fields have a significant impact on the
unstable orbits, shadows and BH evaporation rate. The spinning BH in KR
gravity can be considered as Sgr A* instead of M87*. Among all plasma fields
considered in this work, it is more likely that the plasma field described by
the Eq. (32) is surrounding Sgr A*. However, by looking at the variation in
shadows, distortion and BH evaporate rate, we may infer that the parameter
$\omega_{c}$ described by the Eq. (33) is more sensitive among all cases.
Following this attempt, numerous potential research directions may be pursued
in the future such as the observation and experimental verification of the
type of the plasma medium present around Sgr A*. It can also be investigated
whether our prediction on the BH being not considered as M87* is true or
false. For the people working in areas of quantum aspects of gravity and high
energy astroparticle physics, it can be useful to consider the KR field
together with a plasma medium to discover and comprehend the subject.

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