# Shadow of Kottler black hole in the presence of plasma for a co-moving
observer

Anish Das<EMAIL_ADDRESS>Ashis Saha<EMAIL_ADDRESS>Department of Physics, University of Kalyani, Kalyani 741235, India Sunandan
Gangopadhyay<EMAIL_ADDRESS>
###### Abstract

In this paper, we discussed the shadow of a static, spherically symmetric
black hole geometry in presence of a positive cosmological constant
$\Lambda>0$. We analysed the black hole shadow both in absence and presence of
plasma. Then we study the black hole shadow from the point of view of a
comoving observer. We display the plots for the angular shadow size as
measured by a comoving observer. The effects of the cosmological constant
$\Lambda$ and plasma parameter $k$ on the angular size of the black hole
shadow have been investigated in detail. Finally, by using the observed
angular size of M87∗ and Sgr A∗ black hole, we constrain the value of the
plasma parameter $k$ with a specific observational value of the cosmological
constant $\Lambda$ (or the Hubble constant $H_{0}$).

##  1 Introduction

Black holes are one of the most interesting predictions of general theory of
relativity. Recent detection of black hole mergers by LIGO and Virgo in 2015
[1] and the images of black hole shadow published by the EHT in 2019 [2] and
2021 [3] significantly triggered research in this direction. Here we take
interest in studying the black hole shadow, which was initiated by the works
of Synge [4]. Synge calculated the escape cones of light in case of
Schwarzschild black hole. The escape cones basically reflect the angular
portion where light can escape. In other words, they complement the black hole
shadow, as we know it. The shadow for the spherically symmetric Schwarzschild
black hole is circular. In general as far as we know most of the astrophysical
objects rotate. So the general case is to consider a rotating black hole, the
simplest of which is the Kerr black hole. The shadow in case of Kerr black
hole was calculated by Bardeen in [5]. In this case, the shadow is found to be
deformed rather than circular due to presence of rotation. The reason being
light gets dragged around the black hole due to rotation. Bardeen calculated
the shadow from the point of view of an observer placed at infinity. In
general the observer resides far away from the black hole, so the
approximation of the observer being at infinity is quite logical. However
while we need to compare our analytically computed results with the
observations (like EHT data), we need to place the observer at a finite
distance from the black hole. The computation of black hole shadow for a
finite distant observer was first performed in [6] and later in [7] where they
consider the presence of plasma. Further the black hole shadows were
calculated by incorporating additional fields and hairs [8]-[19] and with
modified gravity theories in [20]-[38]. Also, there are some recent
interesting works in this direction which introduces new techniques to
calculate photon spheres and black hole shadows [39]-[41]. Besides, there is a
recent review work on analytical studies of black hole shadows [42].
It is a well-known fact that the universe is expanding and so any two objects
in the universe are moving apart from one another. Although for nearby objects
the effect is either less pronounced or superseded by gravitational
attraction, still the effect persists. So from theoretical perspective
inclusion of the cosmic expansion in the context of black hole shadow is quite
important. The reason is that this would address the question of how does the
black hole shadow depend on time. Furthermore, it can be readily mentioned
that the size of any object increases due to cosmological expansion [43],
[44]. Inspired by this we consider a system which includes the cosmological
expansion. The criteria that this system needs to satisfy is that it must be
Schwarzschild like at nearby regions and for far away regions, it must
reproduce the FLRW metric. The first of such model was considered by Einstein
and Strauss [45]. In this model, one calculates the geodesics in different
regions and match them at a specific radius, namely the Schücking radius [46].
On the other hand, the McVittie metric [47] efficiently interpolates between
the two mentioned regions. It satisfies the criteria of being Schwarzschild
like in small distances and FLRW like at far distance. However the geodesics
are not completely integrable analytically. Thus we must search for metrics
where there exists sufficient constants of motion and the equations become
completely integrable. A special case is that of Kottler or Schwarzschild de-
Sitter metric as given in [48]. The Kottler metric is a solution of Einstein
equation with a positive cosmological constant $\Lambda>0$. Here the system
basically describes a Schwarzschild black hole being embedded in a de-Sitter
universe. In this case, the geodesics are completely integrable and thus we
can proceed to perform analytical computations. Besides a general
transformation as proposed in [49] can transform the Kottler metric to the
FLRW metric where the expansion is driven by the cosmological constant
$\Lambda$. Some recent works related to the case of expanding universe can be
found in [50]-[57].
Another thing to be noted is that the black hole is in general surrounded by
material media. The material around a black hole is at a very high temperature
due to the immense gravitational field of the black hole. The nature of this
material medium suggests that one can treat it like a highly dense plasma.
Besides, the importance of considering a plasma medium lies in the fact that
in a practical scenario, black holes are surrounded by a dispersive matter
medium. Keeping this motivation in mind several studies have been done
[58]-[61]. This plasma medium can either be magnetic or non-magnetic. The
analysis of magnetic plasma is complicated analytically. So for the sake of
simplicity, we restrict ourselves to non-magnetised plasma where the material
is dust like with pressure $P=0$. Numerous works on black hole shadow
considering plasma can be found in literature. We list a few of them here
[62]-[76]. In our analysis, we aim to study black hole shadow in presence of
plasma for a static and co-moving observer. We would like to study the impact
of cosmological constant $\Lambda$ as well as the plasma parameter $k$ on the
photon sphere as well as black hole shadow from the point of view of a co-
moving observer moving away from the black hole due to cosmic expansion. We
try to constraint the value of the plasma parameter on the basis of the
observational results of M87∗ and Sgr A∗ black holes. Despite of the fact that
the supermassive black holes are rotating still the comparison of the data
with a non-rotating black hole can give us crude results which will help us
get some insight into the plasma medium around the black hole.

The paper is organised as follows. In section 1, we gave an introduction and
overview of the works on black hole shadow and reason for indulging in this
work. In section 2, we discuss black hole shadow in presence of plasma. In
section 3, we study the shadow of black hole with cosmological constant in
presence of plasma. Then, in section 4, we discuss the black hole shadow from
the point of view of a co-moving observer. Then, in section 5, we compare our
results with observation in order to constrain the plasma parameter. Finally,
we conclude in section 6. We also have an Appendix. We use geometric units for
our calculation that is $c=G=\hbar=1$.

##  2 Black hole shadow in presence of plasma

We start with a brief review of black hole shadow in the presence of plasma.
The metric of a static, spherically symmetric black hole spacetime in
$(3+1)$-dimensions is given by

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\leavevmode\nobreak\ .$ (1)

To approach the problem, we first need to analyse the spacetime itself. The
lapse function goes to zero at the event horizon $r=r_{h+}$, that is,
$\Big{(}f(r)\Big{|}_{r=r_{h+}}=0\Big{)}$. This depends upon the spacetime
parameters. The observer resides outside the event horizon. The black hole
shadow boundary is formed by the photons that encircle the black hole. To be
precise, the boundary is formed by light rays that are unstable which can
either fall into the black hole or escape to infinity upon encountering any
small radial perturbation. In order to determine the black hole shadow, we
need to evaluate the geodesics of light rays by using a Lagrangian
($\mathcal{L}$) or Hamiltonian ($\mathcal{H}$). In general due to spherical
symmetry, all planes are identical, so we wish to fix the plane of choice as
$\theta=\frac{\pi}{2}$. Thus the geodesic for $\theta$ becomes
$\theta=\frac{\pi}{2}=$constant or $\dot{\theta}=0$. Next we move on to
calculate the other geodesics for $t$, $\phi$ and $r$. For this we use the
Hamilton’s equations of motion

$\dot{x}^{\mu}=\frac{\partial\mathcal{H}}{\partial
p_{\mu}}\leavevmode\nobreak\ \leavevmode\nobreak\ ;\leavevmode\nobreak\
\leavevmode\nobreak\ \dot{p}_{\mu}=-\frac{\partial\mathcal{H}}{\partial
x^{\mu}}\leavevmode\nobreak\ .$ (2)

We want to continue our discussion further by considering plasma. The reason
being incorporation of plasma will make the case realistic. The plasma medium
can be magnetised or non-magnetised. Consideration of magnetised plasma will
resist us from performing an analytical study and we have to take help of
numerical methods. So in order to study the system analytically we consider
cold, dust-like ($P=0$) and non-magnetised plasma medium which corresponds to
a Hamiltonian ($\mathcal{H}$) of the following form [77]

$\mathcal{H}=\frac{1}{2}\Bigg{[}g^{\mu\nu}p_{\mu}p_{\nu}+\omega_{p}^{2}\Bigg{]}\leavevmode\nobreak\
$ (3)

where $\omega_{p}$ represents the electron plasma frequency. The refractive
index ($n$) of the plasma medium depends on the plasma frequency $\omega_{p}$
as well as the photon frequency ($\omega$), as measured by any arbitrary
observer and the relation between them stands to be [64]

$n^{2}=1-\Bigg{(}\frac{\omega_{p}}{\omega}\Bigg{)}^{2}\leavevmode\nobreak\
\leavevmode\nobreak\ ;\leavevmode\nobreak\ \leavevmode\nobreak\
\omega=\frac{\omega_{0}}{\sqrt{-g_{00}}}$ (4)

where the second equation gives the gravitational redshift relation with
$\omega_{0}$ being the frequency of the photons as measured by a stationary
observer at infinity. By setting $n=1$, we get back the case of spherically
symmetric spacetime devoid of plasma. Since the metric given in eq.(1) is
diagonal so its inverse can be computed easily. We can write

$\omega=\frac{\omega_{0}}{\sqrt{-g_{00}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\hbar\omega=\frac{\hbar\omega_{0}}{\sqrt{-g_{00}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\hbar\omega=\frac{E_{0}}{\sqrt{-g_{00}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\hbar\omega=\frac{p_{0}}{\sqrt{-g_{00}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\hbar\omega=p_{0}\sqrt{-g^{00}}\leavevmode\nobreak\ \leavevmode\nobreak\ .$
(5)

For $\hbar=1$, we have $\omega=p_{0}\sqrt{-g^{00}}$. Using this relation and
eq.(4) in eq.(3), we have

$\mathcal{H}=\frac{1}{2}\Bigg{[}g^{\mu\nu}p_{\mu}p_{\nu}+(n^{2}-1)g^{00}p_{0}^{2}\Bigg{]}\leavevmode\nobreak\
.$ (6)

The Hamiltonian can be recast as

$\mathcal{H}=\frac{1}{2}\Bigg{[}n^{2}g^{00}p_{0}^{2}+g^{ij}p_{i}p_{j}\Bigg{]}$
(7)

where $i,j$ runs from 1 to 3. The metric and hence the Hamiltonian
$\mathcal{H}$ is independent of $t$ and $\phi$. Also by using the Hamilton’s
equation of motion $\dot{p}_{\mu}=-\frac{\partial\mathcal{H}}{\partial
x^{\mu}}$ and using the fact that $\omega_{p}=\omega_{p}(r)$, we get the
constants of motion as $p_{0}=$constant$=-E$ and $p_{3}=$constant$=L$. Here
$E$ and $L$ correspond to the energy and angular momentum of photons as
measured by an observer stationed at infinity. Using the Hamilton’s equations
of motion given in eq.(2) and by restricting ourselves to the equatorial
plane, we get the geodesics for $t$, $\phi$ and $r$ as

$\frac{dt}{d\lambda}=\frac{n^{2}E}{f(r)}$ (8)
$\frac{d\phi}{d\lambda}=\frac{L}{r^{2}}$ (9)
$\Big{(}\frac{dr}{d\lambda}\Big{)}^{2}=n^{2}E^{2}-\frac{L^{2}}{r^{2}}f(r)\leavevmode\nobreak\
.$ (10)

Here $\lambda$ corresponds to the affine parameter. Taking the ratio of
eq.(s)(9) and (10), we have

$\Big{(}\frac{dr}{d\phi}\Big{)}^{2}=r^{4}\Bigg{[}\frac{n^{2}E^{2}}{L^{2}}-\frac{f(r)}{r^{2}}\Bigg{]}\leavevmode\nobreak\
.$ (11)

The solution of the above equation gives the trajectory of photons as
$r=r(\phi)$ in presence of plasma. Eq.(11) can be used to evaluate the
unstable photon orbits by imposing the conditions
$\frac{dr}{d\phi}=\frac{d^{2}r}{d\phi^{2}}=0$.

The first condition constraints the constants of motion ($E$, $L$) in terms of
the photon sphere radius $r_{p}$ and the second condition helps to determine
the photon sphere radius $r_{p}$ in terms of spacetime parameters. The
conditions take the following form

$\displaystyle\frac{n^{2}r^{2}}{f(r)}\Bigg{|}_{r=r_{p}}$ $\displaystyle=$
$\displaystyle\frac{L^{2}}{E^{2}}$ (12)
$\displaystyle\Big{(}2n^{\prime}rf(r)+2nf(r)-nrf^{\prime}(r)\Big{)}\Bigg{|}_{r=r_{p}}$
$\displaystyle=$ $\displaystyle 0\leavevmode\nobreak\ .$ (13)

Figure 1:  Pictorial representation of the event horizon, photon sphere and
the black hole shadow along with the trajectory of light ray.

The second equation gives a fixed value of $r_{p}$ for fixed values of
spacetime parameters and the refractive index $n$ thereby giving a constant
value for $\frac{L^{2}}{E^{2}}$. So we see that along a photon trajectory
$\frac{L^{2}}{E^{2}}$ is constant.
Next we wish to calculate the angular size of the black hole shadow. A ray of
light from the background source upon getting deflected due to the immense
gravitational pull of the black hole, makes an angle $\alpha$ with respect to
the observer’s position $r_{0}$ as given in Fig.1. The Figure shows a dark
disk which represents the black hole being surrounded by photon sphere (shown
in dotted red colour) situated at the distance $r=r_{p}$. Light from photon
sphere travels in curved trajectory and reaches the observer at an angle
$\alpha$ where the tangent gives the measured position of the light ray. If we
consider all rays at the boundary of the cone of angle $\alpha$, it forms the
shadow boundary with radius $R_{s}$ greater than $r_{p}$, that is
$R_{s}>r_{p}>r_{h+}$.

In the above, we have assumed the plasma distribution to have only radial
dependence ($n=n(r)$) and $n^{\prime}=\frac{dn}{dr}$. This is due to the
presence of spherical symmetry. In order to carry out calculations explicitly,
we need to assume a certain form of the plasma frequency ($\omega_{p}$) and
thereby refractive index ($n$). Keeping this in mind, we consider the
following form for the plasma frequency111It is to be noted that as we are
working with an axially symmetric black hole it is justified to assume the
form of $\omega_{p}$ as $\omega_{p}^{2}=\frac{f(r)}{r^{h}}$. However, for a
rotating black hole one needs to consider the form of $\omega_{p}$ as
$\omega_{p}^{2}=\frac{f(r)+f(\theta)}{r^{2}+a^{2}\cos\theta}$ as shown in
[75]. [64]

$\omega_{p}(r)^{2}=\frac{4\pi e^{2}}{m_{e}}N(r)\leavevmode\nobreak\
\leavevmode\nobreak\ ;\leavevmode\nobreak\ \leavevmode\nobreak\
N(r)=\frac{N_{0}}{r^{h}}$ (14)

where $e$ and $m_{e}$ represents the electronic charge and mass, $N(r)$ gives
the number density of electrons in the plasma medium and $N_{0}$ is a
constant. Substitution of the above relation in the expression of refractive
index $n$ in eq.(4) gives

$n(r)^{2}=1-f(r)\frac{k}{r^{h}}\leavevmode\nobreak\ \leavevmode\nobreak\
;\leavevmode\nobreak\ \leavevmode\nobreak\ k=\frac{4\pi
e^{2}}{m_{e}\omega_{0}^{2}}N_{0}\leavevmode\nobreak\ .$ (15)

Figure 2:  Light travelling into past from observer ($r_{0}$) making angle
$\alpha$.

Next we try to evaluate the angular size of black hole shadow in presence of
plasma. In Fig.2 we find that the ray of light makes an angle $\alpha$ with
the radial line that satisfies

$\tan\alpha=\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}\leavevmode\nobreak\
.$ (16)

From our metric we find that in the desired limit, the angular size becomes

$\tan\alpha=\sqrt{\frac{f(r)}{r^{2}\frac{n(r)^{2}E^{2}}{L^{2}}-f(r)}}\Bigg{|}_{r=r_{o}}$
(17)

where $r_{o}$ is the position of the observer. The angular size of black hole
shadow in terms of $\sin\alpha$ takes the form

$\sin^{2}\alpha_{sp}=\frac{f(r)}{r^{2}\frac{n(r)^{2}E^{2}}{L^{2}}}\Bigg{|}_{r=r_{o}}=\frac{z(r_{p})^{2}}{z(r_{0})^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ ;\leavevmode\nobreak\ \leavevmode\nobreak\
z(r)^{2}=\frac{n(r)^{2}r^{2}}{f(r)}\leavevmode\nobreak\ .$ (18)

##  3 Shadow of black hole with a positive cosmological constant in plasma

We now consider the presence of a positive cosmological constant $\Lambda$.
The positive cosmological constant ($\Lambda>0$) introduces an additional
horizon, namely, the cosmological horizon ($r_{C}$). The cosmological horizon
constraints the observers existence within a certain region from the event
horizon $r_{h+}$ to cosmological horizon $r_{C}$. Also, analysing the lapse
function $f(r)$ we can see that the position of the event horizon is shifted
outward. The interesting thing we can observe is that the photon sphere radius
$r_{p}$ remains unchanged implying no effect of cosmological constant
$\Lambda$ [50]. However the angular shadow size not only depends on the photon
sphere radius $r_{p}$ but also explicitly depends on the specification of the
geometry (the lapse function) of the black hole which in turn depends on the
cosmological constant $\Lambda$. This observation can be verified from
eq.(18).

Now we wish to study the Schwarzschild de-Sitter metric or the Kottler metric.
The metric is given as

$ds^{2}=-\Big{(}1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}\Big{)}dt^{2}+\frac{dr^{2}}{\Big{(}1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}\Big{)}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}$ (19)

with the cosmological constant $\Lambda$ related to the Hubble’s constant
$H_{0}$ as $\frac{\Lambda}{3}=\frac{H_{0}^{2}}{c^{2}}$. From the lapse
function $f(r)$ we observe that the horizons lie in the ranges $2M<r_{h+}<3M$,
$3M<r_{C}<\infty$ and $\bar{r}<0$, where $r_{h+}$, $r_{C}$ and $\bar{r}$
respectively give the event horizon, cosmological horizon and the additional
horizon which is unphysical. Also setting $f(r)=0$ provides us a cubic
equation of the form

$r^{3}-\frac{3}{\Lambda}r+\frac{6M}{\Lambda}=0$ (20)

which constraints $\Lambda$ as $0<\Lambda<\frac{1}{9M^{2}}$. The geodesics are
evaluated at the equatorial plane ($\theta=\frac{\pi}{2}$) by incorporating
plasma. The geodesics for $t$, $\phi$ and $r$ take the following forms

$\frac{dt}{d\lambda}=\frac{n^{2}E}{f(r)}=\frac{n^{2}E}{\Big{(}1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}\Big{)}}$
(21) $\frac{d\phi}{d\lambda}=\frac{L}{r^{2}}$ (22)
$\Big{(}\frac{dr}{d\lambda}\Big{)}^{2}=n^{2}E^{2}-\frac{L^{2}}{r^{2}}f(r)=n^{2}E^{2}-\frac{L^{2}}{r^{2}}\Big{(}1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}\Big{)}\leavevmode\nobreak\
.$ (23)

The condition for unstable circular null geodesics gives the photon sphere
radius $r_{p}$ and also it gives a condition on the constants $E,L$ and
$\Lambda$ as

$\frac{L^{2}}{E^{2}}=\frac{n^{2}r^{2}}{f(r)}\Bigg{|}_{r=r_{p}}$ (24)

which simplifies to

$\frac{E^{2}}{L^{2}}=\frac{1}{27M^{2}}-\frac{\Lambda}{3}$ (25)

for $n=1$ as was found in [50]. The angular size of the black hole shadow in
presence of plasma takes the form

$\sin^{2}\tilde{\alpha}_{sp}=\frac{z(r_{p})^{2}}{z(r_{0})^{2}}=\frac{n(r_{p})^{2}}{n(r_{0})^{2}}\frac{r_{p}^{2}}{r_{0}^{2}}\frac{\Big{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Big{)}}{\Big{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Big{)}}.$
(26)

The general equation for deriving the photon sphere radius $r_{p}$ for static
spherically symmetric metric with cosmological constant $\Lambda$ and plasma
medium with refractive index $n$ is given as

$\Big{(}2n^{\prime}(r)rf(r)+2n(r)f(r)-n(r)rf^{\prime}(r)\Big{)}\Bigg{|}_{r=r_{p}}=0\leavevmode\nobreak\
.$ (27)

Let us solve try to solve the equation for different cases.

Case I
Here we consider Schwarzschild de Sitter metric or Kottler metric without
plasma. The lapse function is of the form

$f(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}$ (28)

and the corresponding derivative is

$f^{\prime}(r)=\frac{df}{dr}=\frac{2M}{r^{2}}-\frac{2\Lambda}{3}r\leavevmode\nobreak\
.$ (29)

Since there is no plasma, hence $n(r)=1$ and thereby $n^{\prime}(r)=0$.
Replacing in eq.(27), we get

$2\Big{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Big{)}-r_{p}\Big{(}\frac{2M}{r_{p}^{2}}-\frac{2\Lambda}{3}r_{p}\Big{)}=0\Rightarrow
r_{p}=3M.$ (30)

So we find that the presence of cosmological constant $\Lambda$ in the metric
does not effect the photon sphere radius $r_{p}$ in absence of plasma. The
corresponding angular shadow size takes the form

$\sin^{2}\tilde{\alpha}_{s}=\frac{r_{p}^{2}}{r_{0}^{2}}\frac{\Big{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Big{)}}{\Big{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Big{)}}=\frac{\Big{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Big{)}}{r_{0}^{2}\Big{(}\frac{1}{27M^{2}}-\frac{\Lambda}{3}\Big{)}}\leavevmode\nobreak\
.$ (31)

Case II

Here we consider Schwarzschild de Sitter or Kottler metric with homogeneous
plasma. The refractive index of homogeneous plasma is given as [59], [67]

$n(r)=\sqrt{1-kf(r)}$ (32)

and the corresponding derivative

$n^{\prime}(r)=-\frac{1}{\sqrt{1-kf(r)}}k\Bigg{(}\frac{M}{r^{2}}-\frac{\Lambda}{3}r^{2}\Bigg{)}\leavevmode\nobreak\
.$ (33)

Replacing in eq.(27), we get after simplifying

$\frac{k\Lambda^{2}}{9}r_{p}^{6}-\frac{2}{3}\Lambda kr_{p}^{4}+\frac{4k\Lambda
M}{3}r_{p}^{3}-(1-k)r_{p}^{2}-(4k-3)Mr_{p}+4kM^{2}=0.$ (34)

The above equation is of sixth order. The solution gives the photon sphere
radius $r_{p}$ which is a function of $k$ and $\Lambda$. In the limit of $k\to
0$, we get back the photon sphere radius $r_{p}$ as 3M. The angular shadow
size takes the form

$\sin^{2}\tilde{\alpha}_{sp}=\frac{n(r_{p})^{2}}{n(r_{0})^{2}}\frac{r_{p}^{2}}{r_{0}^{2}}\frac{\Big{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Big{)}}{\Big{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Big{)}}=\frac{\Bigg{(}1-k\Bigg{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Bigg{)}\Bigg{)}}{\Bigg{(}1-k\Bigg{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Bigg{)}\Bigg{)}}\frac{r_{p}^{2}}{r_{0}^{2}}\frac{\Big{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Big{)}}{\Big{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Big{)}}.$
(35)

Case III

Here we consider Schwarzschild de Sitter or Kottler metric with inhomogeneous
plasma. The refractive index of inhomogeneous plasma is given as [59], [67]

$n(r)=\sqrt{1-\frac{k}{r}f(r)}$ (36)

and the corresponding derivative

$n^{\prime}(r)=\frac{1}{2\sqrt{1-\frac{k}{r}f(r)}}\Bigg{(}\frac{k}{r^{2}}-\frac{2kM}{r^{3}}-\frac{\Lambda
k}{3}-\frac{2kM}{r^{3}}+\frac{2}{3}k\Lambda\Bigg{)}\leavevmode\nobreak\ .$
(37)

Replacing in eq.(27), we get after simplifying

$\frac{k\Lambda^{2}}{9}r_{p}^{6}-\frac{2}{3}\Lambda
kr_{p}^{4}+\Bigg{(}\frac{4k\Lambda
M}{3}-2\Bigg{)}r_{p}^{3}+(6M+k)r_{p}^{2}-4kr_{p}M+4kM^{2}=0.$ (38)

The above equation is of sixth order. The solution gives the photon sphere
radius $r_{p}$ which is a function of $k$ and $\Lambda$. In the limit of $k\to
0$, we get back the photon sphere radius $r_{p}$ as 3M. The angular shadow
size takes the form

$\sin^{2}\tilde{\alpha}_{sp}=\frac{n(r_{p})^{2}}{n(r_{0})^{2}}\frac{r_{p}^{2}}{r_{0}^{2}}\frac{\Big{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Big{)}}{\Big{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Big{)}}=\frac{\Bigg{(}1-\frac{k}{r_{p}}\Bigg{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Bigg{)}\Bigg{)}}{\Bigg{(}1-\frac{k}{r_{0}}\Bigg{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Bigg{)}\Bigg{)}}\frac{r_{p}^{2}}{r_{0}^{2}}\frac{\Big{(}1-\frac{2M}{r_{0}}-\frac{\Lambda}{3}r_{0}^{2}\Big{)}}{\Big{(}1-\frac{2M}{r_{p}}-\frac{\Lambda}{3}r_{p}^{2}\Big{)}}.$
(39)

(a) Variation of photon sphere radius $\frac{r_{p}}{M}$ with restpect to the
plasma parameter $k$ (homogeneous plasma). The plots are shown for different
values of cosmological constant $\Lambda M^{2}$ as 0.0075 (black), 0.0300
(red) and 0.0675 (blue).  (b) Variation of photon sphere radius
$\frac{r_{p}}{M}$ with respect to the plasma parameter $\frac{k}{M}$
(inhomogeneous plasma). The plots are shown for different values of
cosmological constant $\Lambda M^{2}$ as 0.0075 (black), 0.0300 (red) and
0.0675 (blue).

Figure 3: Variation of photon sphere radius $\frac{r_{p}}{M}$ with plasma
parameter.

In Fig.3, we show the variation of the photon sphere radius $r_{p}$ with
respect to the plasma parameter $k$. The left plot is for homogeneous plasma
and the right one is for inhomogeneous plasma. In both the cases, we find that
the photon sphere radius $r_{p}$ increases with increase in plasma parameter
though the effect is more pronounced in case of homogeneous plasma. Further,
we have chosen three different values of the cosmological constant $\Lambda$
for both of the plots. We find that with increase in the value of the
cosmological constant, the value of the photon sphere radius decreases for a
fixed value of $k$. Also, we find that all the plots start from the same
common point at $k=0$ implying the fact that in absence of plasma, the
cosmological constant has no effect on the photon sphere radius $r_{p}$. This
observation can also be verified from eq.(s) (34) and (38).

(a) Angular size of the shadow ($\alpha_{stat}$) of the Schwarzschild black
hole, with respect to the position of a static observer ($r_{0}$).  (b)
Angular size of the shadow ($\alpha_{stat}$) of the Schwarzschild de-Sitter
black hole, with respect to the position of a static observer ($r_{0}$). We
have set the value of positive cosmological constant $\Lambda M^{2}=0.03$.

Figure 4: Variation in angular size of the shadow with respect to the position
of the observer $r_{0}$. The plots are shown for different values of the
plasma parameter $\frac{k}{M}$ as 0.0 (black), 0.2 (violet) and 0.4 (red).

(a) Angular size of the shadow ($\alpha_{stat}$) with respect to the position
of a static observer ($r_{0}$) for Schwarzschild black hole (zoomed).  (b)
Angular size of the shadow ($\alpha_{stat}$) with respect to the position of a
static observer ($r_{0}$) for Schwarzschild de-Sitter black hole with $\Lambda
M^{2}=0.03$ (zoomed).

Figure 5: Variation in angular size of the shadow with respect to the position
of the observer $r_{0}$. The plots are shown for different values of plasma
parameter $\frac{k}{M}$ as 0.0 (black), 0.2 (violet) and 0.4 (red).

In Fig.4, we show the variation of the angular size ($\alpha$) of the black
hole shadow. The plots are shown for different values of inhomogeneous plasma
parameter $\frac{k}{M}$. We calculated the angular shadow size in terms of
$\sin\alpha$.

As the observer moves from infinity (in case of an asymptotically flat black
hole) or from the cosmological horizon $r_{C}$ (in case of an asymptotically
de-Sitter black hole) towards the black hole, the angular shadow size starts
to increase from 0 to $\frac{\pi}{2}$ and then falls back to zero. However, to
plot the angular shadow in the range $\alpha\in[0,\pi]$, we shall take
$\alpha=\sin^{-1}Q$ in the range $r_{p}<r_{0}<r_{C}$ and
$\alpha=\pi-\sin^{-1}Q$ in the range $r_{h}<r_{0}<r_{p}$ [50]. Here $Q$
corresponds to the value of $\sin\alpha$ as a function of radial distance
$r_{0}$.
The left plot in Fig.4 is for Schwarzschild black hole where we found that the
position of the observer can extend upto infinity. On the other hand, the
right plot in Fig.4 shows the variation of the angular shadow size with
observer distance $r_{0}$ in case of a Schwarzschild black hole embedded in a
de-Sitter universe. Due to this embedding, the observer can reach upto a
certain limiting radius, that is the cosmological horizon $r_{0}=r_{C}$. In
this case, the shadow size is zero at the cosmological horizon $r_{C}$ and
increases to $\frac{\pi}{2}$ at the photon sphere $r_{p}$. After that as the
observer comes closer to the black hole, the shadow size grows and becomes
$\pi$ at the event horizon.

The plots in Fig.5 are the zoomed plots of the plots in Fig.4. They display
the variation of the angular shadow size with variation in inhomogeneous
plasma parameter $\frac{k}{M}$ as observed by a distant observer. We find that
the shadow size decreases with increase in plasma parameter $\frac{k}{M}$.
Also we mention that the plots are shown for plasma frequency $\omega_{p}^{2}$
having only radial dependence as $\omega_{p}^{2}\sim\frac{1}{r}$.

(a) Angular size of the shadow ($\alpha_{stat}$) with respect to the position
of a static observer ($\frac{r_{0}}{M}$) for Schwarzschild de-Sitter black
hole with plasma parameter $\frac{k}{M}=0$.  (b) Angular size of the shadow
($\alpha_{stat}$) with respect to the position of a static observer
($\frac{r_{0}}{M}$) for Schwarzschild de-Sitter black hole with inhomogeneous
plasma parameter $\frac{k}{M}=0.2$.

Figure 6: Variation in the angular size of the shadow with respect to the
position of the observer $\frac{r_{0}}{M}$. The plots are shown for different
values of cosmological constant $\Lambda M^{2}$ as 0.0000 (black), 0.0075
(violet), 0.0300 (red) and 0.0675 (orange).

The plots given in Fig.6 shows the variation of the black hole shadow for
different values of the cosmological constant $\Lambda$ or in other words the
Hubble’s constant $H_{0}$. The plots are shown for Schwarzschild de-Sitter
black hole with the values of the cosmological constant ($\Lambda M^{2}$) as
0.0000 (black), 0.0075 (violet), 0.0300 (red) and 0.0675 (orange). With
increase in the value of $\Lambda$, we find that the shadow size decreases
more quickly to zero as the cosmological horizon ($r_{C}$) comes closer to the
black hole event horizon ($r_{h+}$). Besides, we also find that the black hole
event horizon shifts outward with increase in the value of the cosmological
constant $\Lambda$. Simply, we can say that $r_{h+}$ increases with increase
in $\Lambda$ whereas $r_{C}$ decreases for the same and thereby they merge
($r_{h+}=r_{C}$) for $\Lambda M^{2}=0.1111$. The left plot is for inhomgeneous
plasma parameter $\frac{k}{M}=0$ and the right plot is for $\frac{k}{M}=0.2$.
The two plots are almost identical since the variation due to plasma gets
reflected in the photon sphere radius $r_{p}$ but no effect of plasma is seen
in case of $r_{h+}$ and $r_{C}$.

##  4 Shadow from the point of view of co-moving observer

Next we try to study the effect of the cosmic expansion on the black hole
shadow. The shadow is viewed by an observer stationed at a certain position in
the domain of outer communication. The cosmological expansion drives the
observer away from the black hole with time and is dictated by the
cosmological constant $\Lambda$. The angular size of the shadow measured by
this co-moving observer is related to that for a static observer by the
aberration relation [79], [80]

$\cos\alpha_{c}=\frac{\cos\alpha_{s}-v}{1-v\cos\alpha_{s}}$ (40)

which in terms of sine function takes the form

$\sin^{2}\alpha_{c}=\Big{(}1-v^{2}\Big{)}\frac{\sin^{2}\alpha_{s}}{\Big{(}1-v\cos\alpha_{s}\Big{)}^{2}}$
(41)

where $v$ is the relative velocity between the two observers. In general, we
consider the co-moving observer is moving away from the static observer with
velocity $v$. This velocity depends on the cosmological constant $\Lambda$ and
thereby the Hubble constant $H_{0}$. Also it must be pointed out that as the
static observer goes from $r_{h+}$ to $r_{p}$, the angular size of the black
hole shadow changes from $\pi$ to $\frac{\pi}{2}$ and then goes from
$\frac{\pi}{2}$ to 0 as the observer goes to $r_{C}$. This observation is
clear from the right panel plot of Fig.4. We have shown numerical values for
the angles. We also point that in our case sine and cosine functions are
allowed to take only positive values in the calculation, hence we put negative
values by hand for a continuation of our results. Using this and by expressing
$\cos\alpha$ in terms of $\sin\alpha$ eq.(41) becomes

$\sin\alpha_{c}=\sqrt{\Big{(}1-v^{2}\Big{)}}\frac{\sin\alpha_{s}}{\Big{(}1\pm
v\sqrt{1-\sin^{2}\alpha_{s}}\Big{)}}$ (42)

which can be rewritten as

$\sin\alpha_{c}=\sqrt{\Big{(}1-v^{2}\Big{)}}\sin\alpha_{s}\frac{\Big{(}1\mp
v\sqrt{1-\sin^{2}\alpha_{s}}\Big{)}}{\Big{(}1-v^{2}(1-\sin^{2}\alpha_{s})\Big{)}}\leavevmode\nobreak\
.$ (43)

The associated graphical representations show very interesting results. If we
consider only the effect of the motion of the observer, we find that the
shadow has certain features quite different from that of the static observer.
In case of a static observer, as the observer moves away from the event
horizon $r_{h+}$ the complete dark sky as seen by him starts to reduce in size
and just at the photon sphere $r_{p}$, he sees half dark and half bright sky.
Further, as he moves outward he eventually sees a completely bright sky. Also
at $r_{p}$ light rays are perpendicular to the observer. But for the co-moving
observer the situation is not the same. For the comoving observer the
effective photon sphere $\tilde{r}_{p}$ is somewhere outward, that is
$\tilde{r}_{p}>r_{p}$. The rest portion is same as that of static observer.
Also the position of the comoving observer is not bound by the cosmological
horizon $r_{C}$ and the observer can readily reach upto infinity.

If we incorporate plasma now, the aberration relation gets modified to a great
extent as given in [81] as

$\cos\alpha_{cp}=\frac{\cos\alpha_{sp}+nv}{\sqrt{\Big{(}n+v\cos\alpha_{sp}\Big{)}^{2}-\Big{(}n^{2}-1\Big{)}\Big{(}1-v^{2}\Big{)}}}$
(44)

which can be rewritten in terms of sine function as

$\sin\alpha_{cp}=\sqrt{\Big{(}1-v^{2}\Big{)}}\frac{\sin\alpha_{sp}}{\sqrt{\Big{(}n\mp
v\sqrt{1-\sin^{2}\alpha_{sp}}\Big{)}^{2}-\Big{(}n^{2}-1\Big{)}\Big{(}1-v^{2}\Big{)}}}\leavevmode\nobreak\
.$ (45)

For $n=1$, we get back eq.(42). The simplified form of the above eq.(45) is
given as

$\sin\alpha_{cp}=\sqrt{\Big{(}1-v^{2}\Big{)}}\sin\alpha_{sp}\frac{\sqrt{\Big{(}1+n^{2}v^{2}-v^{2}\sin^{2}\alpha\Big{)}\mp
2nv\sqrt{\Big{(}1-\sin^{2}\alpha\Big{)}}}}{\sqrt{\Big{(}1+n^{2}v^{2}-v^{2}\sin^{2}\alpha\Big{)}^{2}-4n^{2}v^{2}\Big{(}1-\sin^{2}\alpha\Big{)}}}\leavevmode\nobreak\
.$ (46)

Using this expression we can find the angular size of the black hole shadow
surrounded by plasma as seen by a comoving observer moving away from the black
hole.

Now, in order to plot the angular size of the black hole shadow, we need to
determine the velocity of the comoving observer with respect to a static
observer. The derivation has been performed by Perlick et al in [50]. Also it
has been done for the general case of any static and spherically symmetric
spacetime by Roy et al in [49]. The velocity of the comoving observer with
respect to the static observer takes the form

$|\vec{v}|=\sqrt{1-\frac{f(r)}{f_{0}(r)}}.$ (47)

with $f(r)$ being the lapse function and $f_{0}(r)=f(r)\Big{|}_{\Lambda=0}$.
In case of Kottler metric, the lapse function takes the form
$f(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}$, which results into the velocity
of comoving observer as

$|\vec{v}|=\sqrt{\frac{\Lambda/3}{1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}}}r=\frac{H_{0}r}{\sqrt{1-\frac{2M}{r}-H_{0}^{2}r^{2}}}.$
(48)

We have briefly discussed the formula for the velocity of the comoving
observer with respect to the static observer in the Appendix.

(a) Angular size ($\alpha_{comov}$) of the Schwarzschild de-Sitter black hole
shadow with respect to the position of a co-moving observer
($\frac{r_{0}}{M}$). We set the plasma parameter $\frac{k}{M}=0$.  (b) Angular
size ($\alpha_{comov}$) of the Schwarzschild de-Sitter black hole shadow with
respect to the position of a co-moving observer ($\frac{r_{0}}{M}$). We set
the plasma parameter $\frac{k}{M}=0.2$.

Figure 7: Variation in the angular size of the shadow with respect to the
position of the co-moving observer $\frac{r_{0}}{M}$. The plots are shown for
different values of Hubble constant $H_{0}M$ as 0.01 (black), 0.05 (violet)
and 0.10 (red).

Using the expression of the comoving velocity of the observer, we can
determine the angular shadow size with respect to a comoving observer. This we
show in Fig.7. The left plot represents the shadow without plasma scenario and
the right plot represents shadow in plasma background associated to the value
$\frac{k}{M}=0.2$. Further, we also show the variation of the angular size of
the black hole shadow for different values of the Hubbble constant $H_{0}$.
From both the plots, for a fixed value of the observer position one can
conclude that the shadow size increases with increase in the value of Hubble
constant $H_{0}$ and the shadow size reduces to a finite value and never
reaches zero size as the observer moves away from the black hole.

(a) Angular size ($\alpha_{comov}$) of the Schwarzschild de-Sitter black hole
shadow with respect to the position of a co-moving observer
($\frac{r_{0}}{M}$) for various values of the plasma parameter $\frac{k}{M}$.
(b) The zoomed version of the left image.

Figure 8: Variation in the angular size of the shadow with respect to the
position of the observer $\frac{r_{0}}{M}$. The plots are shown for different
values of inhomogeneous plasma parameter $\frac{k}{M}$ as 0.0 (black), 0.2
(violet) and 0.4 (red) with Hubble constant $H_{0}M=0.10$.

In Fig.8, we probe the effect of the plasma parameter on the angular shadow
radius, as seen by a comoving observer. The plots have been shown for a fixed
value of the Hubble constant $H_{0}M=0.1$ and for inhomogeneous plasma
parameter values $\frac{k}{M}=0.0,0.2$ and $0.4$. The plot in right is the
zoomed version of the left one. We find that the increase of the value of the
inhomogeneous plasma parameter $\frac{k}{M}$ reduces the angular size of the
black hole shadow similar to that in case of a static observer.

(a) Silhouette of the black hole for different values of inhomogeneous plasma
parameter $\frac{k}{M}$ with the observer positioned at $r_{0}=5M$.  (b) The
zoomed version of the left image.

Figure 9: Black hole shadow with variation of inhomogeneous plasma parameter
$\frac{k}{M}$. The plots are shown for different values of inhomogeneous
plasma parameter $\frac{k}{M}$ as 0.0 (black), 0.2 (red) and 0.4 (blue) with
Hubble constant $H_{0}M=0.10$. The observer is positioned at $r_{0}=5M.$

(a) Silhouette of the black hole for different values of inhomogeneous plasma
parameter $\frac{k}{M}$ with the observer positioned at $r_{0}=7M$. (b) The
zoomed version of the left image.

Figure 10: Black hole shadow with variation of inhomogeneous plasma parameter
$\frac{k}{M}$. The plots are shown for different values of inhomogeneous
plasma parameter $\frac{k}{M}$ as 0.0 (black), 0.2 (red) and 0.4 (blue) with
Hubble constant $H_{0}M=0.10$. The observer is positioned at $r_{0}=7M.$

In Fig.(s) 9 and 10, we show the silhouette of the black hole shadow in the
celestial plane. We have set the position of the observer at $r_{0}=5M$ in
Fig. 9 and $r_{0}=7M$ in Fig.10. Since we are considering a spherically
symmetric black hole, so the black hole shadow formed in the celestial plane
($\alpha$-$\beta$ plane) is circular. The radius of the shadow can be
geometrically determined from Figure 1, that is,
$|R_{s}|=|r_{0}\tan\alpha|\approx|r_{0}\sin\alpha|$. The shadow size gets
reduced with increase in plasma parameter $\frac{k}{M}$ in both cases which
can be clearly observed from the zoomed plots. Also, from the axes of the
above Figures, it is clear that the shadow radius $R_{s}$ is larger in case of
the observer positioned at $r_{0}=7M.$ Finally, Figures 9 and 10 also show the
combined effect of plasma parameter $k$ and the cosmological constant
$\Lambda$ on the shadow size, with $k$ having a de-magnification effect and
$\Lambda$ having a magnification effect.

##  5 Comparison with EHT observations

In this section, we compute the angular size of the black hole shadow and
compare it with the observational results. We wish to compare the theoretical
results with observations of supermassive black hole situated at the centre of
M87 galaxy and our galaxy. We discuss them casewise.

###  5.1 Case I: M87∗ supermassive black hole

The observed angular size of the black hole shadow at the centre of M87 galaxy
is (42$\pm$3) $\mu$as [2]. Thus the angular shadow size varies in the range
(39-45) $\mu$as. For the purpose of our calculation, we require the distance
of the supermassive black hole from earth which is (16.8 $\pm$ 0.8) MPc [2].
The mass of the black hole as measured by the $EHT$ collaboration is ($6.5\pm
0.7$) $\times 10^{9}$ $M_{\odot}$ [2]. These are required to calculate the
mass parameter $m$ given as $m=\frac{GM}{c^{2}}$ where $G$ is Newton’s
gravitational constant and $c$ is the velocity of light in vacuum having the
standard values 6.674$\times$10-11 m3Kg-1s-2 and 3$\times$ 108 ms-1
respectively.

Also, our primary interest is in calculating the black hole shadow as observed
by comoving observer. The comoving observer moves away with the cosmic
expansion. In this case, we consider that the cosmic expansion is driven by
the cosmological constant $\Lambda$ which is related to the Hubble constant as
$\frac{\Lambda}{3}=\frac{H_{0}^{2}}{c^{2}}$. The angular shadow size as given
in eq.(s) (45) and (46) depend on the recession velocity of the comoving
observer. The velocity of the comoving observer is given in (48) which is
dependent on Hubble constant $H_{0}$. So, in order to determine the value of
the angular black hole shadow size as measured by the comoving observer, we
need the value of the Hubble’s constant $H_{0}$. The value of $H_{0}$ as
measured by the Planck Telescope is ($67.4\pm 0.5$) km s-1MPc-1 [82].

Again, we calculate the shadow of a black hole surrounded by plasma medium.
The plasma is assumed to be spherically symmetric and the plasma frequency
$\omega_{p}$ and thereby the refractive index $n$ has only radial dependence.
The refractive index takes the form $n=\sqrt{1-\frac{k}{r^{h}}f(r)}$. For
homogeneous plasma, we set $h=0$ and for inhomogeneous plasma we consider the
simplest case setting $h=1$. For these two cases we calculate the angular
shadow size using (46). From $\sin\alpha$, we calculate $\alpha$. This
$\alpha$ gives the angular radius which is doubled to yield the angular
diameter $2\alpha$. This value is compared to the observational result to
yield constraints on the value of the plasma parameter $k$.

Also, we convert the distance of black hole from earth $r_{0}$ as well as the
photon sphere radius $r_{p}$ into metres. The value of $r_{0}$ is 5.184
$\times 10^{23}$ m. And the value of the photon radius for the Schwarzchild
black hole both with and without cosmological constant $\Lambda$ is $3m$, that
is, 2.892 $\times 10^{13}$ m. with mass parameter $m$ having value of 9.64
$\times 10^{12}$ m. Also, the cosmological horizon $r_{C}$ of the black hole
spacetime is at 1.3736 $\times 10^{26}$ metre, where we have used the mass of
the M87∗ supermassive black hole to be ($6.5\pm 0.7$) $\times 10^{9}$
$M_{\odot}$ [2] and the value of the Hubble constant as ($67.4\pm 0.5$) km
s-1MPc-1 [82]. Thus, the observer is located well inside the horizon.

$k$ |  $\alpha_{stat}(\mu$as)
---|---
0.00 | 39.8643
0.05 | 40.7999
0.10 | 41.3061
0.15 | 42.1243
0.20 | 43.0210
0.25 | 44.0093
0.295 | 44.9907
0.296 | 45.0136

$k$ |  $\alpha_{comov}(\mu$as)
---|---
0.00 | 40.0150
0.05 | 40.9502
0.10 | 41.4542
0.15 | 42.2710
0.20 | 43.1633
0.25 | 44.1533
0.289 | 44.9971
0.29 | 45.0197

Table 1: The Table shows the angular shadow size with change in plasma
parameter $k$ for homogeneous plasma frequency $\omega_{p}^{2}=k=constant$.
The left one is for static observer and right one is for comoving observer.

Also we would like to point out that in the overall analysis, we have used a
general expression of the velocity of the co-moving observer eq.(48). For
numerical analysis in cosmological scales, we need to consider the effective
velocity given as [83]

$v_{eff}=cz=H_{0}D+v_{p}\implies v_{p}=cz-H_{0}D$ (49)

where $v_{eff}$ gives the effective velocity which is given in terms of the
redshift factor $z$. Also, $H_{0}$ gives the Hubble constant, $D$ is the
distance of the observer from the black hole and $v_{p}$ is the peculiar
velocity. Now the redshift factor $z$ of the M87∗ black hole with respect to
earth is 0.00428 [84], the Hubble’s constant $H_{0}$ is 67.4 km s-1MPc-1 [82]
and $D$ is 16.8 $Mpc$ [2]. Using these values, we obtain the peculiar velocity
$v_{p}$ to be 178.64 km s-1. However, the velocity due to cosmic expansion is
1105.36 km s-1 which can be obtained using eq.(48). In general we must have
used $v_{eff}$ for our calculations, but the numerical results reveal that the
co-moving velocity is quite large compared to the peculiar velocity and hence
we have neglected it in our work. Thus, our consideration of the earth as a
co-moving frame neglecting the peculiar velocity gets valid.

Table 1 shows the values of the angular size of the black hole shadow for
different values of the plasma parameter $k$. The plasma medium considered
here is of homogeneous nature with frequency $\omega_{p}^{2}=k=$constant. The
values of the angular shadow size varies within the range 39-45 $\mu$as. The
values of the plasma parameter compatible with this range of the angular
shadow size give us the bound on the parameter $k$. We find that the plasma
parameter is bounded as $0\leq k<0.296$ for the shadow observed by the static
observer and $0\leq k<0.29$ in case of the comoving observer. The left table
in 1 shows a list of values for $k$ compatible with the angular shadow size as
observed by a static observer. On the other hand the right table shows the
compatible range of values of $k$ with respect to a comoving observer. We find
that the range of the values of plasma parameter $k$ are almost same in case
of both the static and comoving observer.

$\frac{k}{m}$ |  $\alpha_{stat}(\mu$as)
---|---
0.00 | 39.8643
0.05 | 39.7534
0.10 | 39.6422
0.15 | 39.5307
0.20 | 39.4189
0.25 | 39.3067
0.30 | 39.1943
0.35 | 39.0815
0.386 | 39.0001

$\frac{k}{m}$ |  $\alpha_{comov}(\mu$as)
---|---
0.00 | 40.0150
0.05 | 39.7921
0.10 | 39.6801
0.15 | 39.6801
0.20 | 39.5679
0.25 | 39.4553
0.30 | 39.3424
0.35 | 39.2292
0.40 | 39.1157
0.4508 | 39.0000

Table 2: The Table shows the angular shadow size with change in plasma
parameter $\frac{k}{m}$ for inhomogeneous plasma frequency
$\omega_{p}^{2}=\omega^{2}_{p}(r)=\frac{k}{r}$. The left one is for static
observer and right one is for comoving observer.

The above Table 2 shows the values of the inhomogeneous plasma parameter
$\frac{k}{m}$ compatible with the values of the angular shadow size lying
within the range 39-45 $\mu$as. The Table is constructed by considering the
inhomogeneous plasma frequency of the form $\omega_{p}^{2}(r)=\frac{k}{r}$.
The left table in 2 is for the angular shadow size as observed by a static
observer and that in the right is for comoving observer. The bound on the
plasma parameter $\frac{k}{m}$ is given as $0\leq\frac{k}{m}\leq 0.386$ for
the static observer and $0\leq\frac{k}{m}\leq 0.4508$ in case of comoving
observer. The ranges of the value of plasma parameter $\frac{k}{m}$ is quite
different for static and comoving observers. The range is extended more in
case of comoving observer than that for the static observer.

###  5.2 Case II: Sgr A∗ supermassive black hole

We wish to constraint the plasma parameter in case of Sgr A∗ supermassive
black hole. Such constraining of other types of parameters have been done in
case of Sgr A∗ supermassive black hole in [85]-[88]. We want to carry out the
analysis by comparing the theoretical and observed values of angular size of
the black hole shadow. The observed value of the angular size of the black
hole shadow is (48.7 $\pm$ 7) $\mu$as [89]. In order to perform the numerical
analysis, we require the black hole mass, which is $4^{+1.1}_{-0.6}\times
10^{6}$ $M_{\odot}$ [89]. The black hole is at a distance $(8277\pm 9\pm
33)pc$ [90] from earth. We use the value of the black hole mass as $4\times
10^{6}$ $M_{\odot}$ and distance to the black hole as 8277 pc for our
calculations. The angular size of the black hole shadow varies in the range
(41.7 - 55.7) $\mu$as.

$k$ |  $\alpha_{stat}(\mu$as)
---|---
0.000 | 49.6868
0.050 | 50.5446
0.100 | 51.4799
0.150 | 52.4996
0.200 | 53.6172
0.250 | 54.8490
0.281 | 55.6789
0.282 | 55.7066

$k$ |  $\alpha_{comov}(\mu$as)
---|---
0.000 | 49.6869
0.050 | 50.5447
0.100 | 51.4800
0.150 | 52.4997
0.200 | 53.6173
0.250 | 54.8491
0.281 | 55.6790
0.282 | 55.7067

Table 3: The Table shows the angular shadow size with change in plasma
parameter $k$ for homogeneous plasma frequency $\omega_{p}^{2}=k=constant$.
The left one is for static observer and right one is for comoving observer.

We are interested in carrying out the analysis for both homogeneous and
inhomogeneous plasma distribution. The refractive index in case of homogeneous
plasma takes the form $n=\sqrt{1-kf(r)}$ with $k$ being the plasma parameter
and $f(r)$ being the lapse function. Table 3 shows the angular size of the
black hole shadow measured by both static and co-moving observers. We present
those values of the angular shadow which lie in the desired range
$(41.7-55.7)\mu as$ along with the compatible values of the plasma parameter
$k$. We find that the plasma parameter is bounded as $0\leq k<0.282$ for both
static and co-moving observer. Also, we notice something significant. The
angular size of the black hole shadow is same for both static and co-moving
observer. This is due to the fact that the supermassive black hole Sgr A∗ and
the observer in this case (earth) belongs to the same galactic frame.

$\frac{k}{m}$ |  $\alpha_{stat}(\mu$as)
---|---
0.00 | 49.6868
0.50 | 48.2802
1.00 | 46.8166
1.50 | 45.2895
2.00 | 43.6907
2.50 | 42.0102
2.58 | 41.7330

$\frac{k}{m}$ |  $\alpha_{comov}(\mu$as)
---|---
0.00 | 49.6869
0.50 | 48.2802
1.00 | 46.8167
1.50 | 45.2896
2.00 | 43.6907
2.50 | 42.0103
2.58 | 41.7331

Table 4: The Table shows the angular shadow size with change in plasma
parameter $\frac{k}{m}$ for inhomogeneous plasma frequency
$\omega_{p}^{2}=\frac{k}{r}$. The left one is for static observer and right
one is for comoving observer.

Then we try to constraint the inhomogeneous plasma parameter $\frac{k}{m}$
using the angular size of black hole shadow. The refractive index in case of
inhomogeneous plasma takes the form $n=\sqrt{1-\frac{k}{r}f(r)}$. Table 4
shows the values of angular size of the black hole shadow and the compatible
values of the plasma parameter $\frac{k}{m}$. We find that the plasma
parameter $\frac{k}{m}$ is bounded as $0\leq\frac{k}{m}\leq 2.58$ in case of
both the static and the co-moving observer. Similar to the previous case, we
find that the black hole shadow size is same for both static and co-moving
observers.

##  6 Conclusion

We now summarize our findings and conclude. In this work, we have performed
the analysis of black hole shadow in presence of plasma. To be precise, we
have calculated the angular shadow radius of a spherically symmetric black
hole in presence of inhomogeneous plasma, as seen by both static and comoving
observer. We start by considering a spherically symmetric metric black hole
metric in (3+1) dimensions with arbitrary lapse function $f(r)$ and then
consider lapse function of the Kottler or Schwarzchild de-Sitter black hole
spacetime, having a positive cosmological constant $\Lambda>0$. This $\Lambda$
is responsible for the expansion of the universe. The important and unique
results in our work is the effect of plasma in the black hole shadow with
$\Lambda>0$. We have evaluated the general equation for determining the photon
sphere radius $r_{p}$ which is valid for both homogeneous and inhomogeneous
plasma background. The equations are non-trivial and can be solved only
numerically. Then we have shown the plots for angular shadow size for
Schwarzschild and Schwarzschild de-Sitter black hole with variation in plasma.
Here, we found that with increase in the inhomogeneous plasma parameter $k$,
the shadow size reduces. We have also shown the effect of plasma on the photon
sphere radius $r_{p}$. Later, we have shown the effect of cosmological
constant $\Lambda$ on the angular shadow size. We observe that at the position
of the observer which is well outside the photon sphere, the angular shadow
size reduces with increase in $\Lambda$. The effect remains the same even in
presence of plasma parameter.

The graphical analysis reveal many interesting results. We observe that the
angular shadow size varies in the range $\pi$ to $0$. The shadow size is $\pi$
at the black hole horizon and goes down to zero at infinity for general
asymptotically flat black holes. In case of asymptotically de-Sitter black
holes, the shadow goes to zero size at the cosmological horizon $r_{C}$. The
angular shadow for co-moving observer is related to that of the static
observer. The interesting feature is that in case of co-moving observer, there
is no cosmological horizon and the observer can reach upto infinity. Further,
in this case the shadow size does not go to zero and always remains finite.
Since the observer is moving with the expansion, hence he never feels any
horizon as such.
Then we plot and analysed the shadow size by considering plasma effects. We
observed that plasma impacts the shadow size greatly. In general, practically
an observer can exist outside the photon sphere $r_{p}$. From his perspective
the shadow size changes with variation in plasma parameter $k$. In case of
homogeneous plasma, we find that the shadow size increases with increase in
plasma for both static and co-moving observer. On the other hand, we observe
that the shadow size decreases with the increase in the value of the plasma
parameter $k$ for inhomogeneous plasma. We have shown the plots only for
inhomogeneous plasma. The new results in our work is the analysis of black
hole shadow with respect to both static and comoving observer in presence of
plasma.

Finally, we have compared the angular shadow size with the observational
results of M87∗ and Sgr A∗ data. Comparing our results with the observed
angular shadow size, we find bounds on the plasma parameter $k$. We find that
for homogeneous plasma, $k$ has an upper bound of $k<0.296$ for static
observer and $k<0.29$ for co-moving observer. In case of inhomogeneous plasma,
we find that $\frac{k}{m}$ has an upper bound of $\frac{k}{m}\leq 0.386$ for
static and $\frac{k}{m}\leq 0.4508$ for co-moving observer. These bounds are
for M87∗ black hole. In case of Sgr A∗, we find that the angular shadow size
is the same both for static and co-moving observers. This is due to the fact
that the supermassive black hole Sgr A∗ and the observer in this case (earth)
belongs to the same galactic frame. The bounds on the homogeneous and
inhomogeneous plasma are given as $0\leq k<0.282$ and $0\leq\frac{k}{m}\leq
2.58$ respectively. The constraints are true for both static and co-moving
observers. We would like to mention that a more accurate estimation of the
constraints on the plasma parameter $k$ would emerge from taking into
consideration the spin parameter for rotating black holes. We wish to carry
out such an analysis in future.

## Appendix:
Determination of the velocity of the comoving observer with respect to the
static observer

In Step I, one considers the asymptotically flat version of the original
metric eq.(1) by dropping the cosmological constant, that is, by setting
$\Lambda=0$. This gives

$ds_{\Lambda=0}^{2}=-f_{0}(r)dt^{2}+\frac{1}{f_{0}(r)}dr^{2}+r^{2}\Big{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\Big{)}$ (50)

where $f_{0}(r)=f(r)\Big{|}_{\Lambda=0}$.

In Step II, one writes the metric in isotropic coordinates as

$ds_{\Lambda=0}^{2}=-k(\tilde{r})dt^{2}+g^{2}(\tilde{r})\Bigg{(}d\tilde{r}^{2}+\tilde{r}^{2}\Big{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\Big{)}\Bigg{)}\leavevmode\nobreak\ .$ (51)

This eq.(51) can be compared with eq.(50) to get a relation between $r$ and
$\tilde{r}$ given as

$dr=\frac{r}{\tilde{r}}\sqrt{f_{0}(r)}d\tilde{r}\leavevmode\nobreak\ .$ (52)

The above relation is then used to replace $r$ in terms of $\tilde{r}$ in the
original spherically symmetric metric given in eq.(1).

In Step III, one can make a transformation from $twiddled$ to the co-moving
coordinate ($r_{c},t_{c}$) as $\tilde{r}=r_{c}\leavevmode\nobreak\ a(t_{c})$
[47] where, $r_{c}$ is the radial coordinate in the co-moving frame and $a$ is
the scale factor of the universe which is the function of time ($t_{c}$) in
the co-moving frame. This will lead us to a differential form of $d\tilde{r}$
in terms of $dr_{c}$ and $dt_{c}$. Similarly, one can assume that $dt$ is of
the form

$dt=B(t_{c},r_{c})dt_{c}+C(t_{c},r_{c})dr_{c}\leavevmode\nobreak\ .$ (53)

These coefficients will be determined by the next steps of the algorithm. Also
due to the spherical symmetry (even though we go from original $\to$ twiddled
$\to$ co-moving frame), $\theta$ and $\phi$ parts remain intact.

In Step IV, one can rewrite the metric in terms of $r_{c},t_{c},\theta,\phi$
coordinates as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\Bigg{[}\frac{B^{2}}{f(r)}-\frac{f_{0}(r)r^{2}}{f(r)}\Big{(}\frac{\dot{a}}{a}\Big{)}^{2}\Bigg{]}dt_{c}^{2}+\Bigg{[}\frac{f_{0}(r)r^{2}}{f(r)r_{c}^{2}}-\frac{C^{2}}{f(r)}\Bigg{]}dr_{c}^{2}+2\Bigg{[}\frac{f_{0}(r)r^{2}}{f(r)r_{c}^{2}}\frac{\dot{a}}{a}-\frac{BC}{f(r)}\Bigg{]}dt_{c}dr_{c}$
(54)
$\displaystyle+g^{2}(\tilde{r})\tilde{r}^{2}\Big{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\Big{)}\leavevmode\nobreak\ .$

Imposing the condition that the final form of the metric is isotropic and
diagonal, we get

$B=r\frac{\dot{a}}{a}\frac{f_{0}(r)}{f(r)}\sqrt{f(r)}\Big{[}\frac{f_{0}(r)}{f(r)}-1\Big{]}^{-1/2}\leavevmode\nobreak\
\leavevmode\nobreak\ ;\leavevmode\nobreak\ \leavevmode\nobreak\
C=\frac{r}{r_{c}}\sqrt{f(r)}\Big{[}\frac{f_{0}(r)}{f(r)}-1\Big{]}^{1/2}\leavevmode\nobreak\
.$ (55)

Thus, one determines $dt$ and $dr$ in terms of $dt_{c}$ and $dr_{c}$. Using
this relation, one readily determines $dt_{c}$ and $dr_{c}$ in terms of $dt$
and $dr$ which in turn gives $\partial_{t_{c}}$ and $\partial_{r_{c}}$.
Finally using the normalisation condition $g_{\mu\nu}U^{\mu}U^{\nu}=-1$ and
also the relation between the four velocity of the co-moving observer with
respect to the static observer as

$g_{\mu\nu}U^{\mu}_{s}U^{\nu}_{c}=-\frac{1}{\sqrt{1-v^{2}}}$ (56)

one can determine the velocity of the co-moving frame with respect to the
static frame.

Using the above algorithm the velocity ($v$) of the comoving observer takes
the form [49]

$|\vec{v}|=\sqrt{1-\frac{f(r)}{f_{0}(r)}}.$ (57)

$f(r)$ is the lapse function with cosmological constant ($\Lambda$), whereas
$f_{0}(r)$ is devoid of that term. Besides $\Lambda$ appears in the lapse
function as $\frac{\Lambda}{3}r^{2}=\frac{H_{0}^{2}}{c^{2}}r^{2}$. For $c=1$,
we get $\frac{\Lambda}{3}r^{2}=H_{0}^{2}r^{2}$. Thus, the velocity of the co-
moving observer becomes

$|\vec{v}|=\frac{H_{0}r}{\sqrt{f_{0}(r)}}.$ (58)

## Acknowledgments

A.D. would like to acknowledge the support of S.N. Bose National Centre for
Basic Sciences for Senior Research Fellowship. A.S. acknowledges the financial
support by Council of Scientific and Industrial Research (CSIR, Govt. of
India). The authors would like to thank the anonymous referee for useful
suggestions and comments.

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