# Quasinormal modes, shadow and thermodynamics of black holes coupled with
nonlinear electrodynamics and cloud of strings

Dharm Veer Singh111Visiting Associate, IUCAA Pune, Maharashtra 411007, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>Department Physics, GLA
University, Mathura 281406 Uttar Pradesh  Aradhya Shukla<EMAIL_ADDRESS>Department Physics, GLA University, Mathura 281406 Uttar Pradesh  Sudhaker
Upadhyay222Corresponding author1<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, K.L.S. College, Magadh
University, Nawada, Bihar 805110, India  School of Physics, Damghan
University, P.O. Box 3671641167,
Damghan, Iran

###### Abstract

We construct an exact black hole solution for the Einstein gravity coupled
with the nonlinear electrodynamics (which corresponds to the Maxwell
electrodynamics in the weak field limit) in the presence of a cloud of strings
as the source. We study the thermodynamical properties of the black hole
solutions and derive the corrected first-law of thermodynamics. The presence
of a cloud of strings does not affect the stability of the present black hole.
However, a second-order phase transition exists for this system at a critical
horizon radius. Furthermore, we study the quasinormal modes and their shadow
radius. In addition, we find that, upon variation, the parameters of the
theory show different aspects of the optical characteristics of the black hole
solutions.

Quasinormal modes; Black holes shadow; Black holes thermodynamics.

##  I Introduction

One of the important aspects of general relativity is to find solutions to
Einstein’s field equation. The metric tensor in arbitrarily chosen spacetimes
may lead to an unphysical stress-tensor. Therefore, obtaining a meaningful
solution to Einstein’s field equation is a cumbersome job [1]. The well-known
vacuum (without matter) solution and with matter solution of Einstein’s field
equation is the Schwarzschild black hole and Reissner-Nordström black hole,
respectively. These spherically symmetric solutions have a central singularity
which is separated by a boundary known as the event horizon. On the other
hand, non-singular solutions of Einstein’s field equations are known as
regular black holes. Indeed, one can have black hole metrics without a
physical singularity. Bardeen was the first to realize the idea of a central
matter core and to propose a regular black hole solution with horizons (but
without a singularity) [2]. The Bardeen regular black hole is a spherically
symmetric solution that violates the strong energy condition. Furthermore,
other spherically symmetric regular solutions were proposed [3, 4, 5]. There
has been tremendous development in the investigations of regular black hole
solutions and their properties [6, 7, 8, 9]. Recently, it is observed that
regular black hole solutions with spherical symmetry violet the weak energy
condition [10].

It has been observed that the linearity of the real electromagnetic field
breaks at high energies due to interactions with other physical fields. In
such circumstances, the most suitable alternative is nonlinear electrodynamics
as a simplified phenomenological description of this interaction. The
properties of the nonlinear electrodynamics can be lucid from the perspective
of gravity since strong vector fields are dominant at the center of black
holes. The behavior of charged particles around black holes may be described
by nonlinear electrodynamics. Hence, it is crucial to emphasize the
gravitational field coupled to nonlinear electrodynamics. Born and Infeld were
the first to propose nonlinear electrodynamics with the motivation to have a
finite self-energy of the point charge [11]. Moreover, a lot of progress have
been made in various contexts [12, 13, 14, 15, 16]. Nonlinear electrodynamics
also gets relevance as a generalized Born-Infeld action appears naturally from
the effective string action [17, 18].

Letelier proposed a model analogous to pressure-less perfect fluid known as
cloud of strings (CS) [19]. From this perspective, a generalized Schwarzschild
black hole solution is obtained in the background of a spherically symmetric
cloud of strings [20, 21]. However, the energy-momentum tensor of the string
manifests a non-null pressure, which therefore leads to astrophysical and
cosmological implications. Further studies of a cloud of strings in the
general relativity and alternative theories of gravity are reported in Refs.
[22, 23, 24, 25, 26, 27]. Since strings are supposed to be fundamental
constituents of the universe rather than point-like particles. This derives us
to study the black hole solutions in presence of CS. More precisely, we
provide an exact solution for the Einstein gravity coupled to nonlinear
electrodynamics in presence of a cloud of strings. We also discuss its
thermodynamics, stability and phase transition.

Let us briefly introduce CS as matter source. The CS is governed by the Nambu-
Goto action, which takes the form

$S_{\text{NG}}=\int_{\Sigma}\;m(-\gamma)^{-1/2}d\lambda^{0}d\lambda^{1}=\int_{\Sigma}\;m\left(-\frac{1}{2}\Sigma^{\mu\nu}\Sigma_{\mu\nu}\right)^{1/2}d\lambda^{0}d\lambda^{1},$
(1)

where constant $m$ characterizes each string, ($\lambda^{0},\lambda^{1}$) are
local coordinates of the string being timelike and spacelike in nature,
respectively [28]. Here, $\gamma$ refers to the determinant of an induced
metric $\gamma_{ab}:=g_{\mu\nu}\frac{\partial
x^{\mu}}{\partial\lambda^{a}}\frac{\partial x^{\nu}}{\partial\lambda^{b}}$ on
the strings world sheet. The bivector related to the string world sheet
$\Sigma$ is written by

$\Sigma^{\mu\nu}=\epsilon^{ab}\frac{\partial
x^{\mu}}{\partial\lambda^{a}}\frac{\partial x^{\nu}}{\partial\lambda^{b}},$
(2)

where second rank Levi-Civita tensor $\epsilon^{ab}$ (anti-symmetric in $a$
and $b$) takes following non-zero values: $\epsilon^{01}=-\epsilon^{10}=1$.

Using definition, $T^{\mu\nu}=2\frac{\partial}{\partial
g_{\mu\nu}}m\left(-\frac{1}{2}\Sigma^{\mu\nu}\Sigma_{\mu\nu}\right)^{1/2}$, we
obtain single string energy-momentum tensor as
$T^{\mu\nu}=\frac{m\Sigma^{\mu\rho}\Sigma_{\rho}^{\phantom{\rho}\nu}}{\sqrt{-\gamma}}$.
Since the CS is characterized by a proper density $\rho$ rather than a single
mass, therefore, the energy-momentum tensor for CS is given by

$T^{\mu\nu}=\frac{\rho\Sigma^{\mu\rho}\Sigma_{\rho}^{\phantom{\rho}\nu}}{\sqrt{-\gamma}}.$
(3)

Here, $\rho\;(\gamma)^{-1/2}$ denotes a gauge-invariant density.

The only surviving component of the bivector $\Sigma$ regarding spherically
symmetric solution is $\Sigma^{tr}=-\Sigma^{rt}$. Thus, conservation law leads
to

$T^{t}_{t}=T^{r}_{r}=\frac{a}{r^{2}},$ (4)

for some real constant $a$, which is related to the global monopole
charge[29].

The proper solutions of the perturbation equations of a black hole belonging
to certain complex characteristic frequencies (which satisfy the boundary
conditions) are described by quasinormal modes (QNMs) [30]. A close connection
exists between QNMs and the shadow radius of a black hole [31]. The QNMs are
determined entirely by the dynamics of black holes. At the same time, the
shadow radius belongs to optical properties, such a connection justifies the
relation between dynamics and optical properties of black holes [32, 33].
Recently, the shadow of the $5D$ AdS Reissner-Nordström black hole is studied
in Ref. [34].

The paper is structured as follows. In section II, we consider an action
describing Einstein’s gravity with nonlinear electrodynamics in the presence
of CS. We obtain an exact black hole solution with coordinate singularity for
this model in section III. Here, we discuss the energy conditions for this
black hole solution. Thermodynamical behavior of this black hole is presented
in section IV. We see that the black hole follows a modified first-law of
thermodynamics. We study the photon radius including shadow radius in section
V. Within the section, we consider a photon moving in a circular orbit. The
photon radius is obtained numerically. The section VI is devoted to the study
of the connection between shadow radius and QNMs. We conclude the results and
their importance in the last section.

##  II Gravity coupled with nonlinear electrodynamics and cloud of strings

Now, the action describing Einstein’s gravity coupled to nonlinear
electrodynamics and surrounded by a CS in $4$ dimensions is given by

$S=\frac{1}{2}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left[R+\mathcal{L}({\mathcal{F}})\right]+S_{\text{NG}},$
(5)

the Lagrangian $\mathcal{L}({\mathcal{F}})$ describes nonlinear
electrodynamics with invariant
${\mathcal{F}}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where
$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ is the electromagnetic
field strength tensor for the gauge potential $A_{\nu}$.

The equations of motion for the action are given by [35]

$\displaystyle G_{\mu\nu}\equiv
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R={\mathcal{T}_{\mu\nu}}+T_{\mu\nu},$ (6)
$\displaystyle\nabla_{\mu}\left(\frac{\partial\mathcal{L(F)}}{\partial{\mathcal{F}}}F_{\mu\nu}\right)=0~{}~{}~{}\text{and}~{}~{}~{}\nabla_{\mu}\left({}^{*}F_{\mu\nu}\right)=0.$
(7)

where $G_{\mu\nu}$ is the Einstein tensor and ${\mathcal{T}_{\mu\nu}}\equiv
2\left[\frac{\partial\mathcal{L}}{\partial{\mathcal{F}}}F_{\mu\rho}F_{\nu}^{\rho}-g_{\mu\nu}\mathcal{L}\right]$
is energy-momentum tensor related to the electromagnetic tensor. The explicit
form of Lagrangian density of the nonlinear electrodynamics is considered as
[36]

$\displaystyle\mathcal{L}({\mathcal{F}})={\mathcal{F}}e^{-\frac{k}{q}(2q^{2}F)^{\frac{1}{4}}},\qquad\text{with}\qquad
k=\frac{q^{2}}{2M}.$ (8)

Here, $q$ corresponds to the nonlinear charge of a self-gravitating magnetic
field and $M$ is the mass parameter, later which is related to the mass of the
black hole. The Lagrangian $\mathcal{L}({\mathcal{F}})$ is the function of
${\mathcal{F}}$, where the Lagrangian
$\partial\mathcal{L}/\partial{\mathcal{F}}\to\infty$ as
${\mathcal{F}}\to\infty$ and $\partial\mathcal{L}/\partial{\mathcal{F}}\to 1$
as ${\mathcal{F}}\to 0$. In weak field limit (${\mathcal{F}}<<1$), the
$\mathcal{L}$ identifies to the Maxwell electrodynamics. Nonetheless, under
the strong field limit, $\mathcal{L}$ vanishes.

We consider the following anstaz for the Maxwell field [37, 38]:

$\displaystyle F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\delta^{\theta_{1}}_{[\mu}\delta^{\theta_{2}}_{\nu]}q\sin\theta.$ (9)

with ${\mathcal{F}}$ as

${\mathcal{F}}=\frac{q^{2}}{2r^{4}}.$ (10)

The energy-momentum tensor, in this case, is calculated by

$\displaystyle{\mathcal{T}}^{t}_{t}$ $\displaystyle=$
$\displaystyle{\mathcal{T}}^{r}_{r}=\frac{2Mke^{-\frac{q^{2}}{2Mr}}}{r^{4}},$
(11)

which satisfies the equations of motion in the case of nonlinear
electrodynamics.

##  III Regular black hole solutions with cloud of strings

In this section, the main motivation is to get a static spherically symmetric
solution of the equation of motion (6) with a CS and nonlinear electrodynamics
as source and investigate its properties. In order to achieve the goal, let us
begin by writing spherically symmetric static metric of the form:

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega^{2},$ (12)

with

$f(r)=1-\frac{{2}m(r)}{r},$ (13)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the metric on $2D$
sphere. Using Eq. (6) with the metric element (13) leads to the Einstein field
equations to have the following form:

$\displaystyle\frac{d}{dr}m(r)=\frac{q^{2}}{2r^{2}}e^{-k/r}+\frac{a}{2}.$ (14)

Integrating the above expression (14), we have the explicit form of single
string mass

$m(r)=Me^{-k/r}+\frac{a}{2}r+C_{1},$ (15)

where $C_{1}$ is constant, then $C_{1}=\lim_{r\to\infty}m(r)-ar=M$ and
substituting the $m(r)$ into $f(r)$, the black hole solution (12) becomes

$ds^{2}=-\left[1-\frac{2M}{r}e^{-k/r}-a\right]dt^{2}+\frac{1}{\left[1-\frac{2M}{r}e^{-k/r}-a\right]}dr^{2}+r^{2}d\Omega^{2}.$
(16)

This is an exact black hole solution in the presence of nonlinear source
$e^{-k/r}$ and CS parameter $a$. This black hole is characterized by the
parameters like $M$, $k$ and $a$. The presence of such a nonlinear source and
CS parameter ensure deviation from Schwarzschild black hole. In the limit
$k=0$, the resulting solution reduces to Letelier solution [21]. However, this
coincides to Schwarzschild black hole solution for $k=0$ and $a=0$. The
solution behaves as the Reissner-Nordström black hole for the case of $r>>k$.
For $a=0$ and $k<0$, $m(r)$ increases exponentially for small $r$. On the
other hand, when $k>0$, $m(r)$ has the following properties:

$\lim_{r\to 0^{+}}e^{-k/r}+a=a,\qquad\qquad\lim_{r\to
0^{-}}e^{-k/r}+a=+\infty.$ (17)

The function is discontinuous at $r=0$. For $r>>k$, the obtained solution
behaves as a Reissner-Nordström black hole with CS parameter $a$ as

$f(r)=1-\frac{2M}{r}+\frac{q^{2}}{r^{2}}-a+{\cal
O}\left(\frac{k^{2}}{r^{2}}\right).$ (18)

The charge $q$ and mass $M$ are related by the relation $q^{2}=2Mk$ as
mentioned in (8). It is not cumbersome to estimate the numerical range of $M$,
$a$ and $k$ for the black hole solution (16). This black hole has two
horizons, the event horizon ($r_{+}$) and the Cauchy horizon ($r_{-}$). These
horizons in terms of the Lambert $W$ function are given by

$r_{\pm}=-\frac{k}{\text{W}\left(-\frac{(1-a)k}{{2}M}\right)}\qquad\implies\qquad
r_{\pm}=\frac{{2}M}{(1-a)}e^{W\left(-\frac{(1-a)k}{M}\right)}.$ (19)

The Lambert $W$ function has two branches $W_{0}$ and $W_{-1}$ and provides
two possibilities,

$\displaystyle W_{0}\left(-\frac{(1-a)k}{{2}M}\right)<0\qquad\implies\qquad
k\in\left(0,\frac{{2}M}{(1-a)q}\right],$ $\displaystyle
W_{-1}\left(-\frac{(1-a)k}{{2}M}\right)<0,\,\,\quad\implies\qquad
k\in\left[-\frac{{2}M}{(1-a)q,0}\right).$ (20)

The value of $k$ lies in this interval as we have well defined coordinate
location for a horizon when taking the $\left(-{(1-a)k}/{M}\right)$ branch of
the Lambert function. The range $W_{-1}$ branch is entirely $-ve$ and only
return output occurs when $\left[-{M}/{(1-a)q,0}\right)$. This means that all
the possible solutions will correspond to $r_{+}>0$. The inner and event
horizons are located at, respectively,

$r_{-}=\frac{{2}M}{(1-a)}e^{W_{-1}\left(-\frac{(1-a)k}{{2}M}\right)},\qquad\text{and}\qquad
r_{+}=\frac{{2}M}{(1-a)}e^{W_{0}\left(-\frac{(1-a)k}{{2}M}\right)}.$ (21)

Only for $k={2}M/(1-a)q$, the inner and event horizon coincides and we have an
extremal black hole. Moreover, for $k>{2}M/(1-a)q$, the Lambert $W$ function
is undefined. In order to study the behavior of $f(r)$ as a function of the
different parameters, we plot figure 1. However, the numerical analysis of the
horizon is tabulated in TABLE 1.

---
Figure 1: The plot of metric function $f(r)$ versus horizon radius $r$ for different value of CS parameter $a$ (in the left plot) and deviation parameter $k$ (in the right plot). |  | $k=0.85$ |  |  |  | $a=0.10$ |
---|---|---|---|---|---|---|---
a | $r_{-}$ | $r_{+}$ | $\delta$ | k | $r_{-}$ | $r_{+}$ | $\delta$
0.13 | 0.875 |  0.875 | 0 | 0.65 | 0.357 |  1.39 | 1.037
0.20 | 0.587 |  1.313 | 0.726 | 0.75 | 0.518 |  1.175 | 0.657
0.25 | 0.518 |  1.509 | 0.991 | 0.813 | 0.904 |  0.904 | 0
Table 1: Radius of inner and outer horizons and $\delta=r_{+}-r_{-}$ for
different values of charge $q$.

From the figure 1, we can see that the metric function of the black hole
increases with increasing CS parameter $(a)$ and the metric of the black hole
decreases with increasing deviation parameter ($k$). It is worthwhile to
mention that the behavior of the CS parameter is opposite to the deviation
parameter. The numerical analysis suggests that the inner horizon decreases
with increasing $a$ and decreasing $k$. However, the outer horizon increases
with increasing $a$ and decreasing $k$.

##  IV Thermodynamics

Now, we analyze the thermodynamical quantities associated with the black hole
solution (16) which is characterized by parameters $M$, $k$ and $a$. The black
hole mass $M$ can be obtained by setting metric function $f(r)=0$ at horizon
radius $r_{+}$. This gives

$M_{+}=\frac{(1-a)}{2}\,r_{+}e^{k/r_{+}}.$ (22)

This expression exactly coincides with the mass of the regular black hole for
$a=0$, however, this matches with the mass of the Schwarzschild black hole for
the vanishing deviation parameter and CS parameter.

The Hawking temperature ($T_{+}$) evaluated at the horizon is directly related
to the surface gravity $\kappa$ given by [39, 40]

$\kappa=\left(-\frac{1}{2}\nabla_{\mu}\xi_{\nu}\nabla^{\mu}\xi^{\nu}\right)^{1/2}=\frac{1}{2}f^{\prime}(r_{+}),$
(23)

and $\xi^{\mu}=\partial/\partial t$ is a Killing vector. This form of Hawking
temperature can also be derived from the tunneling method [41]. Corresponding
to black hole solution (16), the Hawking temperature is calculated by

$T_{+}=\frac{(1-a)}{4\pi r_{+}}\left[1-\frac{k}{r_{+}}\right].$ (24)

From the above expression, it is obvious that, for $k=0$ and $a=0$, the
Hawking temperature reduces to $T_{+}=1/4\pi r_{+}$ which is exactly
identified with the case of Schwarzschild black holes.

Next, we would like to study the entropy of our black hole solution. In
general, the explicit form of the entropy $S_{+}$ can be deduced from the
first-law of thermodynamics [42, 43]

$dM=T_{+}\,dS.$ (25)

In fact, this leads to

$\displaystyle S_{+}=\int\,\frac{1}{T_{+}}\frac{\partial M}{\partial
r_{+}}dr_{+}.$ (26)

Plugging the values of Mass (22) and temperature (24) into (26), we obtain the
entropy of our black hole as

$S_{+}=\pi\left(r_{+}(k+r_{+})e^{k/r_{+}}-k^{2}\text{Ei}\left[\frac{k}{r_{+}}\right]\right),$
(27)

where the last term is characterized by exponential integral. This entropy
does not match with the area-law. However, in the absence of $k$, this
expression exactly matches with the entropy of Schwarzschild black hole. From
the expression (27), it is obvious that the area-law is no longer valid for
non-singular black holes for large value of $k$. Now, we can also cross-check
the expression of Hawking temperature evaluated from the first-law of
thermodynamics (25). Here, we have

$T_{+}=\frac{\partial M_{+}}{\partial S_{+}}=\frac{(1-a)}{4\pi
r_{+}}\Big{[}1-\frac{k}{r_{+}}\Big{]}e^{-k/r_{+}}.$ (28)

Here, we see that this expression is not in agreement with one calculated by
area-law or tunneling method in (24). Since expressions of black hole
temperature (24) and (28) are estimated by two different methods. This means
that the first-law of thermodynamics (25) is not appropriate for deriving the
black hole temperature for regular black holes. The modified form of first-law
for black holes is presented in Refs. [44, 45]. This modification depends on
the general structure of the energy-momentum tensor. In fact, the conventional
first-law modifies with an additional factor if the black hole mass parameter
is included in the energy-momentum tensor. The modified first-law is,
therefore, given as [41, 44, 45]

$\mathcal{J}(M,r_{+})\,dM=T_{+}\,dS,$ (29)

where correction factor $\mathcal{J}(M,r_{+})$ is defined in terms of energy
density $T^{0}_{0}$ as

$\mathcal{J}(M,r_{+})=1+4\pi\int_{r_{+}}^{\infty}r_{+}^{2}\frac{\partial
T^{0}_{0}}{\partial M}dr_{+}=e^{-k/r_{+}}.$ (30)

The correction factor in our case is calculated by

$\mathcal{J}(M,r_{+})=e^{-k/r}.$ (31)

The entropy following the corrected first-law of thermodynamics (29) is
derived as

$S_{+}=\pi r_{+}^{2}=\frac{A}{4}.$ (32)

Obviously, this entropy is in the agreement of the area-law and matches
exactly with the entropy of black holes.

The thermodynamical stability of black holes can be understood by studying the
heat capacity of the black hole and, therefore, this is calculated by [40, 42]

$\displaystyle C_{+}=\frac{\partial M_{+}}{\partial T_{+}}=-4\pi
r_{+}^{2}e^{k/r_{+}}\frac{(1-k)}{2k-r_{+}}.$ (33)

It is interesting to note that the heat capacity obtained in the above
equation is independent of a CS parameter. This suggests that the
stability/instability of the present black hole is not affected by the
presence of CS. We also check a second-order phase transition (Davies point)
in the above-obtained solution, where the temperature of the black hole is
maximum $(dT_{+}/dr_{+}=0)$. The heat capacity shows two opposite behaviors:
one of them is negative in the region $r_{+}<r_{C}$ (critical radius) which
justifies the thermodynamic instability of black hole and, for the region
$r_{+}>r_{C}$, this is positive which warrants the black hole stability. In
fact, the heat capacity is discontinuous at $r_{+}=r_{C}$ and divergence
occurs, which conforms the existence of second-order phase transition [46,
47]. If the Davies point appears then the relationship between QNMs and phase
transition still exists [33].

---
Figure 2: The plot of specific heat versus horizon radius $r_{+}$ for
different value of deviation parameter $k$.

The Davies point occurs where the denominator of the heat capacity (33)
ceases. This gives $r_{+}=2k$. This confirms that the Davies point occurs
where the values of deviation parameter is half of the horizon radius.

##  V Null geodesics and photon sphere

Null geodesics are very important to explain the QNMs of a black hole solution
[48]. Here, we follow the photon ring/QNM correspondence [49, 50] to calculate
QNMs of this black hole solution in the eikonal limit. To compute the
geodesics in spacetime, we first focus on the motion of photons in the black
hole solution (16). The Lagrangian for a photon motion limited to equatorial
plane $(\theta=\pi/2)$ can be expressed as

${\cal{L}}=-g_{tt}{\dot{t}}^{2}+g_{rr}{\dot{r}}^{2}+g_{\theta\theta}{\dot{\theta}^{2}}+g_{\phi\phi}{\dot{\phi}}^{2},$
(34)

and the corresponding Hamiltonian is written as

${\cal{H}}=p_{t}\dot{t}+p_{r}\dot{r}+p_{\phi}\dot{\phi}-{\cal{L}}$ (35)

where over-dot refers to the derivative with respect to an affine parameter.
The generalized momenta are given by

$\displaystyle p_{t}=\frac{\partial{\cal H}}{\partial{\dot{t}}}\equiv
E=\text{constant},$ (36) $\displaystyle P_{\phi}=\frac{\partial{\cal
H}}{\partial{\dot{\phi}}}\equiv-J=\text{constant},$ (37) $\displaystyle
p_{r}=\frac{\partial{\cal H}}{\partial{\dot{r}}}=g_{rr}{\dot{r}}$ (38)

The equation of motion associated with the photon can be calculated using the
Hamiltonian formalism, which can be derived as follow

${\dot{t}}=\frac{\partial{\cal H}}{\partial
p_{t}}=-\frac{p_{t}}{g_{tt}},\qquad\qquad{\dot{\phi}}=\frac{\partial{\cal
H}}{\partial
p_{\phi}}=\frac{p_{\phi}}{r^{2}},\quad\text{and}\quad{\dot{r}}=\frac{\partial{\cal
H}}{\partial p_{r}}=-\frac{p_{r}}{g_{rr}}.$ (39)

The Hamiltonian is independent of time $(t)$ and azimuthal coordinate
$(\phi)$. The null geodesics equation is given by

${\dot{r}}^{2}+V_{eff}(r)=0,\qquad\text{with}\qquad
V_{eff}=f(r)\left(\frac{J^{2}}{r^{2}}-\frac{E^{2}}{f(r)}\right).$ (40)

For circular null geodesic which describes the radius of the photon sphere,
the effective potential must follow the conditions:

$V_{eff}=\frac{\partial V_{eff}}{\partial r}=0.$ (41)

The equation of the photon radius ($r_{p}$) is calculated as

$\frac{2(kM-r_{p}(3M-(1-a)e^{k/r}r))}{r^{2}_{p}(2M-(1-a)e^{k/r}r)}=0.$ (42)

This equation can not be solved analytically, so we solve it numerically.

We can determine the angular radius of the black hole shadow by setting $r$ to
$r_{p}$ (circular orbit of the photon) in Eq. (41). The shadow radius for this
black hole is calculated by

$r_{s}=\sqrt{\alpha^{2}+\beta^{2}}=\frac{L_{p}}{E}=\frac{r}{\sqrt{f(r)}}|_{r=r_{p}}.$
(43)

Substituting the value of $f(r)$ from Eq. (16) into Eq. (43), then we plug the
numerical value of $r_{p}$ in the obtained equation and the numerical results
are presented in TABLE 2 and TABLE 3. It is noticed from the TABLES that the
photon radius increases with the deviation parameter and decreases with the CS
parameter.

The shadow of the black hole can be visualized with the help of celestial
coordinates $\alpha$ and $\beta$ [51, 52, 53, 54]. For the obtained black hole
solution the $\alpha$ and $\beta$ are given by

$\displaystyle\alpha=\lim_{r\to\infty}\left(\frac{rp^{\phi}}{p^{t}}\right),\qquad\text{and}\qquad\beta=\lim_{r\to\infty}\left(\frac{rp^{\theta}}{p^{t}}\right).$
(44)

|  | $r_{p}$ |  |  |  |  | $r_{s}$ |
---|---|---|---|---|---|---|---|---
$k$ | $a=0.1$ | $a=0.2$ | $a=0.3$ | $a=0.4$ | $a=0.1$ | $a=0.2$ | $a=0.3$ | $a=0.4$
$0.0$ | 3.333 | 3.751 | 4.285 |  5.001 | 0.544 |  0.516 | 0.482 |  0.447
$0.1$ | 3.195 | 3.612 | 4.148 |  4.863 | 0.541 |  0.511 | 0.478 |  0.443
$0.2$ | 3.047 | 3.466 | 4.004 |  4.720 | 0.534 |  0.505 | 0.474 |  0.440
$0.3$ | 2.886 | 3.309 | 3.851 |  4.570 | 0.529 |  0.497 | 0.468 |  0.436
$0.4$ | 2.710 | 3.139 | 3.687 |  4.412 | 0.513 |  0.480 | 0.461 |  0.431
$0.5$ | 2.512 | 2.953 | 3.511 |  4.244 | 0.497 |  0.477 | 0.453 |  0.425
$0.6$ | 2.283 | 2.745 | 3.318 |  4.065 | 0.476 |  0.463 | 0.443 |  0.418
$0.7$ | 1.999 | 2.505 | 3.105 |  3.872 | 0.441 |  0.440 | 0.431 |  0.410
Table 2: The numerical values of photon radius corresponding to the deviation parameter $(k)$ with fixed CS parameter $(a)$. |  | $r_{p}$ |  |  |  |  | $r_{s}$ |
---|---|---|---|---|---|---|---|---
$a$ | $k=0.1$ | $k=0.2$ | $k=0.3$ | $k=0.4$ | $k=0.1$ | $k=0.2$ | $k=0.3$ | $k=0.4$
$0.0$ | 2.8615 | 2.711 | 2.587 |  2.453 | 0.570 |  0.560 | 0.556 |  0.553
$0.1$ | 3.1954 | 2.041 | 2.926 |  2.790 | 0.541 |  0.532 | 0.530 |  0.528
$0.2$ | 3.6126 | 3.460 | 3.348 |  3.220 | 0.511 |  0.505 | 0.502 |  0.501
$0.3$ | 4.1488 | 4.004 | 3.889 |  3.764 | 0.478 |  0.474 | 0.472 |  0.471
$0.4$ | 4.8636 | 4.720 | 4.619 |  4.490 | 0.444 |  0.440 | 0.439 |  0.438
$0.5$ | 5.8641 | 5.721 | 5.612 |  5.490 | 0.405 |  0.402 | 0.402 |  0.401
$0.6$ | 7.3647 | 7.225 | 7.117 |  7.005 | 0.363 |  0.361 | 0.361 |  0.361
$0.7$ | 9.8652 | 9.727 | 9.621 |  9.513 | 0.315 |  0.313 | 0.313 |  0.313
Table 3: The numerical values of photon radius corresponding to the CS
parameter $(a)$ with fixed deviation parameter $(k)$ .

The shadow of the obtained solution for different values of CS parameter $(a)$
and the different values of deviation parameter $(k)$ is depicted in Fig. 3.

---
Figure 3: The plot of black hole shadow corresponding to the CS parameter
$(a)$ with fixed deviation parameter $(k)$ (upper panel) and corresponding to
the deviation parameter $(k)$ with fixed CS parameter $(a)$ (lower panel).
---
Figure 4: Upper panel: the real and imaginary part of QNMs with respect to $a$
for different $k$. Lower panel: the real and imaginary part of QNMs with
respect to $k$ for different $a$.

From the figure we see that the size of the shadow decreases with the
increasing CS parameter and deviation parameter (as evident from TABLE 2 and
TABLE 3 also). As we know the black hole in the limit $r>>k$, the given black
hole corresponds to the Reissner-Nordström black hole with CS parameter. So,
we can easily comment that the shadow of the resulting black hole with large
horizon radius and small value of $k$ will only match to the shadow of the
Reissner-Nordström black hole. However, as the value of deviation parameter
will increase the shadow radius of the given black hole will get smaller
comparative to the Reissner-Nordström black hole case.

##  VI Connection beteween shadow radius and QNMs

The QNMs encode the details about the stability of the black holes under small
perturbation and these are described by the $\omega=\omega_{R}+i\omega_{I}$.
The signature of the $\omega_{I}$ characterizes the stability of the black
hole. For instance, $\omega_{I}>0$ corresponds to stable modes of black holes
and the rest corresponds to unstable modes. The real part and imaginary part
of the QNMs in the eikonal limit are associated with the angular velocity and
the Lyapunov exponent of (unstable) circular null geodesics, respectively [32,
55]. Such a connection also occurs for the QNMs and gravitational lensing
[56]. The real part of QNMs in the eikonal limit is associated with the radius
of the black hole shadow [57, 58]. Such connections had already been explored
to various black holes [59, 60, 61].

The QNMs frequency $\omega$ can be estimated through photon sphere as

$\omega=l\Omega-i\left(n+\frac{1}{2}\right)|\Lambda|,$ (45)

where $n$ is the overtone number and $l$ is the angular quantum number. Here,
the angular velocity $(\Omega)$ and the Lyapunov exponent $(\Lambda)$ of the
photon sphere are given, respectively, by

$\displaystyle\Omega=\frac{\sqrt{f(r_{p})}}{r_{p}}=\frac{1}{L_{p}},$ (46)
$\displaystyle\Lambda=\frac{\sqrt{f(r_{p})(2f{r_{p}}-r^{2}_{p}f^{\prime\prime}(r_{p}))}}{\sqrt{2}r_{p}}.$
(47)

The QNMs are characterized by complex numbers. The imaginary part of the QNMs
tells about the stability of the black hole such that when $Im\ \omega>0$ the
system is stable and when $Im\ \omega<0$ the system is unstable. The real and
imaginary parts of the QNMs for the given black hole (16) for different values
of deviation parameter $(k)$ and CS parameter $(a)$ are depicted in the figure
4. From the figure, we make the following observations:

  1. 1.

The upper panel of figure 4 tells that the real (imaginary) part of the QNMs
frequency decreases (increases) with increasing the CS parameter. This
indicates that the scalar field perturbation in the presence of deviation
parameter oscillates faster and decays slower as compared to Reisner-Nordström
black hole [62].

  2. 2.

The lower panel of figure 4 suggests that both the real and imaginary parts of
the QNMs frequency increase with the deviation parameter. This means that the
scalar field perturbation in the presence of CS parameter oscillates and
decays faster as compared to Reisner-Nordström black hole [62].

Here, it is worthwhile to mention that the effects of $k$ and $a$ are opposite
to each other. From the right panel of figure 4, we notice that the
$Im\,\omega$ is negative for both $k$ and $a$. Therefore, the obtained black
hole solution is stable.

##  VII Conclusions

The nonlinear electrodynamics as a source of gravity has the possibility to
create regular black holes and star (or soliton) like configurations of
interest. In this work, we have focused on a gravity model coupled with the
nonlinear electrodynamics (which resembles the electrodynamics in the weak
field limit) in the presence of a cloud of strings as the source. The dynamics
of such the gravity model is described by a suitable action and corresponding
equation of motions.

Furthermore, we have discussed the thermodynamics of the black hole solution.
The effects of the deviation parameter and CS parameter on the temperature are
emphasized. The temperature is a decreasing function of the deviation
parameter and CS parameter. Hawking temperature of the black hole is
Interestingly, here we have found that instead of the standard first-law of
thermodynamics this black hole follows a modified first-law of thermodynamics.
From the specific heat, we have found that there is no effect of surrounding
CS on the stability of black hole. However, the second-order phase transition
occurs at certain point (Davies point), where deviation parameter is half of
the value of horizon radius. The existence of Davies point confirms the
relationship between QNMs and phase transition. QNMs for this particular black
hole solution in the eikonal limit are calculated under the consideration of
photon ring/QNM correspondence. The photon radius is obtained numerically.
There are many interesting opportunities that can be studied further in future
works. For example, it will be interesting to explore the quantum effects on
the thermodynamics and stability of the regular black hole solution.

We also study the relation between the shadow radius and QNMs of the solution
(16). From the figures 3 and 4, we found that the real and imaginary part of
the QNMs decreases and increases with the increasing the CS parameter,
respectively. However, and the shadow radius decreases with the increasing the
CS parameter. However, the real and imaginary part of QNMs increases with
increasing deviation parameter and shadow radius decreases with increasing
deviation parameter. The shadow behavior and photon sphere in the presence of
black hole parameter $k$ and $a$ are also studied. The numerical results (cf.
TABLE 2) confirmed that the photon sphere $(r_{p})$ and shadow radius
$(r_{s})$ are decreasing with the deviation parameter. However, with the CS
parameter (cf. TABLE 3), the photon radius ($r_{p}$) is increasing and shadow
radius ($r_{s}$) is decreasing.

## Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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