# Optical Properties of an Axially Symmetric Black Hole in the Rastall Gravity

Bakhtiyor Narzilloev<EMAIL_ADDRESS>Ulugh Beg Astronomical
Institute, Astronomy St. 33, Tashkent 100052, Uzbekistan  Akfa University,
Milliy Bog’ Street 264, Tashkent 111221, Uzbekistan  National University of
Uzbekistan, Tashkent 100174, Uzbekistan  Ibrar Hussain
<EMAIL_ADDRESS>School of Electrical Engineering and Computer
Science, National University of Sciences and Technology, H-12, Islamabad,
Pakistan  Ahmadjon Abdujabbarov<EMAIL_ADDRESS>Shanghai Astronomical
Observatory, 80 Nandan Road, Shanghai 200030, P. R. China  Ulugh Beg
Astronomical Institute, Astronomy St. 33, Tashkent 100052, Uzbekistan
Institute of Nuclear Physics, Ulugbek 1, Tashkent 100214, Uzbekistan  National
University of Uzbekistan, Tashkent 100174, Uzbekistan  Institute of
Fundamental and Applied Research, National Research University TIIAME, Kori
Niyoziy 39, Tashkent 100000, Uzbekistan  Bobomurat Ahmedov<EMAIL_ADDRESS>Ulugh Beg Astronomical Institute, Astronomy St. 33, Tashkent 100052,
Uzbekistan  National University of Uzbekistan, Tashkent 100174, Uzbekistan
Institute of Fundamental and Applied Research, National Research University
TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan

###### Abstract

This work is devoted to the study of the optical properties of the charged-
rotating-NUT-Kiselev (CRNK) black hole in the Rastall theory of gravity. By
investigating the motion of photons in the CRNK black hole spacetime in the
Rastall gravity we show that the deflection angle of photons due to the
gravitational lensing is mostly influenced by the NUT charge, parameter of the
equation of state for the quintessence and the quintessential intensity. We
observe that the effects of the rest of the spacetime parameters are
negligible on the deflection angle. We present the shape of the shadow cast by
the CRNK black hole in the Rastall gravity for various values of the spacetime
parameters. We demonstrate that spin parameter $a$ of the black hole changes
the position and size of the shadow of the black hole very little. While
changing the values of the rest of the spacetime parameters can alter the
shape and the size of the observed shadow of the black hole significantly. We
observe that the radius of the black hole shadow and the deflection angle of
photon are bigger for the CRNK black hole in the Rastall gravity, as compared
with the case of the Kerr black hole. Lastly, we show how the spacetime
parameters can change the observables $R_{s}$ and $\delta_{s}$, being the
average radius of the shadow and the distortion parameter that measures the
deviation of the shape of the shadow from a perfect circle with radius
$R_{s}$, respectively. Our results show that in most scenarios, for bigger
values of $R_{s}$ we get smaller values of the parameter $\delta_{s}$.

###### pacs:

04.70.Bw, 04.50.Kd, 04.70.-s

##  I Introduction

To understand the gravitational fields of the ultra compact objects like black
holes, the investigation of the geodesics of test particles and photons in
black hole spacetimes is of interest from the astrophysical point of view. The
photon motion around compact objects has a direct connection with one of the
predictions of the Einstein theory of General Relativity (GR) about the light
bending. According to GR, due to the curvature produced by a compact object
light bends and due to this bending of light or photon beam by a gravitating
source, it fabricates an effect very similar to that of a lens, called
Gravitational Lensing (GL) Liebes (1964); Wambsganss (1998). The phenomena of
GL was first observed in the solar eclipse of 1919 by Arthur Eddington. After
the discovery of quasars in 1963, which are very bright and distinct objects,
the GL effects are studied with deeper interest Schmidt (1963). The first ever
GL effect of a quasar was observed in 1979 Walsh  _et al._ (1979). The GL
which distorts the image of some background galaxy in its line of sight, can
also amplify its light and can provide images of fainter and more distant
galaxies with the help of telescopes. Thus GL is a useful tool to study the
nature of galaxies too far away from us to be seen with the use of technology.
The GL may be also helpful in detecting the mysterious dark matter in the
Universe (see for example Massey  _et al._ (2010)).

Using the GL as an important tool various properties of static and rotating
black holes have been studied in the GR and other modified theories of gravity
Bozza (2003); Bozza and Mancini (2004); Beckwith and Done (2005a); Schee  _et
al._ (2017); Narzilloev  _et al._ (2021a). The impact of different fields
including magnetic field and quintessential field on the GL have been
thoroughly investigated in the literature Sereno (2005); Sharif and Iftikhar
(2015); Younas  _et al._ (2015); Azreg-Aïnou  _et al._ (2017); Turimov  _et
al._ (2019). An analytic method for the GL in the spacetime field of the
Schwarzschild black hole was first studied by Boazza et. al Bozza  _et al._
(2001). Their technique was later generalized for all spherically symmetric
and statics spacetimes Bozza (2002). The effects of the spin of a black hole
on the GL have been studied in a separate work by Boazza Bozza (2010). A
comparative study of the GL by the Schwarzschild black hole and naked
singularity has been presented by Virbhdra and Keeton, where they have
observed a qualitative difference in the lensing characteristics in the two
cases Virbhadra and Keeton (2008). An approach for the study of the GL in the
Kerr spacetime has been provided by Beckwith and Done Beckwith and Done
(2005b). The effects of charge of black holes on the GL have also been studied
in rotating and non rotating spacetimes to understand the gravitational fields
and the spacetime geometry in these cases Fernando and Roberts (2002); Kumar
_et al._ (2020a); Hsieh  _et al._ (2021).

The Event Horizon Telescope (EHT) collaboration released the first ever image
of a black hole at the centre of a nearby galaxy in 2019, which has further
enhanced the interest of researches in studying the black hole features. When
photon or light passes in the close vicinity of a black hole, due to the
deflection by the gravitational field of the black hole the photons can move
in circular orbits or rings, known as light rings (LRs). Due to the LRs the
central black hole appears as a dark disc in the sky. This dark discs is known
as the black hole shadow. The idea about the observation of the black hole
shadow was first introduced by Falcke et al. Falcke  _et al._ (2000). The
masses of supermassive and stellar mass black holes have been measured
observationally during the last decade of the previous century Brügmann  _et
al._ (2001). Another essential parameter of a black hole is assumed to be its
spin, whose measurement is a crucial challenge in the present day
astrophysics. To estimate the black hole spin, some methods have been put
forwarded by different people (see for example Saida (2017); Riaz  _et al._
(2020)). It has been proposed that the black hole shadows may be useful in
estimating the spin of the background black holes Falcke  _et al._ (2000);
Riaz  _et al._ (2020). Black hole shadows have been investigated in different
black hole spacetimes from different point of views (for example Takahashi
(2004); Atamurotov and Ahmedov (2015); Abdujabbarov  _et al._ (2016); Chang
and Zhu (2021)).

The generalization of the very first black hole solution of the Einstein field
equations, known as the Schwarzschild black hole solution, in the presence of
some physical parameters has been remain a topic of interest. After the
introduction of charge and spin in the Schwarzschild black hole solution,
people have further studied this solution in the presence of some other
quantities including magnetic field and some dark energy candidates. In this
regard Kiselev has been obtained the Schwarzschild black hole solution in the
quintessence background Kiselev (2003). The charged and rotating versions of
the Kiselev black hole solution have also been reported in the literature
Kiselev (2003); Toshmatov  _et al._ (2017). The rotating version of the
charged Kiselev black hole solution has been further generalized in the
presence of the NUT parameter and has been analysed in different context in
the literature Sakti  _et al._ (2019); Molla and Debnath (2021). In the recent
years the charged rotating version of the Kiselev black hole with the NUT
parameter has further been studied in the Rastall theory of gravity Sakti  _et
al._ (2020a). Recently particle dynamics have been studied in the spacetime of
the charged-rotating Kiselev black hole with the NUT parameter in the Rastall
theory of gravity Narzilloev  _et al._ (2021b). Further the effects of the
Rastall parameter and other spacetime parameters have been investigated on the
fundamental frequencies of test particles at small distances from the stable
circular orbits, in the charged-rotating Kiselev black hole wit the NUT
parameter in the background of the Rastall theory of gravity Narzilloev  _et
al._ (2021b). The spacetime properties of rotating black holes and exotic
compact objects in the Kerr/CFT correspondence within the Rastall gravity have
been studied in Sakti  _et al._ (2020a). In Ref. Mota  _et al._ (2022) the
Rastall gravity is generalized to the case when the conservation law of the
energy-momentum tensor is altered. Exact cylindrical symmetric solutions of
the field equations in the Rastall theory of gravity have been obtained in
Farasat Shamir  _et al._ (2021). The properties of rapidly rotating compact
stars in the Rastall gravity are studied in da Silva  _et al._ (2021).
Gravastar solution within the framework of the Rastall gravity has been
explored in Debnath (2021) . The effects of the generalized Rastall gravity on
the properties of compact objects have been analyzed in Mota  _et al._ (2020).
It has been shown that for a suitable choice of the coupling parameter of the
Rastall gravity in the presence of the quintessence parameter, the original
results in the standard GR can be retrieved Shahzad and Abbas (2020). Authors
of Sakti  _et al._ (2020b) have obtained a new twisted rotating black hole
solution applying Demianski-Newman-Janis algorithm to the electrically and
dyonically charged black hole with quintessence in the Rastall theory of
gravity. Other axisymmetric solutions of the field equations of the Rastall
gravity have been obtained in Refs. Mustafa  _et al._ (2020); Moradpour  _et
al._ (2019); Abbas and Shahzad (2018); Spallucci and Smailagic (2018). In a
recent work shadow cast by a black hole surrounded by an anisotropic fluid in
the Rastall theory of gravity has been studied Kumar  _et al._ (2021).
Interested readers may see our previous works devoted to the investigation of
spacetime properties around several massive compact objects Narzilloev  _et
al._ (2019, 2020, 2020a, 2020b, 2020c, 2021); Hakimov  _et al._ (2017);
Shaymatov  _et al._ (2021); Narzilloev  _et al._ (2021a); Rayimbaev  _et al._
(2021); Narzilloev  _et al._ (2021b). It has been demonstrated in Visser
(2018) that at the global scale the Rastall gravity can be treated as an
essential and trivial re-arrangement of the matter sector in the standard
Einstein gravity. If one compares the Einstein and the Rastall theories of
gravity then one may observe that gravity parts of the both theories are
equivalent to each other. Particularly, the non-generic case of the Rastall
gravity is nothing else but the standard Einstein gravity implemented with the
cosmological constant. This fact also drives the motivation to study the
Rastall gravity in the sense of studying the matter sector of the theory,
deeply using the spacetime properties in the close vicinity of gravitational
compact objects.

In the present work we are keen in the investigation of photon motion and
related phenomena of GL and black hole shadows in the spacetime of the
charged-rotating-NUT-Kiselev (CRNK) black hole in the Rastall gravity. In
particular, we are interested to look at the effects of the Rastall parameter
and the other spacetime parameters on the GL and shadow cast by the CRNK black
hole.

This paper is organized as follows: In the next Section we discuss the
spacetime structure of the CRNK black hole in the Rastall theory of gravity.
In the Section III the GL of the CRNK black hole in the Rastall gravity is
studied. Section IV is devoted to the analysis of shadow of the CRNK black
hole in the Rastall gravity. In the last Section we give a conclusion of our
work done here.

##  II Charged-Rotating-NUT-Kiselev spacetime in the Rastall gravity

The GR is a promising theory of gravity and has already passed some
observational tests including the deflection of light by gravitational sources
and the generation of gravitational waves from a binary system of black holes
Abbott (2016). There are still some mysteries exist, for instance the
quantization of gravity on a curved background and the accelerated expansion
of the Universe, which cannot be well explained in the framework of GR.
Therefore, alternative approaches for a well accepted theory of gravity have
been proposed Paul  _et al._ (2009). One such proposal is the Rastall theory
of gravity which is a modification of the GR such that the matter field is
non-minimally coupled with the geometry Rastall (1972). In the Rastall theory
of gravity rotating and non-rotating black hole solutions have been reported
in the literature Lobo  _et al._ (2018); Kumar and Ghosh (2018). Here we
mention that it was claimed by Visser Visser (2018), that the Rastall theory
is equivalent to the GR. Later Darbari et. al Darabi  _et al._ (2018), have
shown that the two theories of gravity i.e the GR and the Rastall theory are
not equivalent and the latter one is an alternative theory of gravity and is
to be considered as an open theory as compared to the GR.

The Kerr black hole, which can be specified by its mass and spin only, and
hence obey the no-hair theorem of black holes, is to be considered as a strong
candidate for an astrophysical black hole. However there is no direct evidence
for it and the astrophysical observations made so far could not proved that
the astrophysical black hole can exactly be described by the Kerr metric.
Therefore the investigation of different properties of rotating black hole
spacetimes in theories of gravity other the GR is a hot area of research.
Charged and as well as uncharged rotating black hole in the Rastall theory of
gravity have recently been discussed in the literature Kumar and Ghosh (2018);
Xu  _et al._ (2018). The recent observation of the image of the M87* black
hole by the EHT collaboration provides a tool to test the features of gravity
in the strong field regime and also to test the hypothesis of the no-hair
theorem of black holes.

Nevertheless the shadow cast by the M87* is almost in agreement with the
shadow of the Kerr black hole, however, deviations from the Kerr black hole
solution of the GR those arise from the different modified theories of gravity
are still in the consideration Bambi  _et al._ (2019); Vagnozzi and Visinelli
(2019); Kumar  _et al._ (2019, 2020b). In this connection the Rastall theory
of gravity is to be considered to have some deviation from the GR in the
strong field regime. intriguingly, the quantum effects arising in the strong
gravity regime Maldacena and Susskind (2013); Giddings (2012), the non-
commutative effects Wei  _et al._ (2015), asymptotic safe behaviour Held  _et
al._ (2019); Kumar  _et al._ (2020c) and fluctuations in the metric
coefficients Giddings and Psaltis (2018) are observed to have some influence
on the shadow cast by a black hole. Motivated from these observations, here we
investigate the optical properties of the CRNK black hole in the Rastall
gravity. The spacetime metric of the CRNK black hole in the Rastall gravity
has the following form and can be checked as a solution of the Rastall
(modified Einstein) field equations using the Mathematica package RGTC (see
Sakti  _et al._ (2020b))

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta}{\Sigma}[dt-\\{a\sin^{2}\theta+2l(1-\cos\theta)\\}d\phi]^{2}$
$\displaystyle+$ $\displaystyle\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}[adt-\\{r^{2}+(a+l)^{2}\\}d\phi]^{2}\,,$

where

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
r^{2}-2Mr+a^{2}+e^{2}+g^{2}-l^{2}-\alpha r^{v},$ (2) $\displaystyle v$
$\displaystyle=$ $\displaystyle\frac{1-3\omega}{1-3\kappa\lambda(1+\omega)},$
(3) $\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle
r^{2}+(l+a\cos\theta)^{2}\ .$ (4)

The CRNK black hole solution in the Rastall gravity contains seven parameters
where $M$ is the total mass and $a$ is the specific angular momentum of the
black hole. The other parameters included in the spacetime metric of the CRNK
black hole in the Rastall gravity are as follows: $l$ is the NUT parameter,
$\alpha$ is the quintessential intensity, $\kappa\lambda$ is the Rastall
gravity parameter, $\omega$ is the parameter of equation of state of the
quintessence, $e$ and $g$ correspond to the electric and magnetic charge of
the black hole, respectively Sakti  _et al._ (2020b). One can introduce new
parameter $q^{2}=e^{2}+g^{2}$ that involves the contribution of both the
electric and the magnetic charges of the black hole to the spacetime
investigated here.

The CRNK black hole solution derived in the Rastall gravity may have two or
three horizons depending on the values of the different spacetime parameters
involved there Sakti  _et al._ (2020b). The event horizon is determined as the
coordinate singularity of the spacetime and is a null hypersurface of constant
$r$. Consequently, the horizons are determined as the roots of the following
algebraic equation

$\displaystyle r^{2}-2Mr+a^{2}+q^{2}-l^{2}-\alpha r^{v}=0\ .$ (5)

For selected values of the parameters $\omega$ and $\kappa\lambda$, the
horizons coincide to have only inner and outer parts or there is no
cosmological horizon. In this specific case of the horizon, the thermodynamic
properties of the black hole are easier to be explored. It is the reason why
in a recent study Xu  _et al._ (2019), only two roots for the horizon equation
are investigated. In the case when the horizon equation has more than two
roots, it becomes very difficult to perform calculations, for example, to
study the entropy product. In the Table 1 those values of the parameters
$\omega$ and $\kappa\lambda$ are presented which give two analytic roots of
the horizon equation and are collected in Sakti  _et al._ (2020b).

Table 1: Selected values of black hole parameters $\omega$ and $\kappa\lambda$ which allow exact analytical solutions for inner and outer horizons presented in the last column. $\omega$, $\kappa\lambda$ |  | Horizon ($r_{\pm}$)
---|---|---
0,0 |  | $\left(M+\frac{\alpha}{2}\right)\pm\sqrt{\left(M+\frac{\alpha}{2}\right)^{2}+l^{2}-a^{2}-q^{2}}$
-1/3,0 |  | $\frac{M}{1-\alpha}\pm\frac{\sqrt{M^{2}-(a^{2}+q^{2}-l^{2})(1-\alpha)}}{1-\alpha}$
0,1/6 |  | $\frac{M}{1-\alpha}\pm\frac{\sqrt{M^{2}-(a^{2}+q^{2}-l^{2})(1-\alpha)}}{1-\alpha}$
-1/3,-1/2 |  | $\left(M+\frac{\alpha}{2}\right)\pm\sqrt{\left(M+\frac{\alpha}{2}\right)^{2}+l^{2}-a^{2}-q^{2}}$
1/3,0 |  | $M\pm\sqrt{M^{2}+l^{2}+\alpha-a^{2}-q^{2}}$

##  III Photon geodesics and gravitational lensing

In this section we discuss motion of photon and the phenomenon of the GL in
the spacetime of CRNK black hole in the Rastall theory of gravity.

###  III.1 Photon motion

Photon geodesics can be explored using the following Hamilton-Jacobi equation
of motion

$\displaystyle\frac{\partial
S}{\partial\lambda}=-\frac{1}{2}g^{\alpha\beta}\frac{\partial S}{\partial
x^{\alpha}}\frac{\partial S}{\partial x^{\beta}},$ (6)

where $\lambda$ defines the affine parameter. When the spacetime is axially
symmetric one can write the action in the following form

$\displaystyle S=-\mathcal{E}t+\mathcal{L}\phi+S(r,\theta)\ ,$ (7)

were $\mathcal{E}$ and $\mathcal{L}$ are the conserved energy and the angular
momentum of photon, respectively, in the axial symmetric spacetime of the CRNK
black hole in the Rastall gravity. It can be easily checked that the Hamilton-
Jacobi equation of motion is separable in the CRNK black hole spacetime and
the components of the four velocity of a photon can be written as

$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\frac{\left((a+l)^{2}+r^{2}\right)\left(\mathcal{E}\left((a+l)^{2}+r^{2}\right)-a\mathcal{L}\right)+\Delta\chi\csc^{2}\theta(\mathcal{L}-\chi\mathcal{E})}{\Delta\Sigma},$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{a\left(\mathcal{E}\left((a+l)^{2}+r^{2}\right)-a\mathcal{L}\right)+\Delta\csc^{2}\theta(\mathcal{L}-\chi\mathcal{E})}{\Delta\Sigma},$
(8) $\displaystyle f(r)=\dot{r}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma}\sqrt{\left(\mathcal{E}\left((a+l)^{2}+r^{2}\right)-a\mathcal{L}\right)^{2}-\Delta\mathcal{K}},$
$\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma}\sqrt{\mathcal{K}-\frac{(a\mathcal{E}\cos
2\theta+4l\mathcal{E}\cos\theta+2\mathcal{L}-\mathcal{E}(a+4l))^{2}}{8(1+\cos\theta)\sin^{2}\frac{\theta}{2}}}\
,$

where $\mathcal{K}$ is the Carter constant, and notation
$\chi=a\sin^{2}\theta+2l(1-\cos\theta)$ is introduced for convenience.

###  III.2 Gravitational lensing

In this subsection we explore the lensing of photons caused by the
gravitational field of the central CRNK black hole in the Rastall gravity.
Assume that a photon is coming from infinity in the equatorial plane of the
CRNK black hole in the Rastall gravity with impact parameter defined as
$b=\mathcal{L}/\mathcal{E}$ and going back to asymptotic infinity. Using the
condition $\dot{r}=0$, one can get the relationship between the radius of
closest approach $r_{0}$ and the impact parameter from the third equation
given in the system (III.1). This relation between $r_{0}$ and $b$ is
demonstrated in Fig.1 for the chosen values of the spacetime parameters. In
the first row of the Fig.1 the lines for the chosen values of the NUT
parameter $l$ and the Rastall parameter $\kappa\lambda$ are presented while
the other spacetime parameters are kept fixed. It can be seen from the
behavior of the lines in the Fig.1 that increase of the NUT parameter reduces
the radius of the closest approach of photons for the fixed value of the
impact parameter. This in turn means that the increase of the NUT parameter
increases the intensity of the gravitational field of the central black hole.
A decrease in the value of the parameter $\kappa\lambda$ however shows that
(from the left figure to the right in the same row) the radius of the closest
approach of photons goes up slightly, reducing the effect of the gravitational
field of the central black hole. In the second row of the Fig.1 the change in
the lines for the selected values of the parameters $\omega$ and
$\kappa\lambda$ are given. It is shown that at large distances the behavior of
the graph is almost linear and for the values of $\omega$ being close to $0$,
the radius of the closest approach is nearly the same as the impact parameter.
In the same figure it can be also seen that a decrease in the value of the
parameter $\kappa\lambda$ makes the values of $\rho_{0}$ and $b$ almost equal
for smaller values of the parameter $\omega$. In the third row of the Fig.1,
the plots correspond to the case of different values of $\alpha$ and
$\kappa\lambda$. It is shown that an increase in the parameter $\alpha$ shifts
the lines towards the smaller radius of the closest approach of photons,
making the gravitational field of the central object stronger. A decrease in
the value of the parameter $\kappa\lambda$ puts the lines closer to each other
and shifts the them up, slightly decreasing the gravitational field of the
black hole. In the first plot of the last row in the Fig.1, the behavior of
the lines for the different values of the spin parameter $a$ is presented. One
can see a slight change in the lines with the increase in the value of the
spin parameter $a$ and the changes become significant only in the close
vicinity of the CRNK black hole in the Rastall gravity. In the second plot of
the last row in the Fig.1, the behavior of the lines for different values of
the electromagnetic charge parameter $q$ is presented and can be explained in
the same way as it is explained above for the different values of the spin
parameter $a$. It means that the behavior of the plots is similar for both
parameters $a$ and $q$. In the last plot of the same figure, the case of
different values of the parameter $\kappa\lambda$ is demonstrated. It shows
that the decrease of the parameter $\kappa\lambda$ indeed makes the radius of
the closest approach $r_{0}$, bigger and hence decreases the gravitational
attraction between the central black hole and photon.

Figure 1: The dependence between the radius of closest approach $r_{0}$ and
the impact parameter $b$ for the different values of the black hole
parameters.

Using the second and third expressions in (III.1) for the case of equatorial
plane ($\theta=\pi/2$), one can get the following dependence between the
radial coordinate $r$ and the angle $\phi$:

$\displaystyle\frac{d\phi}{dr}=\frac{\Sigma\left(a\left(a^{2}+a(2l-b)+\left(l^{2}+r^{2}\right)\right)+\Delta((-a-2l)+b)\right)}{\sqrt{\left(a^{2}+a(2l-b)+\left(l^{2}+r^{2}\right)\right)^{2}-\Delta(a+2l-b)^{2}}}.$
(9)

In order to simplify the further calculations one can introduce a new variable
being equal to the inverse of the radial coordinate as $u=1/r$. For this new
variable the expression above can be written in the following form

$\frac{d\phi}{du}=-\frac{d\phi}{dr}\frac{1}{u^{2}}$

. By integrating this expression one can obtain the dependence between the
bending angle of photon and the inverse of the radius of closest approach
$u_{0}=1/r_{0}$ for different values of the spacetime parameters as

$\displaystyle\delta=2\int_{0}^{u_{0}}\frac{d\phi}{du}du-\pi.$ (10)

It is difficult to evaluate this integral analytically. However, one can
evaluate the above integral numerically and obtain the desired dependence as
presented in Fig. 2. Notice that since $u_{0}$ is the inverse of the radius of
the closest approach, bigger values of this parameter corresponds to the
photon that approaches closer to the central black hole. One can see from the
first plot in the first row of the Fig. 2, that bigger values of the NUT
parameter cause the bending angle to get bigger values, that in turn
corresponds to the stronger gravitational field of the CRNK black hole in the
Rastall gravity. The second plot of the same row in the same figure shows the
dependence of the bending angle of photons for the change in the parameter
$\omega$. It can be seen that smaller values of this parameter make the
gravitational field stronger i.e. the trajectory of photon is going to be more
twisted in the spacetime of the CRNK black hole in the Rastall gravity. The
third plot in the first row of the same figure demonstrates how the change in
the parameter $\kappa\lambda$ modifies the bending angle of photons
propagating in the spacetime of the CRNK black hole in the Rastall gravity.
One can observe a very slight decrease in the values of the bending angle with
the decrease of the parameter $\kappa\lambda$. The change of the bending angle
with the increase of the parameter $\alpha$ is shown in the first plot of the
second row of the Fig. 2. One can see that increase of this parameter bends
the light stronger which means that the gravitational field becomes stronger
for bigger values of the parameter $\alpha$. In the second and third pictures
of the second row of the Fig. 2, one can see a tiny decrease of the bending
angle of photons with the rise of the spin and electromagnetic charge
parameters $a$ and $q$, respectively. In all plots the obtained results are
compared with the case of the Kerr spacetime and it is seen that the CRNK
black hole in the Rastall gravity deflects the light considerably stronger
than the Kerr black hole with the same mass parameter $M$.

Figure 2: The dependence between the bending angle of photons $\delta$ and the
parameter of closest approach $u_{0}$.

Before starting the investigation of the shadow formed around the CRNK black
hole in the Rastall gravity let us first explore the dependence of the photon
sphere on the spacetime parameters. This can be achieved by solving the
following conditions

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle 0\ ,$ (11) $\displaystyle
f^{\prime}(r)$ $\displaystyle=$ $\displaystyle 0\ ,$ (12)

where the form of the function $f(r)$ is given in (III.1) and prime denotes
the derivative with respect to the radial coordinate $r$. The dependence of
photon sphere on the spacetime parameters is demonstrated in Fig. 3. In the
first plot of the first figure one can see an increase in the radius of the
photon sphere with the increase of $\alpha$ and the NUT parameter $l$ which is
equivalent to the strengthening of the gravitational field around the CRNK
black hole in the Rastall gravity. Indeed, when the gravity is stronger
photons can approach closer to the central black hole forming the circular
trajectories and the strengthening of the gravitational field would increase
the radius of such trajectories. Comparing this plot with the second and third
plots in the same row of the Fig. 3, one can see a decrease in the radius of
the photon sphere with the decrease of the parameter $\kappa\lambda$. This
decreasing behaviour ensures once again that a decrease in the parameter
$\kappa\lambda$ weakens the gravitational effects around the CRNK black hole
in the Rastall gravity. In the second row of the same figure (from left to
right) the change in the radius of the photon sphere, with the change of the
parameter $\omega$, for fixed values of the NUT parameter $l$ and for
different values of the parameter $\kappa\lambda$ can be observed. It is
clearly demonstrated that decrease of the parameter $\omega$ makes the radius
of the photon sphere to grow up exponentially when $\omega$ approaches
$\omega\simeq-0.8$. In the third row of the same figure (from left to right)
the dependence of the radius of the photon sphere on the parameter
$\kappa\lambda$ is shown for some fixed values of the parameter $\alpha$ and
for the different values of the parameter $\omega$. One can see that when
$\omega$ is close to 0, then there is a small change in the radius of the
photon sphere. It is also evident that if the values of the parameter $\omega$
are decreasing then the change in the radius of the photon sphere with the
change of the parameter $\kappa\lambda$ becomes considerable only for bigger
values of the parameter $\alpha$. In the first plot of the last row in the
Fig. 3, the decrease in the radius of the photon sphere with the increase in
the values of the spin parameter $a$ is plotted and then it is compared with
the Kerr spacetime for different values of of the parameter $\omega$. One can
see similar behavior of the lines when $\omega$ is not far from $0$. It is
observed that when $\omega$ gets small enough, the difference from the Kerr
case becomes apparent. In the second plot of the same row in the Fig. 3 the
dependence of the radius of the photon sphere on the electromagnetic charge
parameter $q$ is presented for the selected values of the parameter $\alpha$.
It can be seen that increase of the electromagnetic charge $q$ reduces the
radius of the photon sphere. In the last figure the increase in the size of
the photon sphere with the rise of the NUT parameter $l$ is shown for the
various values of the parameter $\kappa\lambda$. The decrease of the radius of
the photon sphere with the decrease of the parameter $\kappa\lambda$ is
clearly observed in this picture. Consequently, the Rastall parameter
$\kappa\lambda$ amplifies the strength of the gravitational field of the CRKN
black hole in the Rastall gravity.

Figure 3: Dependence of the photon sphere radius on the spacetime parameters
of the CRNK black hole in the Rastall gravity.

In this section we observed that the increase of the black hole parameters
$l,\alpha,$ and decrease of the parameter $\omega$ make the interaction
between the central black hole and photon stronger. On the contrary, the
increase of the parameters $a$, $q$ and decrease of the parameter
$\kappa\lambda$ makes it weaker. Now we plan to investigate how these
spacetime parameters affect the shadow around the CRNK black hole in the
Rastall gravity.

##  IV Black hole shadow

In order to describe the shadow of the black hole one may use the celestial
coordinates defined as (see for example Vazquez and Esteban (2004)):

$\displaystyle
x=\lim_{r_{0}\rightarrow\infty}\left(-r_{0}^{2}\sin\theta_{0}\frac{d\phi}{dr}\right),$
(13)

and

$\displaystyle y=\lim_{r_{0}\rightarrow\infty}r_{0}^{2}\frac{d\theta}{dr},$
(14)

where $r_{0}$ is the distance between an observer and a black hole and
$\theta_{0}$ is the angular position of the axis of rotation of the black hole
with respect to the line of sight of the observer. The schematic
representation of the celestial coordinates is shown in Fig. 4. In Fig. 4 the
coordinates $x$ and $y$ describes the position of the points of the image with
respect to the axis of symmetry and equatorial plane, respectively. For the
spacetime (II), using $d\phi/dr$ and $d\theta/dr$ given in ((III.1)), one may
obtain the following expressions for the coordinates $x$ and $y$ as functions
of the impact parameter $b$ and parameter $l$ as

$\displaystyle x$ $\displaystyle=$
$\displaystyle-(b_{1}-2l+2l\cos\theta_{0})\csc\theta_{0},$ (15) $\displaystyle
y$ $\displaystyle=$ $\displaystyle\pm\sqrt{b_{2}-\frac{(a\cos
2\theta-a+2b_{1}+4l\cos\theta-4l)^{2}}{8(\cos\theta+1)\sin^{2}\frac{\theta}{2}}}.\
\ $ (16)

In Eqs. (15) and (16) the parameters $b_{1}=\mathcal{L}/\mathcal{E}$ and
$b_{2}=\mathcal{K}/\mathcal{E}^{2}$ are the impact parameters for general
orbits around the black hole (we refer the readers to Vazquez and Esteban
(2004) for the detailed calculations of these parameters for the Kerr
spacetime). These parameters can be found from the conditions
$\dot{r}=0=\partial_{r}\dot{r}$ and have the following form

$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2ar(M-r)+a\alpha
vr^{v}}(2r(M(a+l)^{2}$ $\displaystyle+$ $\displaystyle
r(a-3l)(a+l)-3Mr^{2}+2q^{2}r+r^{3})$ $\displaystyle+$ $\displaystyle\alpha
r^{v}\left(v(a+l)^{2}+r^{2}(v-4)\right)),$ $\displaystyle b_{2}$
$\displaystyle=$ $\displaystyle\frac{16r^{4}\left(a^{2}-l^{2}-2Mr+q^{2}-\alpha
r^{v}+r^{2}\right)}{\left(2r(M-r)+\alpha vr^{v}\right)^{2}}.$ (18)

Now one may introduce two observables in order to characterize the shape of
the shadow of a rotating black hole. We choose three points on the line
describing the border of the shadow, namely (A), (B), and (C) corresponding to
top, bottom, and rightmost positions of the image presented in the Fig. 4.
Note that the point (C) corresponds to the unstable retrograde equatorial
circular orbits observed by a distant observer. The radius of the reference
circle passing through these three points can be defined as the radius of the
black hole shadow $R_{s}$. Using the dent in the left hand side of the shadow
one may define the deflection $D_{cs}$ representing the difference between the
left endpoints of the reference circle and of the image of the black hole
shadow (see the Fig. 5). Using the above definitions one may introduce the
dimensionless distortion parameter as $\delta_{s}=D_{cs}/R_{s}$. We may now
adopt $R_{s}$ and $\delta_{s}$ as two observables describing the shadow of
rotating black holes Hioki and Maeda (2009). If the distant observer located
on the equatorial plane then the inclination angle is $\theta_{0}=\pi/2$.
Consequently, one may get the maximum gravitational effects on the shadow of
the black hole due to the rotation. The inclination angle corresponding to
supermassive black hole at the center of our Galaxy is also expected to be
close to $\pi/2$. In this particular case we have

$\displaystyle x$ $\displaystyle=$ $\displaystyle-b_{1}+2l,$ (19)
$\displaystyle y$ $\displaystyle=$
$\displaystyle\pm\sqrt{b_{2}-(a-b_{1}+2l)^{2}}.$ (20)

Figure 4: The gravitational lens object between the source and an observer. A
reference coordinate system with the black hole at the origin is set up by a
distant observer. The Boyer-Lindquist coordinates tend to the Cartesian
coordinate system at infinity. The black hole is rotating around the $z$ axis
from the selected reference frame as seen by a distant observer at infinity.
In the chosen system, the line joining the origin with the observer is
orthogonal to the $xy$-plane. The vector being tangent to an incoming light
ray defines a straight line intersecting the $xy$-plane at the point ($x$,
$y$). Figure 5: The apparent shape of a black hole is described by two
observables as the shadow radius $R_{s}$ and the distortion parameter
$\delta_{s}$, approximated by a distorted circle. The parameter $D_{cs}$ is
equal to the difference between the shadow and the left endpoints of the
circle Amarilla  _et al._ (2010).

Using the expressions for $x$ and $y$ one may easily get the shape of the
shadow of the CRNK black hole in the Rastall gravity. We present the shadows
of CRNK black hole in the Rastall theory of gravity in Fig. 6. In the top left
plot of Fig. 6 the change in the shape and size of the shadow around CRNK
black hole in the Rastall gravity for the different values of the NUT charge
$l$ is shown. In the same plot one can see that increase of the parameter $l$
increases the average radius of the shadow. Here one can also see that for the
same spin and mass parameters the shadow cast by the CRNK black hole in the
Rastall gravity is considerably larger than that of the Kerr black hole,
corresponding to the fixed values of the rest of the spacetime parameters of
the CRNK black hole in the Rastall gravity. In the top right plot of Fig. 6
one can see the increase in the size of the shadow of the CRNK black hole in
the Rastall gravity for the decreasing values of the parameter $\omega$. It is
also noticeable that for the values of this parameter $\omega$ close to zero,
the size of the shadow formed around the CRNK black hole in the Rastall
gravity is almost the same as seen in the case of the Kerr black hole. In the
left plot of the second row in the same figure the shape and size of the
shadow of the CRNK black hole in the Rastall gravity for the various values of
the parameter $\kappa\lambda$ are presented. From the same plot it can be
easily noticed that for smaller values of this parameter $\kappa\lambda$, the
size of the shadow of the CRNK black hole in the Rastall gravity coincides
with the one for the Kerr black hole. This and previous plots show that for
smaller values of the parameter $\kappa\lambda$, for the values of the
parameter $\omega$ close to zero, it becomes difficult to distinguish the
apparent shadow of the CRNK black hole in the Rastall gravity from the shadow
of the the Kerr black hole. One can see strong dependence of the size of the
shadow of the CRNK black hole in the Rastall gravity on the parameter
$\alpha$, as it increases considerably with the increase of this parameter
$\alpha$ and is shown in the second plot of the second row of the Fig. 6. It
is demonstrated that for the values of the parameter $\alpha$ close to zero
the size of the shadow of the CRNK black hole in the Rastall gravity becomes
identical to the shadow of the Kerr black hole. In the left plot of the last
row of Fig. 6 the shadow of the CRNK black hole in the Rastall gravity is
compared with the shadow of the Kerr black hole for the different values of
the spin parameter $a$. It is mentioned here that in the Fig. 6 solid lines
correspond to the Kerr case and the dashed lines correspond to the CRNK black
hole case in the Rastall gravity. Even though the values of the spin parameter
$a$ are shown only for the Kerr case, the dashed lines with the same color
represent the shadow with the same spin parameter $a$ for the CRNK black hole
in the Rastall gravity. One can easily see that increase of the spin parameter
$a$ does not change the size of the shadow much and only shifts the center of
the circle of the shadow to the right in both the Kerr and CRNK spacetimes. In
the last plot of the Fig. 6, it is shown how the electromagnetic charge $q$
affects the shadow around the CRNK black hole in the Rastall gravity. One can
see that increase in the parameter $q$ reduces the size of the shadow. From
the discussion presented here we conclude that when the gravitational field
around a black hole becomes stronger the size of the shadow increases and when
it becomes weaker the size of the shadow decreases.

Figure 6: The shape of the shadow of the CRNK black hole in the Rastall
gravity observed at infinity is compared with the shadow of the Kerr black
hole.

One can obtain the observable $R_{s}$ using the equation

$\displaystyle R_{s}=\frac{(x_{C}-x_{A})^{2}+y_{A}^{2}}{2(x_{C}-x_{A})}$ (21)

and the other observable $\delta_{s}$ can be calculated using the relation

$\displaystyle\delta_{s}=\frac{D_{cs}}{R_{s}}.$ (22)

The observables $R_{s}$ and $\delta_{s}$ are presented in Figs. 7 and 8 (plots
on the left for $R_{s}$ and plots on the right for $\delta_{s}$) for the
different values of the spacetime parameters of the CRNK black hole in the
Rastall gravity. In the top left plot of the Fig. 7 it is shown that the
average shadow radius $R_{s}$ increases with the increase of the NUT parameter
$l$. There is a negligible decrease with the increase of the spin parameter
$a$ in $R_{s}$. In the right plot of the first row of the Fig. 7 one can see
deviation of the shape of the shadow around CRNK black hole in the Rastall
gravity from a circle. This deviation becomes more noticeable for smaller
values of the NUT charge $l$ and for faster rotation of the CRNK black hole in
the Rastall gravity. From the left plot of the second row of the Fig. 7 one
can see that the effect of the parameter $\alpha$ on the average radius of the
shadow of the CRNK black hole in the Rastall gravity is much stronger as
compared to the effects of the NUT parameter $l$. This is due to the fact that
$R_{s}$ goes up faster with the increase of $\alpha$ than the increase in the
NUT parameter $l$. However, with the increase of parameters $\alpha$ and $l$
the deviation of the shape of the shadow from a circle becomes smaller as
shown in the right plot of the same row in the same figure. In the left plot
of the third row of the Fig. 7 we have shown how the average radius of the
black hole shadow $R_{s}$ changes with the change of the spin parameter $a$
and the charge parameter $q$. One can easily see that the effect of the latter
is stronger as compared to the effect of the spin of the CRNK black hole in
the Rastall gravity. In the right plot of the same row in the same figure it
is clearly seen that for bigger values of the parameters $q$ and $a$ the
deviation of the shape of the shadow from a circle becomes more visible.

Figure 7: The dependence of $R_{s}$ and $\delta_{s}$ on the spacetime
parameters.

In the first plot of Fig. 8 we have demonstrated how the observable $R_{s}$
changes due to the change in parameters $\alpha$ and $\omega$. One can see
that smaller values of $\omega$ and bigger values of $\alpha$ produce bigger
average radius of the shadow around the CRNK black hole in the Rastall
gravity. In the right plot of the first row of the Fig. 8 it is shown that for
smaller values of the parameter $\alpha$, the effect of the parameter $\omega$
on the observable $\delta_{s}$ is negligible. Moreover, we have shown that the
distortion in the black hole shadow becomes more apparent only for the bigger
values of the parameter $\alpha$. In the left plot of the second row of the
Fig. 8 it is shown that for smaller values of $\omega$ and bigger values of
$\kappa\lambda$ the observable $R_{s}$ takes bigger values. In the right plot
of the same row in the Fig. 8 it is clearly seen that the second observable
$\delta_{s}$ becomes smaller in the corresponding range of values of the
spacetime parameters for the CRNK black hole in the Rastall gravity. In the
last row of the Fig. 8 the change of the observables $R_{s}$ and $\delta_{s}$
in the $\kappa\lambda-q$ plane is demonstrated. It can be seen that for
smaller values of $q$ and bigger values of $\kappa\lambda$ the observable
$R_{s}$ becomes bigger while the observable $\delta_{s}$ becomes smaller i.e.
the average radius of the shadow of the CRNK black hoe in the Rastall gravity
increases and the black hole shadow almost assumes the shape of a perfect
circle.

Figure 8: The dependence of $R_{s}$ and $\delta_{s}$ on the spacetime
parameters as in the previous figure.

##  V Conclusion

In this work we have investigated photon motion, the GL and the black hole
shadow in the spacetime of the CRNK black hole in the Rastall theory of
gravity. For the photon motion we have used the Hamilton-Jacobi equation of
motion for massless particles. From the relation between the radius of closest
approach of photons to the central black hole and the impact parameter of
photons we have found that this relation becomes approximately linear for the
bigger values of these quantities. We have shown that for the fixed value of
the impact parameter, with the increase of the values of the parameters $l$,
$\alpha$, and $\kappa\lambda$ the radius of the closest approach of photons to
the central object goes down. We have observed the opposite effect on the
radius of the closest approach with the increase in the values of the rest of
the spacetime parameters. We have evaluated the bending angle of photons due
to the gravitational field of the central black hole and have shown that the
effect of the spacetime parameters $l$, $\omega$, and $\alpha$ is much
stronger as compared to the effect of the parameters $a$, $\kappa\lambda$ and
$q$. We have demonstrated that increase in $l$ and $\alpha$ increases the
bending angle while increase in $\omega$ reduces it. In spite of the fact that
the effects of the parameters $a$, $q$, and $\kappa\lambda$ are negligible, we
have demonstrated that increase of the first two parameters reduces the
bending angle and increase of the third parameter increases it. We have also
studied the influence of the spacetime parameters on the photon sphere formed
around the CRNK black hole in the Rastall gravity. It is shown that i)
increase of the parameter $\alpha$ makes the radius of photon sphere greater
and this increase in the radius of the photon sphere becomes faster with the
increase of the NUT parameter $l$. On the contrary the radius of the photon
sphere decreases with the decrease of the Rastall parameter $\kappa\lambda$;
ii) the radius of the photon sphere goes up exponentially when the parameter
$\omega$ approaches the value $\sim-0.8$; iii) the effect of the Rastall
parameter $\kappa\lambda$ on the radius of the photon sphere strongly depends
on the parameters $\omega$ and $\alpha$. Further we have observed that for
bigger values of $\alpha$ and smaller values of $\omega$ the effect becomes
stronger; iv) the effect of the parameters $a$ and $q$ on the radius of the
photon sphere is similar and increase in the values of both the parameters
reduces the radius of the photon sphere; v) the radius of the photon sphere
becomes bigger for bigger values of the parameters $l$ and $\kappa\lambda$.

Then we have investigated the shadow formed around the CRNK black hole in the
Rastall gravity. We have constructed the shape of the shadow in comparison
with the Kerr black hole case. We have obtained the following results i) the
size of the shadow slightly increases with the increase of the NUT parameter
$l$ and it decreases with the increase of the electromagnetic charge parameter
$q$; ii) decrease in the value of the spacetime parameter $\omega$ makes the
size of the shadow bigger; iii) decrease in the value of the parameter
$\kappa\lambda$ makes the size of the black hole shadow smaller. Further the
difference in the sizes of the shadows for the values of the parameter
$\kappa\lambda$ between -2 and -4 is negligible. The size of the shadow in the
case $\kappa\lambda=-0.4$ is much bigger than these two values of the
$\kappa\lambda$ i.e. -2 and -4; iv) the influence of the parameter $\alpha$ on
the size of the shadow is much stronger as compared to the effect of the rest
of the spacetime parameters. The increase of the parameter $\alpha$ causes
increase in the size of the shadow; v) the change of the spin parameter $a$
only changes the position of the black hole shadow. The change in the size of
the shadow of the black hole is negligible for change in the parameter $a$. As
the final step we have studied the dependence of the two observable
quantities, $R_{S}$ and $\delta_{s}$ on the spacetime parameters. We have
demonstrated that the effect of the spin parameter $a$ on the size of the
shadow is indeed negligible but can change the distortion parameter
$\delta_{s}$ considerably for smaller values of the NUT parameter $l$ and for
the bigger values of the electromagnetic charge parameter $q$. We have seen
that for bigger values of the parameter $l$ the observable $R_{s}$ becomes
bigger but the second observable $\delta_{s}$ becomes smaller. Similar
behavior of the two observable is observed for different values of the
parameters $\alpha$ and $\kappa\lambda$. We have seen that the bigger values
of $R_{s}$ corresponds to smaller values of the $\delta_{s}$ and this effect
is more sensitive to the values of the parameters $\omega$ and $\kappa\lambda$
in comparison to the other spacetime parameters.

One could ask the question whether is it possible to get constraints on the
parameters of a CRNK black hole in the Rastall gravity using the M87* black
hole shadow observations by the EHT collaboration. The low precision of the
M87* black hole shadow image observation indicates, if we assume that it is
the rotating Kerr black hole, the black hole spin measurement uncertainty will
be in the big range within $0.5\leq\alpha_{*}\leq 0.94$, where
$\alpha_{*}=J/M^{2}$ ($J$ is the total angular momentum of the black hole). It
means that with the current accuracy of the shadow observations the CRNK black
hole in the Rastall gravity is not distinguishable from the Kerr black hole.

It is worth noticing that the results presented in this work are consistent
with each other i.e. the effect of all the spacetime parameters on the GL,
shadow cast by the black hole and two observables $R_{s}$ and $\delta_{s}$ are
in agreement. In the next stage we would like to investigate the shadow cast
by the CRNK black hole in the Rastall gravity by comparing the images of the
shadow formed from the radiation of an accretion disc around the black hole,
obtained by two different ray tracing codes namely, ”GYOTO” Vincent  _et al._
(2011) and ray tracing code by Ayzenberg et. al Ayzenberg and Yunes (2018).

## Acknowledgments

This research is supported by Grants F-FA-2021-432 and MRB-2021-527 of the
Uzbekistan Ministry for Innovative Development and by the Abdus Salam
International Centre for Theoretical Physics under the Grant No. OEA-NT-01.

## References

  * Liebes (1964) S. Liebes, Phys. Rev. 133, B835 (1964).
  * Wambsganss (1998) J. Wambsganss, Living Rev. Rel. 1, 12 (1998), arXiv:astro-ph/9812021 .
  * Schmidt (1963) M. Schmidt, Nature 197, 1040 (1963).
  * Walsh  _et al._ (1979) D. Walsh, R. F. Carswell, and R. J. Weymann, Nature 279, 381 (1979).
  * Massey  _et al._ (2010) R. Massey, T. Kitching, and J. Richard, Rept. Prog. Phys. 73, 086901 (2010), arXiv:1001.1739 [astro-ph.CO] .
  * Bozza (2003) V. Bozza, Phys. Rev. D 67, 103006 (2003), arXiv:gr-qc/0210109 .
  * Bozza and Mancini (2004) V. Bozza and L. Mancini, Astrophys. J. 611, 1045 (2004), arXiv:astro-ph/0404526 .
  * Beckwith and Done (2005a) K. Beckwith and C. Done, Mon. Not. Roy. Astron. Soc. 359, 1217 (2005a), arXiv:astro-ph/0411339 .
  * Schee  _et al._ (2017) J. Schee, Z. Stuchlík, B. Ahmedov, A. Abdujabbarov, and B. Toshmatov, Int. J. Mod. Phys. D 26, 1741011 (2017).
  * Narzilloev  _et al._ (2021a) B. Narzilloev, S. Shaymatov, I. Hussain, A. Abdujabbarov, B. Ahmedov, and C. Bambi, Eur. Phys. J. C 81, 849 (2021a), arXiv:2109.02816 [gr-qc] .
  * Sereno (2005) M. Sereno, Mon. Not. Roy. Astron. Soc. 356, 937 (2005), arXiv:astro-ph/0410441 .
  * Sharif and Iftikhar (2015) M. Sharif and S. Iftikhar, Adv. High Energy Phys. 2015, 854264 (2015).
  * Younas  _et al._ (2015) A. Younas, M. Jamil, S. Bahamonde, and S. Hussain, Phys. Rev. D 92, 084042 (2015).
  * Azreg-Aïnou  _et al._ (2017) M. Azreg-Aïnou, S. Bahamonde, and M. Jamil, Eur. Phys. J. C 77, 414 (2017), arXiv:1701.02239 [gr-qc] .
  * Turimov  _et al._ (2019) B. Turimov, B. Ahmedov, A. Abdujabbarov, and C. Bambi, Int. J. Mod. Phys. D 28, 2040013 (2019), arXiv:1802.03293 [gr-qc] .
  * Bozza  _et al._ (2001) V. Bozza, S. Capozziello, G. Iovane, and G. Scarpetta, Gen. Rel. Grav. 33, 1535 (2001), arXiv:gr-qc/0102068 .
  * Bozza (2002) V. Bozza, Phys. Rev. D 66, 103001 (2002), arXiv:gr-qc/0208075 .
  * Bozza (2010) V. Bozza, Gen. Rel. Grav. 42, 2269 (2010), arXiv:0911.2187 [gr-qc] .
  * Virbhadra and Keeton (2008) K. S. Virbhadra and C. R. Keeton, Phys. Rev. D 77, 124014 (2008).
  * Beckwith and Done (2005b) K. Beckwith and C. Done, Mon. Not. Roy. Astron. Soc. 359, 1217 (2005b), arXiv:astro-ph/0411339 .
  * Fernando and Roberts (2002) S. Fernando and S. Roberts, Gen. Rel. Grav. 34, 1221 (2002).
  * Kumar  _et al._ (2020a) R. Kumar, S. U. Islam, and S. G. Ghosh, Eur. Phys. J. C 80, 1128 (2020a), arXiv:2004.12970 [gr-qc] .
  * Hsieh  _et al._ (2021) T. Hsieh, D.-S. Lee, and C.-Y. Lin, Phys. Rev. D 103, 104063 (2021).
  * Falcke  _et al._ (2000) H. Falcke, F. Melia, and E. Agol, Ap. J. Lett. 528, L13 (2000), arXiv:astro-ph/9912263 [astro-ph] .
  * Brügmann  _et al._ (2001) B. Brügmann, A. M. Ghez, and J. Greiner, Proceedings of the National Academy of Sciences 98, 10525 (2001), https://www.pnas.org/content/98/19/10525.full.pdf .
  * Saida (2017) H. Saida, PTEP 2017, 053E02 (2017), arXiv:1606.04716 [astro-ph.HE] .
  * Riaz  _et al._ (2020) S. Riaz, D. Ayzenberg, C. Bambi, and S. Nampalliwar, Astrophys. J. 895, 61 (2020), arXiv:1911.06605 [astro-ph.HE] .
  * Takahashi (2004) R. Takahashi, J. Korean Phys. Soc. 45, S1808 (2004), arXiv:astro-ph/0405099 .
  * Atamurotov and Ahmedov (2015) F. Atamurotov and B. Ahmedov, Phys. Rev. D 92, 084005 (2015), arXiv:1507.08131 [gr-qc] .
  * Abdujabbarov  _et al._ (2016) A. Abdujabbarov, B. Toshmatov, Z. Stuchlík, and B. Ahmedov, Int. J. Mod. Phys. D 26, 1750051 (2016), arXiv:1512.05206 [gr-qc] .
  * Chang and Zhu (2021) Z. Chang and Q.-H. Zhu, Journal of Cosmology and Astroparticle Physics 2021, 003 (2021).
  * Kiselev (2003) V. V. Kiselev, Class. Quant. Grav. 20, 1187 (2003), arXiv:gr-qc/0210040 .
  * Toshmatov  _et al._ (2017) B. Toshmatov, Z. Stuchlík, and B. Ahmedov, Mod. Phys. Lett. A 32, 1775001 (2017), arXiv:1707.00403 [gr-qc] .
  * Sakti  _et al._ (2019) M. F. A. R. Sakti, A. Suroso, and F. P. Zen, Eur. Phys. J. Plus 134, 580 (2019), arXiv:1909.06595 [hep-th] .
  * Molla and Debnath (2021) N. U. Molla and U. Debnath, International Journal of Modern Physics A 36, 2150210 (2021), https://doi.org/10.1142/S0217751X21502109 .
  * Sakti  _et al._ (2020a) M. F. A. R. Sakti, A. Suroso, and F. P. Zen, Annals Phys. 413, 168062 (2020a), arXiv:1901.09163 [gr-qc] .
  * Narzilloev  _et al._ (2021b) B. Narzilloev, I. Hussain, A. Abdujabbarov, B. Ahmedov, and C. Bambi, Eur. Phys. J. Plus 136, 1032 (2021b), arXiv:2110.01772 [gr-qc] .
  * Mota  _et al._ (2022) C. E. Mota, L. C. N. Santos, F. M. da Silva, G. Grams, I. P. Lobo, and D. P. Menezes, International Journal of Modern Physics D 31, 2250023-76 (2022).
  * Farasat Shamir  _et al._ (2021) M. Farasat Shamir, I. Yaqoot, and G. Mustafa, New. Astr. 89, 101624 (2021).
  * da Silva  _et al._ (2021) F. M. da Silva, L. C. N. Santos, and C. C. Barros, Classical and Quantum Gravity 38, 165011 (2021), arXiv:2010.00086 [astro-ph.HE] .
  * Debnath (2021) U. Debnath, European Physical Journal Plus 136, 442 (2021).
  * Mota  _et al._ (2020) C. E. Mota, L. C. N. Santos, F. M. da Silva, G. Grams, I. P. Lobo, and D. P. Menezes, arXiv e-prints , arXiv:2007.01968 (2020), arXiv:2007.01968 [gr-qc] .
  * Shahzad and Abbas (2020) M. R. Shahzad and G. Abbas, European Physical Journal Plus 135, 502 (2020).
  * Sakti  _et al._ (2020b) M. F. A. R. Sakti, A. Suroso, and F. P. Zen, Annals Phys. 413, 168062 (2020b), arXiv:1901.09163 [gr-qc] .
  * Mustafa  _et al._ (2020) G. Mustafa, M. R. Shahzad, G. Abbas, and T. Xia, Modern Physics Letters A 35, 2050035 (2020).
  * Moradpour  _et al._ (2019) H. Moradpour, Y. Heydarzade, C. Corda, A. H. Ziaie, and S. Ghaffari, Modern Physics Letters A 34, 1950304 (2019), arXiv:1910.07878 [physics.gen-ph] .
  * Abbas and Shahzad (2018) G. Abbas and M. R. Shahzad, European Physical Journal A 54, 211 (2018), arXiv:1811.00878 [physics.gen-ph] .
  * Spallucci and Smailagic (2018) E. Spallucci and A. Smailagic, International Journal of Modern Physics D 27, 1850003 (2018), arXiv:1709.05795 [gr-qc] .
  * Kumar  _et al._ (2021) R. Kumar, B. P. Singh, M. S. Ali, and S. G. Ghosh, Physics of the Dark Universe 34, 100881 (2021).
  * Narzilloev  _et al._ (2019) B. Narzilloev, A. Abdujabbarov, C. Bambi, and B. Ahmedov, Phys. Rev. D 99, 104009 (2019), arXiv:1902.03414 [gr-qc] .
  * Narzilloev  _et al._ (2020) B. Narzilloev, D. Malafarina, A. Abdujabbarov, and C. Bambi, Eur. Phys. J. C 80, 784 (2020), arXiv:2003.11828 [gr-qc] .
  * Narzilloev  _et al._ (2020a) B. Narzilloev, J. Rayimbaev, A. Abdujabbarov, and C. Bambi, The European Physical Journal C 80, 1074 (2020a), arXiv:2005.04752 [gr-qc] .
  * Narzilloev  _et al._ (2020b) B. Narzilloev, J. Rayimbaev, S. Shaymatov, A. Abdujabbarov, B. Ahmedov, and C. Bambi, Phys. Rev. D 102, 044013 (2020b), arXiv:2007.12462 [gr-qc] .
  * Narzilloev  _et al._ (2020c) B. Narzilloev, J. Rayimbaev, S. Shaymatov, A. Abdujabbarov, B. Ahmedov, and C. Bambi, Phys. Rev. D 102, 104062 (2020c), arXiv:2011.06148 [gr-qc] .
  * Narzilloev  _et al._ (2021) B. Narzilloev, J. Rayimbaev, A. Abdujabbarov, B. Ahmedov, and C. Bambi, Eur. Phys. J. C 81, 269 (2021), arXiv:2103.11090 [gr-qc] .
  * Hakimov  _et al._ (2017) A. Hakimov, A. Abdujabbarov, and B. Narzilloev, International Journal of Modern Physics A 32, 1750116 (2017).
  * Shaymatov  _et al._ (2021) S. Shaymatov, B. Narzilloev, A. Abdujabbarov, and C. Bambi, Phys. Rev. D 103, 124066 (2021).
  * Narzilloev  _et al._ (2021a) B. Narzilloev, D. Malafarina, A. Abdujabbarov, B. Ahmedov, and C. Bambi, Phys. Rev. D 104, 064016 (2021a), arXiv:2105.09174 [gr-qc] .
  * Rayimbaev  _et al._ (2021) J. Rayimbaev, B. Narzilloev, A. Abdujabbarov, and B. Ahmedov, Galaxies 9 (2021), 10.3390/galaxies9040071.
  * Narzilloev  _et al._ (2021b) B. Narzilloev, J. Rayimbaev, A. Abdujabbarov, and B. Ahmedov, Galaxies 9 (2021b), 10.3390/galaxies9030063.
  * Visser (2018) M. Visser, Physics Letters B 782, 83 (2018).
  * Abbott (2016) B. P. e. a. Abbott (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).
  * Paul  _et al._ (2009) B. C. Paul, P. S. Debnath, and S. Ghose, Phys. Rev. D 79, 083534 (2009).
  * Rastall (1972) P. Rastall, Phys. Rev. D 6, 3357 (1972).
  * Lobo  _et al._ (2018) I. P. Lobo, H. Moradpour, J. P. Morais Graça, and I. G. Salako, Int. J. Mod. Phys. D 27, 1850069 (2018), arXiv:1710.04612 [gr-qc] .
  * Kumar and Ghosh (2018) R. Kumar and S. G. Ghosh, Eur. Phys. J. C 78, 750 (2018), arXiv:1711.08256 [gr-qc] .
  * Darabi  _et al._ (2018) F. Darabi, H. Moradpour, I. Licata, Y. Heydarzade, and C. Corda, Eur. Phys. J. C 78, 25 (2018), arXiv:1712.09307 [gr-qc] .
  * Xu  _et al._ (2018) Z. Xu, X. Hou, X. Gong, and J. Wang, The European Physical Journal C 78 (2018), 10.1140/epjc/s10052-018-5991-x.
  * Bambi  _et al._ (2019) C. Bambi, K. Freese, S. Vagnozzi, and L. Visinelli, Phys. Rev. D 100, 044057 (2019).
  * Vagnozzi and Visinelli (2019) S. Vagnozzi and L. Visinelli, Phys. Rev. D 100, 024020 (2019).
  * Kumar  _et al._ (2019) R. Kumar, S. G. Ghosh, and A. Wang, Phys. Rev. D 100, 124024 (2019).
  * Kumar  _et al._ (2020b) R. Kumar, S. G. Ghosh, and A. Wang, Phys. Rev. D 101, 104001 (2020b).
  * Maldacena and Susskind (2013) J. Maldacena and L. Susskind, Fortsch. Phys. 61, 781 (2013), arXiv:1306.0533 [hep-th] .
  * Giddings (2012) S. B. Giddings, Phys. Rev. D 85, 044038 (2012).
  * Wei  _et al._ (2015) S.-W. Wei, P. Cheng, Y. Zhong, and X.-N. Zhou, JCAP 08, 004 (2015), arXiv:1501.06298 [gr-qc] .
  * Held  _et al._ (2019) A. Held, R. Gold, and A. Eichhorn, JCAP 06, 029 (2019), arXiv:1904.07133 [gr-qc] .
  * Kumar  _et al._ (2020c) R. Kumar, B. P. Singh, and S. G. Ghosh, Annals Phys. 420, 168252 (2020c), arXiv:1904.07652 [gr-qc] .
  * Giddings and Psaltis (2018) S. B. Giddings and D. Psaltis, Phys. Rev. D 97, 084035 (2018).
  * Xu  _et al._ (2019) Z. Xu, Y. Liao, and J. Wang, International Journal of Modern Physics A 34, 1950185 (2019), https://doi.org/10.1142/S0217751X19501859 .
  * Vazquez and Esteban (2004) S. E. Vazquez and E. P. Esteban, Nuovo Cim. B 119, 489 (2004), arXiv:gr-qc/0308023 .
  * Hioki and Maeda (2009) K. Hioki and K.-I. Maeda, Phys. Rev. D 80, 024042 (2009).
  * Amarilla  _et al._ (2010) L. Amarilla, E. F. Eiroa, and G. Giribet, Phys. Rev. D 81, 124045 (2010).
  * Vincent  _et al._ (2011) F. H. Vincent, T. Paumard, E. Gourgoulhon, and G. Perrin, Class. Quant. Grav. 28, 225011 (2011), arXiv:1109.4769 [gr-qc] .
  * Ayzenberg and Yunes (2018) D. Ayzenberg and N. Yunes, Class. Quant. Grav. 35, 235002 (2018), arXiv:1807.08422 [gr-qc] .