# Force Free Magnetic Field and Modifications in Kerr Black Hole

Haidar Sheikhahmadi h.sh.ahmadi<EMAIL_ADDRESS>School of
Astronomy, Institute for Research in
Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran

###### Abstract

A recent research, presented a solution for a Schwarzschild black hole with a
force-free magnetic field showing how the canonical form of background metric
changes, (Sheikhahmadi, 2021). Therefore, it is logical that this research
seeks the modified solutions for the Kerr black hole as well. The goal of this
paper is, thereby, to pinpoint an exact solution for the Kerr black holes
perturbed by force-free magnetic fields. However, to do this we use a well
known tetrad formalism and obtain an explicit expression for the
electromagnetic strength tensor in the background metric, that is Kerr, based
on the tangent space calculations. Analyzing the stress-energy tensor reveals
the perturbed factors in the metric that allow us to understand better the
physics of the disks around massive and supermassive black holes. These
results show that it is not enough to depend solely on perturbation theory set
against the background metric, without considering the backreaction of the
force-free magnetufluid on it. This research indicates the necessity of
studying the impact of force-free magnetic fields on the structure and
evolution of mysterious cosmic objects.

pacs:97.60.Lf, 52.35.B, 52.65.Kj,04.25.Nx

Kerr Black Hole, Force Free Magnetic Field, Accretion Disc, Blandford-Znajek
Mechanism, Tetrad Formalism

††journal: ApJ

##  1 Introduction

Scientists and astrophysicist are still fascinated by the influence of Kerr
black hole—remarkable cosmic singularities endowed with rotational property,
in spacetime (Kerr, 1963), see also (Newman & Janis, 1965),(Boyer & Lindquist,
1967). With the growing comprehension of these puzzling objects, there are
numerous intriguing aspects that are observed nearby (Akiyama et al., 2019).
Out of these phenomena, the appearance of force-free magnetic fields (FFMF)s
surrounding Kerr black holes may be one of the most interesting aspects that
illustrates how different phenomena like gravity, electromagnetism, and
magnetohydrodynamics work together in the severe environment of astrophysics
(Chandrasekhar & Elbert, 1953),(Teukolsky & Press, 1974),(Chandrasekhar,
1961),(Semenov et al., 2004), (Thorne & MacDonald, 1982), (Brennan et al.,
2013). Blandford and Znajek developed a mathematical model which assumes that
rotating black holes interact with a magnetic plasma, and extract the energy
of rotation in the process. It is a working scenario that enables us to
explain why there are strong jets within active galactic nuclei (AGN) and
other astrophysical processes. This process involves using the rotational
energy from a black hole for accelerating a collimated outflow or jet by means
of the Blandford-Znajek mechanism (Blandford & Znajek, 1977). The reason for
this is due to interaction of the black hole’s large and plasma magnetic
fields. The FFMF is when the energy inferred from a plasma’s inertial masses
is much smaller than that attributed to its electromagnetic energies (Priest,
2014),(Marsh, 1996). In this regime the charged bodies are closely linked to
the magnetic fields and move alongside them with speeds that rapidly approach
speed of light (lüst & Schlütter, 1954),(Teukolsky & Press, 1974). In other
words, near rotating black holes the rotation energy of black hole forms
strong and organized magnetic field, which deep affect on local plasma
behavior (Aschwanden, 2004). Investigating FFMF surrounding a Kerr black hole
offers insight into the complex geometry associated with high rotations rates
(Uzdensky, 2005),(Contopoulos et al., 2013),(Wand & Ritz, 2014),(Lupsasca &
Rodriguez, 2015). Understanding the relationship between the gravity field of
a black hole, its rotation, and the behavior of magnetized plasma in the
surroundings will help scientists understand how astrophysical jets are
generated; why some accretion disks remain stable while others collapse; and
most importantly, how energy can be extracted, inwarded and/or outwarded, from
the Kerr black hole (Wilson & Ritz, 2015),(Pan & Yu, 2015). There are advances
in numerical simulation and theoretical models over the past few years which
make it possible for scientists to study more on the FFMF of a Kerr black
holes (Grignani et al., 2018),(Zajacek et al., 2018),(Misner et al., 1998).
Such studies illuminate the basic physics of magnetized plasma within the
frameworks of strong gravity and rotation of the black holes. Here we can
refer the reader to a significant list of literature on FFMF in Kerr black
holes (Yuan et al., 2019),(Grignani et al, 2020), (Camilloni et al, 2020) and
many other outstanding literature therein. This study will explore the FFMF
surroundings for Kerr black holes and the associated physics behind rotating
cosmological singularities. We present the modifications arising due to the
presence of the FFMF near Kerr black holes. The study of such details of FFMF
around Kerr black holes will help us understanding better how such phenomena
of gravity, electromagnetism and plasma take place at the most critical places
of the Universe. Then via these modifications, one may seek to aid in a larger
scientific quest of unravelling the intricacies of Kerr black holes as well as
gaining insights into the underlying physical principles that underpin such
sophisticated phenomenons deeply more. Inspired by a recent paper on FFMF in
Schwarzschild black holes (Sheikhahmadi, 2021), one may wonder whether a
similar study can be undertaken for Kerr black holes as a much more physical
object.
This paper is organised as follows:
In Sec. 2 the necessary mathematical tools, including flat and curve
electromagnetic strength tensors, the background metric, tangent space, and
their relationships, will be presented. In Sec. 3, the energy momentum tensor
in the Kerr background will be obtained, then considering it as a source for
the perturbation, in Sec. 4 the different components of the perturbed metric
will be obtained in detail. In Sec. 5 some discussions will take place, and it
will be devoted to concluding remarks as well.

##  2 Mathematics of Force-Free Magnetic Fields

Here, we shall consider the consequences of the presence of free-force
electrodynamics in the Kerr black hole. In our analysis, we compute the
various elements of FFMF in a spherical Cartesian metric. Then, by employing
the tetrad formalism (Cartan, 1922),(Newman & Penrose, 1962), we transform the
electromagnetic-field tensor to the Kerr black hole background. For instance,
by taking advantage of force-free electrodynamics, we can study any possible
scenario, which includes the Blandford-Znajek mechanism. The significance of
this mechanism is considerable concerning several astronomic effects including
the existence of outer disks around massive black holes in galaxies as it
occurs in the M87 galaxy centre. As a result, it shed some lights into how
jetting and GRBs formulate. The essential construction for such phenomena is
the same, yet our attitude towards their study may differ somewhat.
Particularly we want to explore what is the effect of the force-free
magnetofluid energy-momentum tensor acting as a source on the Kerr black
hole’s metric. The examination of the influence of the FFMF on geometry
deformation associated with the Kerr black hole can provide useful information
about the changes in the geometry-induced interplay.

###  2.1 Force-Free Field in Spherical Coordinate

Here we will obtain different components of the force-free field
electromagnetism in a spherical coordinate, that is

$d{S^{2}}=-d{t^{2}}+d{r^{2}}+{r^{2}}d{\theta^{2}}+{r^{2}}{\sin^{2}}\theta
d{\varphi^{2}}\,.$ (1)

The master equation that governs magnetohydrodynamics reads

${\nabla^{2}}B+{\alpha^{2}}B=0\,,$ (2)

where $B$ reads magnetic field, and $\alpha$ stands for a scalar function, one
notices that by considering $\alpha$ as a constant the problem reduces to an
exact solvable solution. There are some techniques, both analytical and
numerical, to solve Eq.(2). Here we will follow the approach appeared in
(Chandrasekhar & Kendall, 1957),(Chandrasekhar, 1956),(Chandrasekhar &
Xanthopoulos, 1979). By introducing a scalar function like $\Psi$, Eq.(2) can
be rewritten as follows

${\nabla^{2}}\Psi+{\alpha^{2}}\Psi=0\,.$ (3)

This equation readily has three independent solutions as

$L_{S}=\nabla\Psi\,,~{}T=\nabla\times(\hat{e}\Psi)\,,~{}S=\frac{1}{\alpha}\nabla\times
T\,,$ (4)

well known as solenoidal, toroidal and poloidal solutions respectively. Here
$\hat{e}$ is a unit vector indicating an arbitrary and specific direction.
Combining the two above equations one obtains

$B=\frac{1}{\alpha}\nabla\times\nabla\times(\hat{e}\Psi)+\nabla\times(\hat{e}\Psi)\,$
(5)

By introducing $\Psi(r,\theta)=R(r)\Theta(\theta)$ aiming at separating $\Psi$
in Eq.(3) and by virtue of Eq.(1), one has

$\displaystyle\frac{1}{{{r^{2}}}}\frac{d}{{dr}}\big{(}{{{r^{2}}}}\frac{{dR(r)}}{{dr}}\big{)}+{\alpha^{2}}R(r)-\frac{{n(n+1)}}{{{r^{2}}}}R(r)$
$\displaystyle=0\,,$ (6)
$\displaystyle\frac{1}{{\sin\theta}}\frac{d}{{d\theta}}\big{(}\sin\theta\frac{{d\Theta(\theta)}}{{d\theta}}\big{)}+n(n+1)\Theta(\theta)$
$\displaystyle=0\,.$ (7)

Then, the function of $\Psi(r,\theta)$ reads

$\Psi(r,\theta)=\sum_{n=1}(kr)j_{n}(kr)(1-\cos\theta)P_{n}^{(1,-1)}(\cos\theta)\,,$
(8)

here $j_{n}$ stands for spherical Bessel functions and $P_{n}^{(1,-1)}$ refers
the well known Jacobi polynomials and
$k=\alpha\left(K^{2}+\Psi^{2}\right)^{1/2}/\Psi$ in which if $K$ is a
constant, e.g., $K=0$ then $\alpha$ to be $k$, (Marsh, 1996; Sheikhahmadi,
2021). As a first candidate, in Eq.(8) one can consider $n=1$, that
immediately reduces to the following expression

$\Psi(r,\theta)=\Big{(}\frac{{\sin(kr)}}{{{{kr}}}}-{{\cos(kr)}}\Big{)}\sin\theta\,.$
(9)

Utilizing all the above results, components of the axial symmetric
$B(r,\theta)$ can be achieved as follows

${B_{r}}=\frac{1}{{{r^{2}}\sin\theta}}{\partial_{\theta}}\Psi\,,~{}{B_{\theta}}=-\frac{1}{{r\sin\theta}}{\partial_{r}}\Psi\,,~{}{B_{\phi}}=\frac{k\Psi)}{{r\sin\theta}}\,.$
(10)

By having the components of $B$, and considering metric (1), both covariant
and contravariant forms of the electromagnetic strength tensor can be obtained
(Sheikhahmadi, 2021)

${\mathfrak{F}_{ab}}=\left({\begin{array}[]{*{20}{c}}0&{{E_{r}}}&{r{E_{\theta}}}&{r\sin\theta{E_{\phi}}}\\\
{-{E_{r}}}&0&{-r{B_{\phi}}}&{r\sin\theta{B_{\theta}}}\\\
{-r{E_{\theta}}}&{r{B_{\phi}}}&0&{-{r^{2}}\sin\theta{B_{r}}}\\\
{-r\sin\theta{E_{\phi}}}&{-r\sin\theta{B_{\theta}}}&{{r^{2}}\sin\theta{B_{r}}}&0\end{array}}\right)\,,$
(11)
${\mathfrak{F}^{ab}}=\left({\begin{array}[]{*{20}{c}}0&{-{E_{r}}}&{-\frac{{{E_{\theta}}}}{r}}&{-\frac{{{E_{\phi}}}}{{r\sin\theta}}}\\\
{{E_{r}}}&0&{-\frac{{{B_{\phi}}}}{r}}&{\frac{{{B_{\theta}}}}{{r\sin\theta}}}\\\
{\frac{{{E_{\theta}}}}{r}}&{\frac{{{B_{\phi}}}}{r}}&0&{-\frac{{{B_{r}}}}{{{r^{2}}\sin\theta}}}\\\
{\frac{{{E_{\phi}}}}{{r\sin\theta}}}&{-\frac{{{B_{\theta}}}}{{r\sin\theta}}}&{\frac{{{B_{r}}}}{{{r^{2}}\sin\theta}}}&0\end{array}}\right)\,,$
(12)

where lower case Latin words $a$ and $b$ refers spherical coordinate.
Obviously for the FFMF problem the electric components are equal to zero,
i.e., ${E_{r}}={E_{\theta}}={E_{\phi}}=0$, see (Brennan et al.,
2013),(Blinder, 2004).

###  2.2 Force-Free Field in Kerr Metric

In this subsection, we will extend the calculations to a Kerr background, in
terms of Boyer and Lindquist coordinate (Boyer & Lindquist, 1967), that is

${g_{\mu\nu}}=\left({\begin{array}[]{*{20}{c}}{\frac{{{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}}{{{a^{2}}-2mr+{r^{2}}}}}&0&0&0\\\
0&{{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}&0&0\\\
0&0&{\frac{{\sin{{(\theta)}^{2}}\left({\left({{a^{4}}+{a^{2}}{r^{2}}}\right)\cos^{2}{{(\theta)}}+2mr{a^{2}}\sin^{2}{{(\theta)}}+{a^{2}}{r^{2}}+{r^{4}}}\right)}}{{{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}}}&{-\frac{{2mra\sin^{2}{{(\theta)}}}}{{{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}}}\\\
0&0&{-\frac{{2mra\sin^{2}{{(\theta)}}}}{{{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}}}&{\bf{\frac{{2mr-{r^{2}}-{a^{2}}\cos^{2}{{(\theta)}}}}{{{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}}}}\end{array}}\right)\,,$
(13)

where $G=1$ and $a:=J/m$ is the Kerr parameter. We will need the contravariant
form of the metric (13) in the next calculations, therefore it is better to
bring it here, that reads

$g^{\mu\nu}=\left({\begin{array}[]{*{20}{c}}{\frac{{{a^{2}}-2mr+{r^{2}}}}{{{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}}}&0&0&0\\\
0&{\frac{1}{{{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}}}&0&0\\\
0&0&{\frac{{\left({{a^{2}}\cos^{2}{{(\theta)}}-2mr+{r^{2}}}\right)\csc^{2}{{(\theta)}}}}{{\left({{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}\right)\left({{a^{2}}-2mr+{r^{2}}}\right)}}}&{-\frac{{2amr}}{{\left({{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}\right)\left({{a^{2}}-2mr+{r^{2}}}\right)}}}\\\
0&0&{-\frac{{2amr}}{{\left({{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}\right)\left({{a^{2}}-2mr+{r^{2}}}\right)}}}&{\frac{{\left({-{a^{4}}-{a^{2}}{r^{2}}}\right)\cos^{2}{{(\theta)}}-2mr{a^{2}}\sin^{2}{{(\theta)}}-{a^{2}}{r^{2}}-{r^{4}}}}{{\left({{r^{2}}+{a^{2}}\cos^{2}{{(\theta)}}}\right)\left({{a^{2}}-2mr+{r^{2}}}\right)}}}\end{array}}\right)\,.$
(14)

Here one should note $\mu$ and $\nu$ run from $1\ldots 4$, to clarify this the
bold face component in Eq.(13) reads the time-time component of the metric,
that is $g_{44}.$
To make a map between Kerr background and Spherical coordinate one needs the
concept of tetrads $e_{\mu}^{a},$ see (Chandrasekhar, 1998), (Newman & Janis,
1965),(Newman & Penrose, 1962), through the following equation

${F_{\mu\nu}}=e_{\mu}^{a}e_{\nu}^{b}{\mathfrak{F}_{ab}}\,.$ (15)

Here we bring some of the tetrads for Kerr metric as

$\begin{array}[]{l}\,e_{\mu=1}^{a=1}=\,\frac{{\sqrt{{a^{2}}{{\cos}^{2}}\left(\theta\right)+{r^{2}}}}}{{\sqrt{{a^{2}}-2mr+{r^{2}}}}};\,e_{\mu=2}^{a=2}=\sqrt{{r^{2}}+{a^{2}}{{\cos}^{2}}\left(\theta\right)};e_{\mu=3}^{a=3}=\frac{{\sin\left(\theta\right)\sqrt{{r^{2}}+{a^{2}}{{\cos}^{2}}\left(\theta\right)}{\mkern
1.0mu}\sqrt{{a^{2}}-2mr+{r^{2}}}}}{{\sqrt{{a^{2}}{{\cos}^{2}}\left(\theta\right)-2mr+{r^{2}}}}};e_{\mu=4}^{a=4}=\frac{{\sqrt{{a^{2}}{{\cos}^{2}}\left(\theta\right)-2mr+{r^{2}}}}}{{\sqrt{{a^{2}}\cos{{\left(\theta\right)}^{2}}+{r^{2}}}}}\,.\end{array}$
(16)

Now using Eqs. (16),(15) and (11) the non-zero covariant components of the
electromagnetic strength tensor are attained as

${F_{12}}=\frac{{\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\left(\theta\right)}\right)\sin\left(\theta\right)\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}}{{\sqrt{{a^{2}}-2mr+{r^{2}}}~{}r}}\,,$
(17)
${F_{13}}=-\frac{{\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\left(\theta\right)}\right)\sin{{\left(\theta\right)}^{2}}\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}}{{\sqrt{{a^{2}}{{\cos}^{2}}\left(\theta\right)-2mr+{r^{2}}}k{r^{2}}}}\,,$
(18)
${F_{23}}=\frac{{2\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\left(\theta\right)}\right)\sin\left(\theta\right)\sqrt{{a^{2}}-2mr+{r^{2}}}\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)\cos\left(\theta\right)}}{{\sqrt{{a^{2}}{{\cos}^{2}}\left(\theta\right)-2mr+{r^{2}}}kr}}\,.$
(19)

To obtain the contravariant form of these expressions one uses the following
rule

${F^{\alpha\beta}}={g^{\alpha\mu}}{g^{\beta\nu}}{F_{\mu\nu}}\,,$ (20)

and consequently one has

${F^{12}}=-\frac{{\sqrt{{a^{2}}-2mr+{r^{2}}}\sin\left(\theta\right)\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}}{{\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\left(\theta\right)}\right)r}}\,,$
(21)
${F^{13}}=-\frac{{\sqrt{{a^{2}}{{\cos}^{2}}\left(\theta\right)-2mr+{r^{2}}}\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}}{{\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\left(\theta\right)}\right)k{r^{2}}}}\,,$
(22)
${F^{23}}=-\frac{{2\sqrt{{a^{2}}{{\cos}^{2}}\left(\theta\right)-2mr+{r^{2}}}\left({-\cos\left({kr}\right)kr+\sin\left({kr}\right)}\right)\cot\left(\theta\right)}}{{\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\left(\theta\right)}\right)\sqrt{{a^{2}}-2mr+{r^{2}}}kr}}\,.$
(23)

Now we are in a position that different components of the energy momentum
tensor can be obtained. We will use these results as a source to perturb the
Kerr, background, metric.

##  3 FFMF energy momentum tensor as a source of perturbations

First we will obtain components of energy momentum tensor, then will solve
perturbed Einstein field equations to obtain the modifications in Kerr metric,
induced by FFMF source.

###  3.1 Energy momentum tensor

The definition of the energy momentum tensor in a general curved background
reads (Jackson, 1999)

${{T}_{\mu\nu}}={F_{\mu\gamma}}F_{\nu}^{\gamma}-\frac{1}{4}g_{\mu\nu}^{(0)}({F_{\alpha\beta}}{F^{\alpha\beta}})\,,$
(24)

where $g_{\mu\nu}^{(0)}$ here is the background Kerr metric, Eq.(13). The
diagonal components, for our problem, read

$\begin{array}[]{l}{T_{44}}=\frac{{-{g^{(0)}_{44}}}}{2}({F_{12}}{F^{12}}+{F_{13}}{F^{13}}+{F_{23}}{F^{23}});\,\,\,\,{T_{11}}=\frac{{{g^{(0)}_{11}}}}{2}({F_{12}}{F^{12}}+{F_{13}}{F^{13}}-{F_{23}}{F^{23}})\,,\\\
\\\
{T_{22}}=\frac{{{g^{(0)}_{22}}}}{2}({F_{12}}{F^{12}}-{F_{13}}{F^{13}}+{F_{23}}{F^{23}});\,\,\,\,{T_{33}}=\frac{{{g^{(0)}_{33}}}}{2}(-{F_{12}}{F^{12}}+{F_{13}}{F^{13}}+{F_{23}}{F^{23}})\,.\end{array}$
(25)

For the off-diagonal components we obtain

$\begin{array}[]{l}{T_{43}}={T_{34}}=-\frac{{{g^{(0)}_{34}}}}{2}({F_{12}}{F^{12}}+{F_{13}}{F^{13}}+{F_{23}}{F^{23}})\\\
{T_{12}}={g^{(0)}_{22}}{F_{13}}{F^{23}};\,\,{T_{21}}={g^{(0)}_{11}}{F_{23}}{F^{13}}\,,\\\
{T_{13}}={g^{(0)}_{33}}{F_{12}}{F^{32}};\,\,{T_{31}}={g^{(0)}_{11}}{F_{32}}{F^{12}}\,,\\\
{T_{23}}={g^{(0)}_{33}}{F_{21}}{F^{31}};\,\,{T_{32}}={g^{(0)}_{22}}{F_{31}}{F^{21}}\,.\end{array}$
(26)

By using the results of Eqs.(17) to (23) one gets

$\begin{array}[]{l}T_{44}=\frac{{\left({2mr-{r^{2}}-{a^{2}}{{\cos}^{2}}\theta}\right){{\sin}^{2}}\theta}}{{2\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\theta}\right)}}\Big{\\{}\frac{{{{\sin}^{2}}\theta{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{r^{2}}}}\\\
\\\
\,\,\,\,\,\,\,\,+\frac{{{{\sin}^{2}}\theta{{\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{4}}}}+\frac{{4{{\cos}^{2}}\theta{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{2}}}}\Big{\\}}\,,\end{array}$
(27)

$\begin{array}[]{l}{T_{11}}=\frac{{\left({2mr-{r^{2}}-{a^{2}}{{\cos}^{2}}\theta}\right){{\sin}^{2}}\theta}}{{\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\theta}\right)}}\Bigg{\\{}\frac{{{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{r^{2}}}}+\frac{{{{\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{4}}}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{1}{2}\left({\frac{{{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{r^{2}}}}+\frac{{{{\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{4}}}}+\frac{{4{{\cot}^{2}}\theta{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{2}}}}}\right)\Bigg{\\}}\,,\end{array}$
(28)

$\begin{array}[]{l}T_{22}={{\left({2mr-{r^{2}}-{a^{2}}{{\cos}^{2}}\theta}\right){{\sin}^{2}}\theta}}\Bigg{\\{}\frac{{{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{r^{2}}}}+\frac{{4{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}{{\cot}^{2}}\theta}}{{{k^{2}}{r^{2}}}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{1}{2}\left({\frac{{{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{r^{2}}}}+\frac{{{{\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{4}}}}+\frac{{4{{\cot}^{2}}\theta{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{2}}}}}\right)\Bigg{\\}}\,,\end{array}$
(29)

$\begin{array}[]{l}T_{33}=\frac{{{{\sin}^{4}}\theta\left({\frac{{2mr{\mkern
1.0mu}{a^{2}}{{\sin}^{2}}\theta}}{{\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\theta}\right)}}+{r^{2}}+{a^{2}}}\right)}}{2}\times\\\
,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left({\frac{{{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{r^{2}}}}+\frac{{{{\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{4}}}}+\frac{{4{{\cot}^{2}}\theta{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{2}}}}}\right)\,.\end{array}$
(30)

Avoiding prolongation of the expressions, we bring only some of the off-
diagonal components, for instance

$\begin{array}[]{l}{T_{43}}={T_{34}}=\frac{{m\,r{\mkern
1.0mu}a{{\sin}^{4}}\theta}}{{2\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\theta}\right)}}\\\
\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left({\frac{{{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{r^{2}}}}+\frac{{{{\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{4}}}}+\frac{{4{{\cot}^{2}}\theta{{\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)}^{2}}}}{{{k^{2}}{r^{2}}}}}\right)\,,\end{array}$
(31)

${T_{13}}={T_{31}}={\sin^{2}}\theta\cos\theta\Big{\\{}\frac{{{{\cos}^{2}}\left({kr}\right)kr}}{2}-\frac{{\cos\left({kr}\right)\sin\left({kr}\right)}}{2}\,+\left({\frac{{m\cos\left({kr}\right)}}{2}-\frac{{\sin\left({kr}\right)}}{{2k}}}\right)k\cos\left({kr}\right)\Big{\\}}\,,$
(32)

${T_{12}}={T_{21}}=-\frac{{\left({{r^{2}}+{a^{2}}{{\cos}^{2}}\theta}\right){{\sin}^{2}}\theta}}{{2{k^{2}}{r^{3}}\sqrt{{a^{2}}-2mr+{r^{2}}}}}\left({{k^{2}}\sin\left({kr}\right){r^{2}}+\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)\left({\cos\left({kr}\right)kr-\sin\left({kr}\right)}\right)\cot\theta\,.$
(33)

###  3.2 FFMF as a source of perturbations on Kerr background

Here, and to study the effects of induced FFMF on the Kerr black hole,we are
going to employ the well established linearized theory of gravity (Weinberg,
1972),(Misner et al., 1998),(Padmanabhan, 2010),(Ryder, 2009). In this
approach one can perturbatively write down the general form of the metric as

$g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu},$ (34)

where the source of $h_{\mu\nu}$ arise from force-free magnohydrodynamics and
$g_{\mu\nu}^{(0)}$ refers the vacuum Kerr solution. We must emphasise here,
$g_{\mu\nu}^{(0)}$ is used to raise and lower indices in linearized gravity
approach. Accordingly, Einstein field equations can be expressed as follows

$G_{\mu\nu}^{(0)}+\delta{G_{\mu\nu}}=0+{T_{\mu\nu}}\,.$ (35)

where

$\delta{G_{\mu\nu}}=\delta{R_{\mu\nu}}-\frac{1}{2}g_{\mu\nu}^{(0)}\delta{R}\,.$
(36)

We consider the first order of perturbation, in doing so we need to calculate
the $1^{st}$ order Christoffel symbols, Riemann and Ricci tensors and
consequently the Ricci scalar.

For the metric,(34), the Christoffel symbols read

$\Gamma_{\mu\nu}^{(1)\eta}=\frac{1}{2}\left(\nabla_{\nu}h_{\mu}^{\eta}+\nabla_{\mu}h_{\nu}^{\eta}-\nabla^{\eta}h_{\nu\mu}\right)\,.$
(37)

By utilizing Eq.(37), one easily gets the Riemann tensor as follows

$\begin{array}[]{l}R_{\mu\nu\sigma}^{(1)\eta}=\frac{1}{2}\left(\nabla_{\nu}\nabla_{\sigma}h_{\mu}^{\eta}+\nabla_{\nu}\nabla_{\mu}h_{\sigma}^{\eta}-\nabla_{\nu}\nabla^{\eta}h_{\mu\sigma}-\nabla_{\sigma}\nabla_{\nu}h_{\mu}^{\eta}-\nabla_{\sigma}\nabla_{\mu}h_{\nu}^{\eta}+\nabla_{\sigma}\nabla^{\eta}h_{\mu\nu}\right),\end{array}$
(38)

then by using $g_{\mu\nu}^{(0)}$ to contract Eq.(38), the Ricci tensor is
obtained as

$R_{\mu\nu}^{(1)}=R_{\mu\eta\nu}^{(1)\eta}=\frac{1}{2}\left(\nabla_{\eta}\nabla_{\nu}h_{\mu}^{\eta}+\nabla_{\eta}\nabla_{\mu}h_{\nu}^{\eta}-\nabla_{\eta}\nabla^{\eta}h_{\mu
k}-\nabla_{\nu}\nabla_{\mu}h\right)\,,$ (39)

where $h$ is the trace of $h_{\mu\nu}$. To obtain the Ricci scalar the
following relation is constructive

$R_{\nu}^{(1)\mu}=g^{\mu\eta(0)}R_{\nu\eta}^{(1)}-h^{\mu\eta}R_{\nu\eta}^{(0)}\,.$
(40)

To simplify our analysis, we find it more convenient to use a modified metric
called the trace-reversed metric, denoted as $\bar{h}{\mu\nu}$. This metric is
related to the perturbed metric $h_{\mu\nu}$, as follows

$\bar{h}_{\mu\nu}\equiv h_{\mu\nu}-\frac{1}{2}g_{\mu\nu}^{(0)}h\,.$ (41)

Additionally, we introduce a small coordinate transformation represented by
$x^{\mu}\longrightarrow x^{\mu}+\xi^{\mu}$ (Padmanabhan, 2010). By imposing
the Lorentz gauge condition, we obtain an important equation that describes
the propagation of an electromagnetic wave on a curved background, given by

$\square\xi^{\mu}=-\nabla_{\nu}\bar{h}^{\mu\nu},.$

This equation describes the relationship between the transformation
$\xi^{\mu}$ and the trace-reversed metric components. It provides valuable
insights into how electromagnetic waves propagate in a curved spacetime. In
fact, we can express it as follows

$\square\bar{h}_{\mu\nu}+2R_{\eta\mu\epsilon\nu}^{(0)}\bar{h}^{\eta\epsilon}=4{T}_{\mu\nu},,$
(42)

Here, $R_{\eta\mu\epsilon\nu}^{(0)}$ represents the Riemann tensor associated
with the background geometry, and ${T}_{\mu\nu}$ corresponds to the energy
momentum tensor. This equation is of great significance in understanding the
effects of the trace-reversed metric and the associated force-free
electromagnetic waves on the Kerr background (Padmanabhan, 2010).

To conduct this study, we consider the prevailing physical conditions that
allow us to focus on terms up to the order of
$\mathcal{O}(\lambda^{2}/L^{2})$. Let’s clarify it in more details; Our
objective is to delve into the equations governing the evolution of a system
under a specific configuration. Specifically, we seek to understand the
disturbances that arise when force-free electrodynamics is present in a
background characterized by a Kerr metric. To streamline our calculations and
gain valuable insights, we introduce two parameters that capture the scales of
perturbations ($\lambda$) and the variations in the background metric ($L$).
This assumption proves to be immensely helpful as it simplifies our analysis
and leads to intriguing findings. By considering the relative magnitudes of
$\lambda$ and $L$, we can effectively discern the dominant effects and the key
features of the system’s evolution. As a consequence, we can simplify equation
(42) by neglecting the second term, resulting in a more manageable form

$\square\bar{h}_{\mu\nu}=4{T}_{\mu\nu}\,,$ (43)

here we suppose $G=1.$ This equation provides us with valuable insights into
the propagation of gravitational waves in the presence of matter. To further
explore this, we can utilize the radiative wave form and establish the
following relationship

$\begin{array}[]{l}\bar{h}_{\mu\nu}=16\pi\int\frac{d^{3}R}{4\pi|r-\mathbf{R}|}T_{\mu\nu}=16\pi\int
d^{3}R\left(\sum_{l,m}T_{\mu\nu}\frac{1}{2l+1}\frac{R^{l}}{r^{l+1}}Y_{l}^{m}(\theta,\varphi)Y_{l}^{*m}(\theta,\phi)\right)\,,\end{array}$
(44)

to calculate different components of $\bar{h}_{\mu\nu}$ in Eq.(44), the
following relations for harmonic oscillators are useful

$\displaystyle Y_{0}^{0}=\frac{1}{\sqrt{4\pi}},\quad
Y_{2}^{0}=\sqrt{\frac{5}{4\pi}}\left(\frac{3}{2}\cos^{2}\theta-\frac{1}{2}\right)\,,$
$\displaystyle\cos^{2}\theta=\frac{2}{3}\left(\sqrt{\frac{4\pi}{5}}Y_{2}^{0}+\frac{1}{2}\right),\quad\sin^{2}\theta=\frac{2}{3}\left(1-\sqrt{\frac{4\pi}{5}}Y_{2}^{0}\right)\,.$
(45)

It is also useful to note that by considering the magnetostatic limits, we can
approximate the inverse distance between two points as (Weinberg,
1972),(Ryder, 2009).

$\displaystyle|\mathbf{r}-\mathbf{R}|=\left(r^{2}-2\mathbf{r}\cdot\mathbf{R}+R^{2}\right)^{1/2}\approx
r\left(1-\frac{\mathbf{r}\cdot\mathbf{R}}{r^{2}}\right)+\cdots\,,$ (46)
$\displaystyle|\mathbf{r}-\mathbf{R}|^{-1}=\left(r^{2}-2\mathbf{r}\cdot\mathbf{R}+R^{2}\right)^{-1/2}\approx\frac{1}{r}\left(1-\frac{\mathbf{r}\cdot\mathbf{R}}{r^{2}}\right)+\cdots\,.$
(47)

These approximations enable us to simplify the mathematical expressions to
obtain the modifications to the background metric satisfactorily.

##  4 FFMF induced perturbations

With the help of extracted energy momentum tensor, we are now in a position,
to obtain both the diagonal and off-diagonal components of $\bar{h}_{\mu\nu}$,
as appearing in Eq.(44). Therefore first by utilizing the Eqs.(27) to (30) we
will obtain the diagonal components of $\bar{h}_{\mu\nu}$.

$\begin{array}[]{l}\bar{h}_{44}\simeq\frac{{32\pi
m}}{{15{k^{2}}r}}\Big{(}\left({9{a^{2}}{k^{2}}-5{k^{2}}}\right)\ln\left({kr}\right)-\left({{a^{2}}{k^{4}}+3{a^{2}}{k^{2}}}\right)\ln\left({kr}\right)-\left({5+\left({-9{a^{2}}+5}\right){k^{2}}}\right)\ln\left(2\right)\\\
\hskip
28.45274pt-\left({-10+3\left({4-3\gamma}\right){a^{2}}+5\gamma}\right){k^{2}}-5\gamma+10\Big{)}-\frac{{32}}{{15}}\pi{k^{2}}m\left({{a^{2}}-\frac{5}{6}}\right)r-\frac{{16}}{{75}}\pi{k^{4}}m\left({{a^{2}}-5}\right){r^{3}}+C_{44}\,,\end{array}$
(48)

$\begin{array}[]{l}\bar{h}_{11}\simeq\frac{{{\rm{224}}\pi
m}}{{15{k^{3}}r}}\Bigg{[}-\frac{{10}}{7}+\left({\left({{a^{2}}-\frac{{20{m^{2}}}}{7}+\frac{5}{7}}\right){k^{2}}}\right)\ln\left({kr}\right)+\left({\frac{{\left({-9{a^{2}}+20{m^{2}}}\right){k^{4}}}}{7}+\left({{a^{2}}-\frac{{20{m^{2}}}}{7}}\right){k^{2}}}\right)\ln\left({kr}\right)\\\
\hskip
56.9055pt+\left({\frac{5}{7}+\left({{a^{2}}-\frac{{20{m^{2}}}}{7}+\frac{5}{7}}\right){k^{2}}}\right)\ln\left(2\right)+\gamma\left({{a^{2}}-\frac{{20{m^{2}}}}{7}+\frac{5}{7}}\right){k^{2}}+\frac{{5\gamma}}{7}\Bigg{]}\\\
\hskip
56.9055pt-\frac{{16}}{{15}}\pi{k^{2}}m\left({14{a^{2}}-40{m^{2}}+5}\right)r+\frac{{64}}{{135}}\left({{a^{2}}-10{m^{2}}+\frac{5}{4}}\right)\pi{k^{2}}{r^{2}}+C_{11}\,,\end{array}$
(49)

$\begin{array}[]{l}\bar{h}_{22}\simeq\frac{{4\pi\sin\left({2kr}\right)\left({\pi{a^{2}}{k^{4}}-7{a^{2}}{k^{4}}-15\,{a^{2}}{k^{2}}+50{k^{2}}+35}\right)}}{{15{k^{5}}r}}+\frac{{8\pi\sin\left({2kr}\right)\left({{a^{2}}{k^{4}}+3{a^{2}}{k^{2}}-40{k^{2}}-35}\right)r}}{{15{k^{3}}}}\\\
\\\ \hskip
28.45274pt\frac{{16\pi\left({3{a^{2}}{k^{2}}+60{k^{2}}{{\cos}^{2}}\left({kR}\right)-25{k^{2}}+60{{\cos}^{2}}\left({kR}\right)-25}\right)}}{{45{k^{2}}}}{r^{2}}+C_{22}\,,\end{array}$
(50)

$\begin{array}[]{l}\bar{h}_{33}\simeq-\frac{{32\pi\sin\left({2kr}\right)\left({-\frac{7}{4}-\frac{{{a^{2}}{k^{4}}}}{2}+\left({\frac{{{a^{2}}}}{4}+4}\right){k^{2}}}\right)}}{{15{k^{5}}r}}-\frac{{32\pi\sin\left({2kr}\right)\left({{a^{2}}{k^{6}}+\left({-\frac{{{a^{2}}}}{2}-8}\right){k^{4}}+\frac{{7{k^{2}}}}{2}}\right)}}{{15{k^{5}}}}r\\\
\\\ \hskip
28.45274pt+\frac{{32\pi\left({-40\left({{k^{4}}-\frac{1}{2}{k^{2}}}\right)k\cos{{\left({kR}\right)}^{2}}+\left({\left({\frac{{5{a^{2}}}}{3}+\frac{{50}}{3}}\right){k^{4}}-\frac{{25{k^{2}}}}{3}}\right)k}\right)}}{{75{k^{5}}}}{r^{2}}+C_{33}\,.\end{array}$
(51)

Some interesting cases can be checked. For instance if one consider $a=0$,
then the corrections reduces to Schwarzschild metric in the presence of FFMF
that shows a good concordance with previous results (Sheikhahmadi, 2021). If
one switch off the FFMF, by putting $k=0$, then immediately problem reduces to
Kerr solution. Now we can calculate the off-diagonal components.

$\begin{array}[]{l}\bar{h}_{34}=\bar{h}_{43}\simeq\frac{{32\pi
ma\left({\left({-1+\frac{9}{7}{a^{2}}{k^{2}}-2{k^{2}}}\right)\left({\gamma+\ln\left(2\right)+\ln\left({kr}\right)}\right)+2-\frac{{4{k^{2}}\left({3{a^{2}}-7}\right)}}{7}+\left({-\frac{2}{7}{a^{2}}{k^{4}}-\frac{3}{7}{a^{2}}{k^{2}}+1}\right)\ln\left({kr}\right)}\right)}}{{15{k^{2}}r}}\\\
\hskip
56.9055pt-\frac{{32}}{{105}}a\left({{a^{2}}-\frac{7}{3}}\right)\pi{k^{2}}mr-\frac{{16}}{{525}}\pi
ma{\mkern 1.0mu}{k^{4}}\left({{a^{2}}-14}\right){r^{3}}+C_{43}\,,\end{array}$
(52)

$\begin{array}[]{l}\bar{h}_{13}=\bar{h}_{31}\simeq\frac{{4\pi\left({15k-30\cos{{\left({kr}\right)}^{2}}k-15\sin\left({2kr}\right)m{k^{2}}}\right)}}{{75{k^{4}}r}}\\\
\hskip
56.9055pt+\frac{{4\pi\left({-10{k^{3}}+20{k^{3}}\cos{{\left({kr}\right)}^{2}}+10\sin\left({2kr}\right)m{\mkern
1.0mu}{k^{4}}}\right)}}{{75{k^{4}}}}r+\frac{{4\pi\left({\frac{{5{k^{5}}m}}{4}+10{k^{4}}\sin\left({2kr}\right)}\right)}}{{75{k^{4}}}}{r^{2}}+C_{13}\,,\end{array}$
(53)

$\begin{array}[]{l}\bar{h}_{12}=\bar{h}_{21}\simeq\frac{{{\pi^{2}}\left({\left({\left({-{a^{2}}m+\frac{5}{2}{m^{3}}}\right){k^{2}}-\frac{{11m}}{2}}\right){k^{2}}{{\cos}^{2}}\left({kr}\right)-2\left({-\frac{9}{4}+\frac{{9{k^{2}}{m^{2}}}}{8}}\right)\sin\left({2kr}\right)k-\frac{{\left({\left({-{a^{2}}m+\frac{5}{2}{m^{3}}}\right){k^{2}}-\frac{{11m}}{2}}\right){k^{2}}}}{2}}\right)}}{{4{k^{4}}r}}\\\
\\\ \hskip 56.9055pt+\frac{{\left({m{\mkern
1.0mu}k{{\cos}^{2}}\left({kr}\right)-2\sin\left({2kr}\right)-\frac{{m{\mkern
1.0mu}k}}{2}}\right){\pi^{2}}}}{{4k}}r+\frac{{\left({{{\cos}^{2}}\left({kr}\right)-\frac{1}{2}}\right){\pi^{2}}}}{4}{r^{2}}+C_{12}\,,\end{array}$
(54)

$\begin{array}[]{l}\bar{h}_{23}=\bar{h}_{23}\simeq\frac{{{\pi^{2}}\left({\left({\left({-{a^{2}}m+\frac{5}{2}{m^{3}}}\right){k^{2}}-\frac{{11m}}{2}}\right){k^{2}}{{\cos}^{2}}\left({kr}\right)-2\sin\left({2kr}\right)\left({-\frac{9}{4}+\frac{{9{k^{2}}{m^{2}}}}{8}}\right)k-\frac{{\pi\left({\left({-{a^{2}}m+\frac{5}{2}{m^{3}}}\right){k^{2}}-\frac{{11m}}{2}}\right){k^{2}}}}{2}}\right)}}{{4{k^{4}}r}}\\\
\hskip
56.9055pt+\frac{{{\pi^{2}}\left({mk\cos{{}^{2}}\left({kr}\right)-2\sin\left({2kr}\right)-\frac{{mk}}{2}}\right)}}{{4k}}r+\frac{{{\pi^{2}}\left({{{\cos}^{2}}\left({kr}\right)-\frac{1}{2}}\right)}}{4}{r^{2}}C_{23}\,.\end{array}$
(55)

In above equations the $C_{\mu\nu}$s are some integration constants introduced
to control the behaviour of $h_{\mu\nu}$ coefficients in their $k\rightarrow
0$ limit. When considering the background Kerr metric, denoted as
$g_{\mu\nu}^{(0)}$, and examining the diagonal expressions for
$\bar{h}_{\mu\nu}$ (given by Eqs. (48) to (51)), an interesting result
emerges: the trace of $\bar{h}_{\mu\nu}$ is found to be zero, i.e.,
$\bar{h}=0$. This implies that for the perturbed metric, we have
$h_{\mu\nu}=\bar{h}_{\mu\nu}$. Upon closer analysis of Eqs. (52) to (51), we
observe that in regions far from the black hole, and at the linearized order
of the theory, the Kerr parameter does not appear explicitly. As the strength
of the force-free magnetofluid diminishes, the associated corrections become
negligible, approaching these $C_{\mu\nu}$s coefficients. Finally in the limit
$k\rightarrow 0$, the solution will turn into the pure Kerr solution.

##  5 Conclusion

As mentioned already, the Blandford-Znajek mechanism is a theoretical model
based on General Relativity and magnetohydrodynamics. Observational data
provide evidences on existing strong jets in many astrophysical systems but
the details of the dynamics governing jet formation and the energetic
processes under jets are still an area of ongoing scientific studies.
To gain deeper insights into the underlying physics of this phenomenon, a
closer investigation of the problem was undertaken. The primary objective was
to address the following question: How does the presence of a force-free
magnetofluid source impact the geometric properties of the background black
hole? Essentially, the aim was to determine an appropriate metric that can
effectively describe a magnetized accretion disk surrounding a black hole.
By delving into this problem, main quest was dedicated to exploring the
interplay between the FFMF source and the underlying geometry of the black
hole. The focus was on discerning the modifications and distortions induced by
the presence of the magnetized accretion disk. The ultimate goal was to obtain
a comprehensive understanding of the metric that accurately represents the
combined black hole and magnetized plasma system. By addressing these key
aspects, a refined metric is obtained, which provides a more comprehensive
description of the magnetized accretion disk’s influence on the black hole’s
geometric structure. Such an achievement would significantly contribute to
advancing our understanding of the intricate physics involved in this complex
astrophysical scenario.
This study has shown that it is not enough to depend solely on perturbation
theory set against the background metric. The perturbation of the relevant
Einshtein equations revealed that further improvements to the metric had to be
effected. These corrections comprise the diagonals and the offsets, thus
giving a wider coverage of the system.
In order to accomplish this, the tetrad formalism provided a necessary linkage
between flat and curved geometries. This formalism helped reproduce the
strength tensor of electromagnetic field intensity components. Such an
approach enabled a better comprehension of the interaction between a warped
spacetime and the surrounding electromagnetic field. The succeeding
calculations involved perturbation theory and up to a first order
approximation of the metric. The use of this approximation enabled the
determination of these corrections that increased the accuracy of the system
examined.
The study is meant to give a clearer picture of how perturbations affect
geometries around the black hole. However, through the analysis of the
modified corrections it can be found that the results depend on how close they
are to the black hole. Two distinct regions can be identified: around the
black hole and also in areas far from that.
Close to the black hole and for relatively small distances the correction term
is dependent on the distance as, $1/r$. Importantly, if both the Kerr and
electromagnetic parameters are equal to zero then the problem returns to the
widely known Schwarzschild case characterized by the gravitational field of
the non rotational black hole. On the contrary, these corrections exhibit a
characteristic $r$ power law going all the way up to the second order, expect
$h_{44}$ and $h_{34}$ containing $r$ and $r^{3}$, for areas situated far away
from the black hole. The manner in which it behaves is almost similar to the
case of a rotating Kerr-de Sitter solution, involving a cosmological term.
The analysis highlights some of the changes incurred when there is a black
hole and subsequent force-free magnetofluid. The behaviour near the black hole
fits the Schwarzschild solution while the behavior farther away reveals how
Kerr-de Sitter solutions interact with rotational, cosmic, and geometrical
parameters in these respective areas.

With regard to future projects, there are two immediate ways forward. It could
be interesting to study charged black holes while there are force-free fields
around them since this could give an opportunity to understand how different
fields interact with each other belonging to a charged black hole and an
accretion disk. Such an investigation might shed light on new physical
manifestations and enrich our understanding of these intricate systems.
Also of interest in this regard is the exploring of shadows, and astrophysical
properties related to black holes in the presence of force-free fields. Study
of shadows due to gravitational bending of the light around the black holes
has sparked a lot of interest. Studying these shadows as well as other
astrophysical features in an environment of force-free fields can contribute a
lot in understanding black holes’ nature and their attributes.

## acknowledgments

H.S. expresses sincere gratitude to Yousef Sobouti for introducing this
problem and engaging in numerous constructive discussions during the early
stages of this work. Additionally, H.S. extends heartfelt appreciation to
Hassan Firouzjahi for engaging in insightful discussions on the physics of
black holes and for valuable suggestions resulted in improvement of the
quality of presentation of the work.

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