# Rotating Kiselev Black Holes in $f(R,T)$ Gravity

Sushant G. Ghosh${}^{a,\;b}$<EMAIL_ADDRESS><EMAIL_ADDRESS>Shafqat Ul
Islam${}^{a,\;b}$<EMAIL_ADDRESS>Sunil D. Maharajb<EMAIL_ADDRESS>a
Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India
b Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics
and Computer Science, University of KwaZulu-Natal, Private Bag 54001, Durban
4000, South Africa

###### Abstract

Exact solutions describing rotating black holes can provide significant
opportunities for testing modified theories of gravity, which are motivated by
the challenges posed by dark energy and dark matter. Starting with a spherical
Kiselev black hole as a seed metric, we construct rotating Kiselev black holes
within the $f(R,T)$ gravity framework using the revised Newman-Janis algorithm
— the $f(R,T)$ gravity-motivated rotating Kiselev black holes (FRKBH), which
encompasses, as exceptional cases, Kerr ($K=0$) and Kerr-Newman ($K=Q^{2}$)
black holes. These solutions give rise to distinct classes of black holes
surrounded by fluids while considering specific values of the equation-of-
state parameter, $w$, for viable choices for the $f(R,T)$ function. From the
parameter space or domain of existence of black holes defined by $a$ and
$\gamma$ for FKRBH, we discover that when $a_{1}<a<a_{2}$, there is a critical
value $\gamma=\gamma_{E}$ which corresponds to extreme value black holes
portrayed by degenerate horizons. When $a<a_{1}$ ($a>a_{2}$), we encounter two
distinct critical values $\gamma=\gamma_{E1},\;\gamma_{E2}$ with
$\gamma_{E1}>\gamma_{E2}$ (or $\gamma=\gamma_{E3},\;\gamma_{E4}$ with
$\gamma_{E3}>\gamma_{E4}$. We delve into the horizon and global structure of
FKRBH spacetimes and examine their dependence on parameters $w$ and $\gamma$.
This exploration is motivated by the remarkable effects of $f(R,T)$ gravity,
which gives rise to diverse and intricate spacetime structures within the
domain where black holes exist.

##  I Introduction

The $f(R)$ gravity theory undoubtedly explains the existence of a late-time
cosmic acceleration of the Universe [1]. Despite $f(R)$ gravity’s potential
advantages, this theory can not explain various observational tests conducted
in the solar system, such as the motion of planets [2, 3]. It also faces
challenges in explaining the cosmic microwave background (CMB) tests, the
effects of strong gravitational lensing, and phenomena on galactic scales [4,
5, 6, 7, 8, 9]. Further, $f(R)$ gravity struggles to account for stable
stellar configurations [10, 11, 12]. These shortcomings have prompted
scientists to further develop and generalize the theory by exploring the
coupling between scalar curvature and matter [13, 14, 15]. By incorporating
this coupling, researchers hope to overcome the limitations and improve the
overall consistency of the theory. Harko et al. [16] have put forward a
generalized form of $f(R)$ gravity theory, namely $f(R,T)$ gravity. The
presence and distribution of matter affect the curvature of spacetime, and
conversely, the curvature of spacetime influences the motion of matter. This
coupling leads to exciting phenomena and potentially explains specific
astrophysical observations. The $f(R,T)$ gravity is an intriguing approach
that extends the Einstein field equations by introducing additional terms
involving the curvature scalar $R$ and the trace of the energy-momentum tensor
$T$. The theory received significant attention in recent years because of its
potential to address various cosmological phenomena and serves as a modified
theory of gravity that can account for dark matter and dark energy (see, e.g.,
[17, 18, 19, 20, 21, 22]). These articles provide an extensive overview of
$f(R,T)$ gravity and discuss its theoretical foundations, mathematical
formalism, and cosmological implications. They cover various topics, including
models, observational constraints, and the impact of $f(R,T)$ gravity on the
early and late universe. This gravitational theory’s dependence on $T$ can be
attributed to quantum effects, imperfect fluids, extra fluids, or an effective
cosmological constant. The application of $f(R,T)$ gravity has produced
exciting results in various regimes, as evidenced in references [23, 24, 25,
26, 27].

For high densities and pressures, the effects of modifying gravity through
$f(R,T)$ theory are expected to become more pronounced. Therefore, it is
natural to investigate the impact of these modifications on compact objects or
black holes. In theories where the Lagrangian density depends on $T$, it is
expected that there will be differences in the solutions compared to general
relativity, particularly when considering non-zero energy momentum tensors.
This implies additional effects arising from the coupling between matter and
geometry. An intriguing system to explore effects is fluids surrounding
spherical matter sources. Motivated by this, Santos et al. [28] discussed
spherical black hole solutions within the framework of $f(R,T)$ gravity, with
the black hole being surrounded by the quintessence fluid discussed by Kiselev
[29]. They investigated specific physical scenarios corresponding to solutions
obtained by carefully selecting appropriate values for the fluid equation of
state parameters. They solely focussed on viable choices for the $f(R,T)$
function to ensure that they could connect the obtained results to an
extension of the Kiselev solution [29]. However, testing astrophysical
observations poses a challenge for spherical black hole models since black
hole spin, i.e., rotating black holes commonly found in nature, significantly
influences astrophysical processes [30]. Furthermore, the lack of rotating
black hole models in the modified gravity, including $f(R,T)$ gravity, hampers
the ability to test the theory through observations [31, 32, 33, 34, 35]. The
Kerr metric [36], which represents rotating black hole solutions in general
relativity resulting from the collapse of massive stars, holds great
significance. Remarkably, the revised Newman-Janis method has been successful
in generating rotating metrics from non-rotating seed metrics in other
modified gravity models [31, 37, 38, 39, 40, 34, 35, 41, 42]. This success has
motivated us to pursue a rotating or axisymmetric extension of the spherical
metric obtained in [28] or to find a Kerr-like metric, referred to as an
$f(R,T)$ motivated rotating Kiselev black hole (FRKBH) metric, which can be
tested using astrophysical observations. We employ the revised Newman-Janis
algorithm starting from a spherical black hole in [28] as a seed metric to
construct a rotating spacetime. We examine the black hole’s various
properties, including its horizon’s structure, and create Penrose and
embedding diagrams for further analysis. To methodically analyze the FRKBH, we
use specific values for the fluid parameter $w$ corresponding to black holes
surrounded by different fields. These fields include dust ($w=0$), radiation
($w=1/3$), quintessence ($w=-2/3$), and phantom ($w=-4/3$) [29, 40].

The paper is organized as follows: In Section II, we review spherical
symmetric black holes in $f(R,T)$ gravity. Then, in Section III, we construct
the rotating counterpart of the spherical seed metric (13), known as the FRKBH
metric. Additionally, within the same section, we discuss the general features
of the FRKBH metric, including horizon structures and energy conditions.
Furthermore, we analyze the phase space of the FRKBH spacetimes, highlighting
their unique properties in Section V. In Section IV, we utilize the spacetime
isometries to determine the conserved mass and angular momentum of the FRKBH
spacetime. Finally, in Section VI, we summarise our main findings.

##  II Black Hole Solution

The $f(R,T)$ modified theory of gravity is considered to be a generalisation
of general relativity [16, 25, 24, 26, 43, 44, 45]. The Einstein-Hilbert
action in the context of $f(R,T)$ gravity takes the form

$S=\frac{1}{16\pi}\int f(R,T)\sqrt{-g}d^{4}x+\int L_{m}\sqrt{-g}d^{4}x,$ (1)

where the function $f(R,T)$ is an arbitary function of the Ricci scalar $R$
and the trace $T$ of the energy-momentum tensor of matter $T_{\mu\nu}$.
$L_{m}$ in Eq. (1) denotes the matter Lagrangian density, which is linked to a
specific energy-momentum tensor. By varying the action with respect to metric
tensor $g_{\mu\nu}$ results [16]

$\displaystyle\delta S=$
$\displaystyle\frac{1}{16\pi}\int\left[f_{R}(R,T)R_{\mu\nu}\delta
g^{\mu\nu}+f_{R}(R,T)g_{\mu\nu}\Box\delta g^{\mu\nu}\right.$ $\displaystyle-
f_{R}(R,T)\nabla_{\mu}\nabla_{\nu}\delta
g^{\mu\nu}+f_{T}(R,T)\frac{\delta(g^{\eta\xi}T_{\eta\xi})}{\delta
g^{\mu\nu}}\delta g^{\mu\nu}$
$\displaystyle\left.-\frac{1}{2}g_{\mu\nu}f(R,T)\delta
g^{\mu\nu}+\frac{16\pi}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}\right]\sqrt{-g}d^{4}x,$ (2)

where $f_{R}(R,T)=\partial f(R,T)/\partial R$ and $f_{T}(R,T)=\partial
f(R,T)/\partial T$. The variation of $T$ with respect to the metrics tensor
yields

$\frac{\delta(g^{\eta\xi}T_{\eta\xi})}{\delta
g^{\mu\nu}}=T_{\mu\nu}+\Theta_{\mu\nu}.$ (3)

where

$\Theta_{\mu\nu}\equiv g^{\eta\xi}\frac{\delta T_{\eta\xi}}{\delta
g^{\mu\nu}}=-2T_{\mu\nu}+g_{\mu\nu}L_{m}-2g^{\eta\xi}\frac{\partial^{2}L_{m}}{\partial
g^{\mu\nu}g^{\eta\xi}}.$ (4)

After integrating the second and third term in Eq. (II), we obtain the field
equations of the $f(R,T)$ gravity as

$\displaystyle f_{R}$
$\displaystyle(R,T)R_{\mu\nu}-\frac{g_{\mu\nu}}{2}f(R,T)+(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})f_{R}(R,T)$
$\displaystyle=8\pi
T_{\mu\nu}-f_{T}(R,T)T_{\mu\nu}-f_{T}(R,T)\Theta_{\mu\nu}\,.$ (5)

When $f(R,T)\equiv f(R)$, Eq. (II) reduces to the field equations in the
context of $f(R)$ gravity [40]. The novel feature that $f(R,T)$ gravity
introduces is the possibility of arbitrary coupling between matter and
geometry. This paper considers a special case of $f(R,T)$ gravity such that
the $f(R,T)$ function is given by

$f(R,T)=R+2f(T),$ (6)

where $f(T)$ is an arbitrary function of the trace of the energy momentum
tensor. On using the Eq. (6) in Eq. (II), the field equations simplifies to

$\displaystyle R_{\mu\nu}-\frac{g_{\mu\nu}}{2}R=$ $\displaystyle 8\pi
T_{\mu\nu}-2f^{\prime}(T)T_{\mu\nu}$
$\displaystyle-2f_{T}^{\prime}(T)\Theta_{\mu\nu}+f(T)g_{\mu\nu},$ (7)

where $f^{\prime}(T)=df(T)/dT$.

We solve Eq. (7) for the quintessence field introduced by Kiselev [29], which
is characterized by the equation of state $p=\omega\rho$. with
$-1/3<\omega<-1$. One possible candidate to explain dark energy is
quintessence field. The $T_{\mu\nu}$ for the quintessence matter reads [29]

$\displaystyle T^{t}_{\>\>\>t}=T^{r}_{\>\>\>r}$ $\displaystyle=\rho(r),$ (8)
$\displaystyle T^{\theta}_{\>\>\>\theta}=T^{\phi}_{\>\>\>\phi}$
$\displaystyle=-\frac{1}{2}\rho(3\omega+1),$ (9)

and $\omega$ is the parameter of the equation of state. Kiselev black holes
[29] have the components of energy-momentum tensor effectively connected to an
anisotropic fluid represented by

$T^{\mu}_{\>\>\>\nu}=\mbox{diag}(\rho,-p_{r},-p_{t},-p_{t}),$ (10)

where $p_{r}=-\rho$ and $p_{t}=\frac{1}{2}\rho(3w+1)$, which can be extracted
from the general form of the anisotropic fluid [46]:

$T_{\mu\nu}=-p_{t}g_{\mu\nu}+(p_{t}+\rho)U_{\mu}U_{\nu}+(p_{r}-p_{t})N_{\mu}N_{\nu},$
(11)

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Figure 1: The parameter space of $a$ and $\gamma$ for FKRBH. When $a_{1}<a<a_{2}$, an extreme value $\gamma=\gamma_{E}$ corresponding extremal value black holes exists. When $a<a_{1}$ ($a>a_{2}$), we have two extremal $\gamma=\gamma_{E1}$ and $\gamma=\gamma_{E2}$ with $\gamma_{E1}>\gamma_{E2}$ (or $\gamma=\gamma_{E3}$ and $\gamma=\gamma_{E4}$ with $\gamma_{E3}>\gamma_{E4}$). The behaviour of horizons at $\omega=0$ with varying black hole parameter $\gamma$ at (i) $a=0.42$ (top right), (ii) $a=0.60$ (bottom left), and (iii) $a=0.90$ (bottom right). |
---|---
|
Figure 2: The parameter space of $a$ and $\gamma$ at $\omega=-2/3$ (top right). The behaviour of horizons at $\omega=-2/3$ with varying black hole parameter $\gamma$ at (i) $a=0.60$ (top right) (ii) $a=0.80$ (bottom left) and (iii) $a=0.90$ (bottom right). The solid blue and black lines correspond to extreme values of parameters. |
---|---
|
Figure 3: The parameter space of $a$ and $\gamma$ at $\omega=-4/3$ (top right). The behaviour of horizons at $\omega=-4/3$ with varying black hole parameter $\gamma$ at (i) $a=0.60$ (top right) (ii) $a=0.86$ (bottom left) and (iii) $a=0.90$ (bottom right). The solid blue and black lines correspond to extreme values of parameters. |
---|---
Figure 4: The parameter space of $a$ and $\gamma$ at $\omega=1/3$ (top right).
The behaviour of horizons at $\omega=-2/3$ and $a=0.80$ with varying black
hole parameter $\gamma$. The solid blue line correspond to extreme value of
parameter $\gamma$.

where $p_{t}(r)$, $\rho(r)$ and $p_{r}(r)$ are the tangential or transverse
pressure, the energy density and the radial pressure of the fluid,
respectively. The quantities $U_{\mu}$ and $N_{\mu}$ represent the four-
velocity and radial unit vector, respectively, such that they obey the
conditions $U_{\nu}U^{\nu}=1$, $N_{\nu}N^{\nu}=-1$ and $U_{\nu}N^{\nu}=0$. The
matter Lagrangian density associated with the anisotropic fluid is given by
$L_{m}=(-1/3)(p_{r}+2p_{t})$ [47]. Ghosh [40] was the pioneer in obtaining the
rotating counterpart of Kiselev black holes. Eq. (4) can be written as

$\Theta_{\mu\nu}=-2T_{\mu\nu}-\frac{1}{3}(p_{r}+2p_{t})g_{\mu\nu}.$ (12)

On using Eq’s (7), (11) and (12), a static spherical symmetric solution of
Einsteins equations reads

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(1-\frac{2M}{r}+\frac{K}{r^{d}}\right)dt^{2}-\frac{dr^{2}}{\left(1-\frac{2M}{r}-\frac{K}{r^{d}}\right)}-r^{2}d\Omega^{2},$
(13)

where

$d=\frac{8(\gamma\omega+\pi(3\omega+1))}{\gamma(3-\omega)+8\pi}.$ (14)

Thus we have a general form of exact spherically symmetric black holes in
$f(R,T)$ gravity describing a Kiselev black hole, i.e., a black hole solution
surrounded by the quintessence matter. The parameter $-1<\omega<-1/3$ for a de
Sitter horizon causes acceleration, and $-1/3<\omega<0$ for an asymptotically
flat solution. Here, $\omega$ is the parameter of the equation of state,
$\gamma$ is the model-dependent parameter from the $f(R,T)$ gravity, $K$ and
$M$ are integration constants. The Kiselev black hole reduces to Schwarzschild
black hole for $K=0$, and to Reisner-Nordstrom black hole for $d=2$ and
$K=Q^{2}$.

##  III Rotating Black hole

The Newman$-$Janis algorithm, initially designed within the framework of
general relativity, has found widespread application in building rotating
black hole solutions from their non-rotating counterparts [48]. Recently, this
algorithm has also been utilized in modified gravity theories to generate
rotating solutions based on non-rotating configurations [49, 50, 51, 52, 53].
However, it is essential to note that when applying the Newman$-$Janis
algorithm to arbitrary non-general relativity spherically symmetric solutions,
specific issues may arise in the resulting axially-symmetric metric [54]. In
rotating black holes derived from modified gravity using the Newman$-$Janis
algorithm, additional sources will probably arise alongside the original ones.
The metric for a rotating black hole in Einstein-Gauss-Bonnet gravity is also
generated by utilizing the modified Newman$-$Janis algorithm, which
incorporates Azreg-Aïnou’s non-complexification procedure [55, 56]. This
procedure has successfully generated other rotating solutions with imperfect
fluid content in Boyer$-$Lindquist coordinates, starting from spherically
symmetric static solutions [57, 58, 31, 32, 33, 34, 35, 59] and it can also
generate rotating Kislev black hole solutions [40]. The resulting rotating
black hole metric, the counterpart of the spherically symmetric solution (13),
governed by parameters $M$, $a$, $\omega$, and $\gamma$ encompasses the Kerr
solution [36] and also generalizes the Kerr-Newman solution [60], which in
Boyer-Lindquist coordinates reads

$\displaystyle ds^{2}=$
$\displaystyle\left(\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\right)dt^{2}-\frac{\Sigma}{\Delta}\,dr^{2}$
$\displaystyle+2a\sin^{2}\theta\left(1-\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\right)dt\,d\phi-\Sigma\,d\theta^{2}$
$\displaystyle-\,\sin^{2}\theta\left[\Sigma+a^{2}\sin^{2}\theta\left(2-\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\right)\right]d\phi^{2},$
(15)

with

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
r^{2}+a^{2}-2rM(r),\quad\Sigma=r^{2}+a^{2}\cos^{2}\theta,$ $\displaystyle
M(r)$ $\displaystyle=$ $\displaystyle M-\frac{K}{2r^{d-1}},$ (16)

with $a$ being the spin parameter. The metric Eq. (15) reverts to Kerr black
holes for the special case $K\to 0$, to Kerr-Newman black holes when $K=Q^{2}$
and $d=2$ or when $\gamma$ and $\omega$ are related via

$\gamma=\frac{-4\pi(3\omega-1)}{5\omega-3}$ (17)

and to spherically symmetric black holes (13) when only $a=0$. For
definiteness, we call the five-parameter metrics (15) $-$ the $f(R,T)$
gravity-motivated rotating Kiselev black holes (FRKBH). To comprehensively
analyse the FRKBH, we employ specific parameter values $w$ corresponding to
black holes surrounded by various fields. These encompass dust ($w=0$),
radiation ($w=1/3$), quintessence field ($w=-2/3$), and phantom field
($w=-4/3$) [29, 40]. Interestingly, similar to the Kerr spacetime, the FRKBH
spacetime metric (15) nevertheless maintains the time-translational and
rotational invariance isometries, which, respectively, entail the existence of
two Killing vector fields $\eta^{\mu}_{(t)}=\left(\frac{\partial}{\partial
t}\right)^{\mu}$ and
$\eta^{\mu}_{(\phi)}=\left(\frac{\partial}{\partial\phi}\right)^{\mu}$.

|
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Figure 5: Behaviour of event horizon (solid lines) and Cauchy horizon (dashed lines) vs spin parameter $a$ for different values of $\gamma$ and $\omega=0,~{}-2/3,~{}-4/3,~{}1/3$.  |
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|
Figure 6: The behavior of the SLS with varying spin $a$ and $\gamma$ for
$\omega=0,~{}-2/3,~{}-4/3,~{}1/3$.

In order to further analyze the source associated with the metric (15), we use
an orthonormal basis in which the energy momentum tensor is diagonal [51, 61,
39]

$e^{(a)}_{\mu}=\left(\begin{array}[]{cccc}\sqrt{\mp(g_{tt}-\Omega
g_{t\phi})}&0&0&0\\\ 0&\sqrt{\pm g_{rr}}&0&0\\\
0&0&\sqrt{g_{\theta\theta}}&0\\\
{g_{t\phi}}/{\sqrt{g_{\phi\phi}}}&0&0&\sqrt{g_{\phi\phi}}\end{array}\right),$
(18)

with $\Omega=g_{t\phi}/{g_{\phi\phi}}$. The components of the energy momentum
tensor in the orthonormal frame read

$T^{(a)(b)}=e^{(a)}_{\mu}e^{(b)}_{\nu}G^{\mu\nu}.$ (19)

Considering the line element (15), we can write the components of the
respective energy momentum tensor as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{(d-1)K}{(a^{2}+r^{2})^{2}r^{d-2}}=-P_{1},$ $\displaystyle
P_{2}$ $\displaystyle=$
$\displaystyle\frac{(d-1)\Big{[}a^{2}(d+2)+dr^{2}\Big{]}}{2(a^{2}+r^{2})^{2}r^{d}}=P_{3},$
(20)

###  III.1 Energy conditions and horizons

To check the weak energy condition, we can choose an appropriate orthonormal
basis [51, 61, 39] in which the energy momentum tensor reads

$T^{(a)(b)}=\mbox{diag}(\rho,P_{1},P_{2},P_{3}).$ (21)

The weak energy condition requires $\rho\geq 0$ and $\rho+P_{i}\geq 0$
($i=1,\;2,\;3$) [62]. Therefore, $K\geq 0$ and $d\geq 1$, which are
assumptions used throughout this work, are necessary for the weak energy
condition to be satisfied. Furthermore, the strong energy condition is given
by

$\rho\geq 0,\quad\rho+P_{i}\geq 0(i=1,\;2,\;3),\quad\rho+P_{r}+2P_{\theta}\geq
0,$ (22)

which are also fulfilled only when $K\geq 0$ and $d\geq 1$.

The event horizon is a stationary null surface that serves as the origin of
null geodesic rays that are projected into the future but are never able to
travel arbitrarily far from the black hole [63, 64], are defined by the
surfaces $g^{\mu\nu}\partial_{\mu}r\partial_{\nu}r=g^{rr}=\Delta=0,$ and thus,
the radii are zeros of

$r^{2}+a^{2}-2Mr+\frac{K}{r^{d-2}}=0.$ (23)

For the special case $d=2$, and $K=Q^{2}$, Eq. (23) reduces to

$r^{2}+a^{2}-2Mr+Q^{2}=0,$ (24)

where $K$ is identified as the charge $Q^{2}$, and solutions of the above
equation give radii of horizons for the Kerr-Newman black hole given by

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}-Q^{2}}.$ (25)

$\omega$ |  $a$ |  $\gamma$ |  $r_{+}$ |  $r_{-}$
---|---|---|---|---
$0$ |  $a=0.42$ |  $-0.50$ |  0.7517 |  0.2778
|  |  $-1.44$ |  0.5587 |  0.5587
|  |  $-5.74$ |  1.3450 |  1.3450
|  |  $-6.00$ |  1.5704 |  1.1549
|  $a=0.60$ |  $-6.36$ |  1.3954 |  1.3954
|  |  $-6.60$ |  1.6014 |  1.2109
|  $a=0.90$ |  $-7.59$ |  1.3105 |  1.3105
|  |  $-7.85$ |  1.4256 |  1.331
|  |  $-9.40$ |  0.8411 |  0.5699
|  |  $-10.40$ |  0.6551 |  0.6551
$-2/3$ |  $a=0.60$ |  $-5.00$ |  0.7293 |  0.2565
|  |  $-5.38$ |  0.4825 |  0.4825
|  |  $-6.30$ |  1.3972 |  1.3972
|  |  $-6.35$ |  1.6071 |  1.2095
|  $a=0.80$ |  $-6.47$ |  1.3793 |  1.3793
|  |  $-6.60$ |  1.5897 |  1.1287
|  $a=0.90$ |  $-6.60$ |  1.3149 |  1.3149
|  |  $-6.70$ |  1.4387 |  1.1077
|  |  $-7.20$ |  0.8734 |  0.5647
|  |  $-7.98$ |  0.6580 |  0.6580
$-4/3$ |  $a=0.60$ |  $-6.24$ |  1.5896 |  1.223
|  |  $-6.27$ |  1.3978 |  1.3978
|  |  $-6.805$ |  0.4872 |  0.4872
|  |  $-6.90$ |  0.6860 |  0.2891
|  $a=0.80$ |  $-6.08$ |  1.3498 |  1.3498
|  |  $-6.00$ |  1.5018 |  1.1364
|  $a=0.90$ |  $-3.82$ |  0.6501 |  0.6501
|  |  $-5.30$ |  0.8507 |  0.5718
|  |  $-5.95$ |  1.4466 |  1.1121
|  |  $-6.04$ |  1.3160 |  1.3160
$1/3$ |  $a=0.80$ |  $-7.76$ |  1.3902 |  1.3902
|  |  $-8.20$ |  1.5644 |  1.1801

Table 1:  Table summarizing the values Event horizons ($r_{+}$), Cauchy
horizons ($r_{-}$) and extremal horizon $T_{+}=r_{-}$ for various $\omega$ and
$\gamma$.

An analysis of Eq. (23) reveals that it has a maximum of two real positive
roots, corresponding to the inner Cauchy horizon ($r_{2}$) and outer event
horizon ($r_{1}$), such that $r_{2}\leq r_{1}$. Two distinct real positive
roots of $\Delta=0$ infer the nonextremal black hole, while no black hole in
the absence of real positive roots of Eq. (23), i.e., no horizon exists. There
exists a particular value of the parameter $\gamma$, $\gamma=\gamma_{e}$, for
which an extremal black hole occurs, such that Eq. (23) admits a double root;
i.e., the two horizons coincide $r_{2}=r_{1}=r_{e}$. We have explicitly shown
that, for fixed values of $a$, $K$ and $\omega$, $r_{1}$ decreases and $r_{2}$
increases with increasing $\gamma$ and eventually coincide for the extremal
value of $\gamma$, i.e., $r_{2}=r_{1}=r_{e}$ for $\gamma=\gamma_{e}$.
Moreover, we infer that it is possible to find extremal values of parameters
$a=a_{e}$ for fixed $\gamma$ and $\omega$, and $\omega=\omega_{e}$ for fixed
$a$ and $\gamma$, for which the algebraic equation $\Delta=0$ has double roots
(cf. Figures 1-4).

#### Case $\omega=0,\;-2/3,\;\text{and}-4/3:$

Intrestingly, when $a<a_{1}$ (e.g., $a_{1}\approx 0.5$ for $\omega=0$) and
$a>a_{2}$ (e.g., $a_{2}=0.809814$ for $\omega=0$), there exists two extreme
values of $f(R,T)$ gravity parameter viz. $\gamma=\gamma_{E1},\gamma_{E2}$ (or
$\gamma_{E3},\gamma_{E4}$) such that $\Delta=0$ admits roots corresponding to
extremal black holes with degenerate horizons, whereas for
$\gamma_{E2}<\gamma<\gamma_{E1}$, one gets an NS. However, when
$a_{1}<a<a_{2}$, there is only one extreme parameter value, i.e.,
$\gamma_{E}$, corresponds to an extremal black hole with degenerate horizons.
In this case, again, $\gamma<\gamma_{E}$ corresponds to the black hole with
two horizons and $\gamma>\gamma_{E}$ leads to naked singularities.

#### Case $\omega=1/3:$

In this case, for a given $a$, a critical value of $\gamma=\gamma_{E}$ exists,
such that the $\Delta=0$ has a double root corresponding to an extremal black
hole with degenerate horizons. When $\gamma<\gamma_{E}$, $\Delta=0$ has two
simple roots and has no zero for $\gamma>\gamma_{E}$. These two cases
correspond to a non-extremal black hole with two horizons and an NS (or no-
horizon spacetime).

Further, the static observers in the stationary spacetime follow the worldline
of the timelike Killing vector $\eta^{\mu}_{(t)}$, such that their four-
velocity is $u^{\mu}\propto\eta^{\mu}_{(t)}$ with the proper normalization
factor. These observers exist as long as $\eta^{\mu}_{(t)}$ is timelike, such
that $\eta^{\mu}_{(t)}\eta_{\mu(t)}=g_{tt}=0$ or

$r^{2}+a^{2}\cos^{2}\theta-2Mr+\frac{K}{r^{d-2}}=0,$ (26)

defines the boundary of the static limit surface (SLS), which, apart from
black hole parameters, also depends on $\theta$ and coincides with the event
horizon only at the poles. For the particular case $K=Q^{2}$ and $d=2$, Eq.
(26) corresponds to the Kerr-Newman black hole as

$r^{2}+a^{2}\cos^{2}\theta-2Mr+Q^{2}=0\,label{gtteqKN}$ (27)

and admits the solutions

$r_{SLS}^{\pm}=M\pm\sqrt{M^{2}-a^{2}\cos^{2}\theta-Q^{2}},$

which can be identified as the SLS radii for the Kerr-Newman black hole.
Equation (26) is solved numerically, and the behaviour of SLS is shown in Fig.
6. It is clear from Fig. 6 that the radii of the SLS decrease with increasing
$K$ and $a$. The two SLS, corresponding to the real positive roots of Eq.
(26), coincide for suitably chosen parameters. However, these extremal values
are different from those for the degenerate horizons. For fixed values of $M$
and $a$, the SLS radii for the FKRBHs are smaller than the Kerr black hole
values. Likewise, the Kerr black hole, apart from $\Delta=0$, which is merely
a coordinate singularity, rotating metric (15) is also singular at $\Sigma=0$,
which is attributed to a ring-shaped physical singularity at the equatorial
plane of the centre of the black hole with radius $a$.

##  IV Komar Mass and Angular Momentum

Moreover, zero angular momentum observers, which are stationary observers with
zero angular momentum concerning spatial infinity, but due to frame dragging,
have the position-dependent angular velocity $\omega$ given by

$\omega=\frac{d\phi}{dt}=-\frac{g_{t\phi}}{g_{\phi\phi}}=\frac{a\left(a^{2}-\Delta+r^{2}\right)}{(a^{2}+r^{2})^{2}-a^{2}\Delta\sin^{2}{\theta}},$
(28)

which increases as the observer approaches the black hole and eventually takes
the maximum value at the event horizon:

$\Omega=\left.\omega\right|_{r=r_{1}}=\frac{a}{(r_{1}^{2}+a^{2})},$ (29)

such that observers are in a state of co-rotation with the black hole. Here,
$\Omega$ is the black hole angular velocity, which has the same form as the
Kerr black hole [64, 65].

The conserved quantities associated with the asymptotically timelike and
spacelike Killing vector fields, respectively, $\eta^{\mu}_{(t)}$ and
$\eta^{\mu}_{(\phi)}$, correspond to the mass and angular momentum assigned to
the stationary, asymptotically flat black hole spacetime. A general argument
for equality of the conserved Arnowitt-Deser-Misner mass [66] and of the Komar
mass [67] for stationary spacetimes having a timelike Killing vector is
established in Refs. [68, 69]. Following the Komar [67] definitions of
conserved quantities, we consider a spacelike hypersurface $\Sigma_{t}$,
extending from the event horizon to spatial infinity, which is a surface of
constant $t$ with unit normal vector $n_{\mu}$ [65]. The two-boundary $S_{t}$
of the hypersurface $\Sigma_{t}$ is a constant $t$ and constant $r$ surface
with unit outward normal vector $\sigma_{\mu}$. The effective mass reads [67]

$M_{\text{eff}}=-\frac{1}{8\pi}\int_{S_{t}}\nabla^{\mu}\eta^{\nu}_{(t)}dS_{\mu\nu},$
(30)

where $dS_{\mu\nu}=-2n_{[\mu}\sigma_{\nu]}\sqrt{h}d^{2}\theta$ is the surface
element of $S_{t}$, $h$ is the determinant of the ($2\times 2$) metric on
$S_{t}$, and

$n_{\mu}=-\frac{\delta^{t}_{\mu}}{|g^{tt}|^{1/2}},\qquad\sigma_{\mu}=\frac{\delta^{r}_{\mu}}{|g^{rr}|^{1/2}},$
(31)

are, respectively, timelike and spacelike unit outward normal vectors. Thus,
mass integral Eq. (30) turns into an integral over closed 2-surface at
infinity

$\displaystyle M_{\text{eff}}=$
$\displaystyle\frac{1}{4\pi}\int_{0}^{2\phi}\int_{0}^{\phi}\frac{\sqrt{g_{\theta\theta}g_{\phi\phi}}}{|g^{tt}g^{rr}|^{1/2}}\nabla^{t}\eta^{r}_{(t)}d\theta
d\phi$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int_{0}^{2\phi}\int_{0}^{\phi}\frac{\sqrt{g_{\theta\theta}g_{\phi\phi}}}{|g^{tt}g^{rr}|^{1/2}}\left(g^{tt}\Gamma^{r}_{tt}+g^{t\phi}\Gamma^{r}_{t\phi}\right)d\theta
d\phi.$ (32)

Using the metric elements Eq. (15), we obtain the effective mass of the
rotating Kiselev black hole

$M_{\text{eff}}=M-\frac{K}{2r^{d-1}}\left[\left(\frac{r}{a}+\frac{a}{r}\right)(d-1)\tan^{-1}\frac{a}{r}+1\right],$
(33)

which is corrected due to the field and goes over to the Kerr black hole case
that is $M_{\text{eff}}=M$, when $K=0$. For the special case $d=2$ and
$K=Q^{2}$, Eq. (33) resembles the effective mass for the Kerr-Newman black
hole with $K$ as the electric charge $Q^{2}$ and reads [70]

$M_{\text{eff}}^{KN}=M-\frac{Q^{2}}{2r^{2}a}\left[(r^{2}+a^{2})\tan^{-1}\left(\frac{a}{r}\right)+ar\right].$
(34)

The effective mass for the spherically symmetric Kiselev black hole ($a=0$) is
obtained from Eq. (33) and reads

$M_{\text{eff}}^{NR}=M-\frac{d~{}K}{2r^{d-1}},$ (35)

which reduces to the Reissner$-$Nordstrom black hole for $K=Q^{2}$ and $d=2$:

$M_{\text{eff}}^{RN}=M-\frac{Q^{2}}{r},$

and to the Schwarzschild black hole $M_{\text{eff}}^{S}=M$, when $K=0$.

Now, we use the spacelike Killing vector $\eta^{\mu}_{(\phi)}$ to calculate
the effective angular momentum [67]

$J_{\text{eff}}=\frac{1}{16\pi}\int_{S_{t}}\nabla^{\mu}\eta^{\nu}_{(\phi)}dS_{\mu\nu},$
(36)

using the definitions of the surface element, Eq. (36) recast as

$\displaystyle J_{\text{eff}}=$
$\displaystyle-\frac{1}{8\pi}\int_{0}^{2\phi}\int_{0}^{\phi}\nabla^{\mu}\eta^{\nu}_{(t)}n_{\mu}\sigma_{\nu}\sqrt{h}d\theta
d\phi$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int_{0}^{2\phi}\int_{0}^{\phi}\frac{\sqrt{g_{\theta\theta}g_{\phi\phi}}}{|g^{tt}g^{rr}|^{1/2}}\left(g^{tt}\Gamma^{r}_{t\phi}+g^{t\phi}\Gamma^{r}_{\phi\phi}\right)d\theta
d\phi.$ (37)

After performing the integration for the rotating Kiselev black hole Eq. (15),
this reads

$\displaystyle J_{\text{eff}}$ $\displaystyle=$ $\displaystyle
Ma+\frac{K}{4r^{d}a^{2}}\Big{[}\left((d-3)a^{2}+r^{2}(d-1)\right)ar$ (38)
$\displaystyle-(d-1)(a^{2}+r^{2})^{2}\tan^{-1}\frac{a}{r}\Big{]}$

which vanishes identically in the limiting case of $a=0$. For the particular
case of $d=2$ and $K=Q^{2}$ it reduces to

$J_{\text{eff}}^{KN}=Ma+\frac{Q^{2}(r^{2}-a^{2})}{4ar}-\frac{Q^{2}}{4a^{2}r^{2}}(r^{2}+a^{2})^{2}\tan^{-1}\left(\frac{a}{r}\right),$
(39)

which can be identified as the Kerr-Newman black hole value [70]. In the
asymptotic limits $r\to\infty$, the effective angular momentum Eq. (38)
restores the value $J_{\text{eff}}^{K}=Ma$, which corresponds to the value for
the Kerr black hole. Thus, the effects of the field subside at a very large
distance from the black hole. Equations (33) and (38) imply that at a finite
radial distance the values of the effective mass and angular momentum get
modified from their asymptotic values and depend on of $\gamma$. Figures 7 and
8, graphically present the normalized values of the effective mass and angular
momentum as functions of $r$ for various $\gamma$ values. It is evident from
the figures that these effective quantities gradually decrease with decreasing
$r$. Introducing parameters $\gamma$ and $\omega$ reduces the effective mass
and angular momentum, specifically $M_{eff}/M\leq 1$ and $J_{eff}/Ma\leq 1$.
Moreover, at a fixed radial coordinate, the normalized values of the effective
quantities for regular black holes ($\gamma\neq 0$) are smaller than those for
Kerr black holes ($K=0$) (See Ref. [34]). The impact of a non-zero $\gamma$ is
significant only in the vicinity of the event horizon $r_{+}$. However, it
diminishes at larger distances from $r_{+}$, resulting in $M_{eff}/M=1$ and
$J_{eff}/Ma=1$ for large $r$ (see Fig’s 7 and 8). Consequently, $M_{eff}$ and
$J_{eff}$ are consistently smaller than their asymptotic values.

|
---|---
|
Figure 7: The behaviour of effective mass with $r$ at $a=0.9$ and with varying $\omega$ and $\gamma$. |
---|---
|
Figure 8: The behaviour of effective angular momentum with $r$ at $a=0.9$ and
with varying $\omega$ and $\gamma$.

It is well known that the Killing vectors $\eta^{\mu}_{(t)}$ or
$\eta^{\mu}_{(\phi)}$ are not the generators of the stationary black hole
horizon; rather, it is their specific linear combination [65]

$\chi^{\mu}=\eta^{\mu}_{(t)}+\Omega\eta^{\mu}_{(\phi)},$ (40)

such that $\chi^{\mu}$ is globally timelike outside the event horizon, though
it is a Killing vector only at the horizon [65]. The Komar conserved quantity
at the event horizon associated with $\chi^{\mu}$ reads as [67]

$\displaystyle J_{\chi}$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi}\int_{S_{t}}\nabla^{\mu}\chi^{\nu}dS_{\mu\nu},$
(41) $\displaystyle=$
$\displaystyle-\frac{1}{8\pi}\int_{S_{t}}\nabla^{\mu}\left(\eta^{\mu}_{(t)}+\Omega\eta^{\mu}_{(\phi)}\right)dS_{\mu\nu}.$

Using Eqs. (33) and (38), we obtain

$\displaystyle J_{\chi}$ $\displaystyle=$ $\displaystyle
M_{\text{eff}}-2\Omega J_{\text{eff}},$ (42) $\displaystyle=$
$\displaystyle\frac{M(r_{1}^{2}-a^{2})}{(r_{+}^{2}+a^{2})}-\frac{\left(r_{+}^{2}-(s-1)a^{2}\right)}{(r_{+}^{2}+a^{2})s}\frac{K}{r_{+}^{-(s-2)/s}}.$

To understand the implication of the above conserved quantity, one must know
the black hole horizon temperature [65]

$\displaystyle T_{+}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2\pi}=\frac{\Delta^{\prime}}{4\pi(r_{+}^{2}+a^{2})},$
(43) $\displaystyle=$ $\displaystyle\frac{r_{+}^{2}-a^{2}}{4\pi
r_{+}(a^{2}+r_{+}^{2})}-\frac{K(d-1)}{4\pi r_{+}(a^{2}+r_{+}^{2})r_{+}^{d-2}}$

whereas entropy is defined as follows

$S_{+}=\frac{A}{4}=\pi(r_{+}^{2}+a^{2}).$ (44)

Thus, we have derived the Komar mass and angular momentum for FKRBH spacetimes
and calculated the other thermodynamical quantities. We can regain the
exceptional cases of the Kerr and Kerr-Newman black holes in the proper
limits.

|
---|---
Figure 9: Left: Plot showing $\Delta(r)$ vs $r$ such that $\Delta(r)=0$ admits two positive root $r_{2}$ (event horizon) and $r_{2}$ (Cauchy horizon) at the values of $\omega$ and $\gamma$ for which $d=0,1,2$. Right: Penrose diagrams of the corresponding spacetime in the parameter space $(M,a,\omega,\gamma)$.  |
---|---
Figure 10: Left: Plot showing $\Delta(r)$ vs $r$ such that $\Delta(r)=0$ admits two positive equal roots $r_{1}=r_{2}$, corresponding to degenerate horizons at the values of $\omega$ and $\gamma$ for which $d=2$. Right: Penrose diagrams of the corresponding spacetime in the parameter space $(M,a,\omega,\gamma)$. |
---|---
Figure 11: Left: Plot showing $\Delta(r)$ vs $r$ such that $\Delta(r)=0$
admits two positive roots $r_{1}$ (event horizon) and $r_{2}$ (Cauchy horizon)
at the values of $\omega$ and $\gamma$ for for which $0<d<2$ and $d\neq 1$.
Right: Penrose diagrams of the corresponding spacetime in the parameter space
$(M,a,\omega,\gamma)$.

##  V Structure of rotating spacetime: Penrose diagrams

The Penrose diagrams are designed to show the entire causal structure of any
given geometry on a 2-D finite sheet. They provide a crucial road map for
moving around a black hole. Time is vertical, space is horizontal, and null
rays are angled at 45$\degree$ to the axes, much like in the Minkowski
diagram. A Penrose diagram, in contrast to a Minkowski diagram, has ”points at
infinity” added by compactification, which helps to visualize the intricate
structure of infinity that results from the union of space and time. Along the
symmetric $z$-axis ($\theta=0\degree$), we have generated diagrams for the
FKRBH spacetime that correspond to each of the many points in the parameter
spaces in Figs. 1-4.

For the case when the value of $\omega$ and $\gamma$ are such that $d=0,1$ or
$2$, the Penrose diagrams, as well as the horizons, are precisely like that of
Kerr black hole as given in Fig. 9. The surface $r=0$ is regular and time-like
that exists well beyond inner horizon $r_{1}$. The Universe is represented by
area I between $r=\infty$ and $r_{1}$, and the black hole is symbolized by
region II between $r_{1}$ and $r_{2}$. Beyond the surface, $r=0$ to
$r=-\infty$ lies the antiverse. The region I’ is a mathematical extension of a
second copy of the region I geometry glued along the anti-horizon in the
opposite direction of time. In addition to a universe, antiverse, or black
hole, the full analytic extension of the spacetime geometry also includes a
parallel universe, a white hole, and a parallel antiverse as given by I’, II’,
and part of region III’ between $r=0$ and $r_{\infty}$, for the extremal black
hole where the inner and outer horizon merge ($r_{1}=r_{2}$), the Penrose
diagram is given in Fig. 10 with $d=0,1$ or $2$. Its Penrose diagram
illustrates that an extremal rotating Kiselev black hole has a different
structure than a typical FKRBH. The spacetime, in this case, has only two
regions, the region I between $r=\infty$ and $r_{1}$ is the universe and the
region II between $r_{1}$ and $r_{-\infty}$ which includes the timelike
surface $r=0$. The region beyond $r=0$ is the antiverse. Finally, the Penrose
diagram and the horizon structure for the case when $0<d<2$ but $d\neq 1$ is
shown in Fig. 11. In this case, the spacetime has a singularity at $r=0$
surface that lies beyond the inner horizon $r_{1}$.

##  VI Conclusion

Recently, Santos et al. [28] made a significant discovery in $f(R,T)$ gravity.
They investigated the parameter of the equations of state, denoted as $w$, and
identified the first spherical black holes within this modified theory. This
modification of the Kiselev black hole in the $f(R,T)$ gravity can reproduce
well known solutions of the Einstein field equation as exceptional cases. They
examined specific values of $w$ associated with black holes surrounded by
various fields, such as dust ($w=0$), radiation ($w=1/3$), quintessence
($w=-2/3$), cosmological constant ($w=-1$), and phantom ($w=-4/3$). In their
study, the authors considered the model $f(T)=\varkappa T^{n}$ and explored
the conditions necessary to generate Kiselev black holes in the context of
$f(R,T)$ gravity. It was found that when $n$ takes on the accepted value of
$1$, several particular values of the parameter $w$ in $f(R,T)$ gravity yield
solutions that deviate from the Kiselev black hole in the general relativity.
However, it is challenging to test spherical black hole models through
astrophysical observations alone, as black hole spin, precisely rotating black
holes commonly observed in nature, play a crucial role in astrophysical
processes. Testing modified gravity theories like $f(R,T)$ without
incorporating rotating black hole models typically hinders observational
verification. The Kerr metric describes astrophysical black holes and remains
the only stationary, vacuum axisymmetric metric that satisfies the Einstein
field equations while avoiding pathologies beyond the event horizon. The
groundbreaking observations made by the Event Horizon Telescope (EHT) provided
images of supermassive black holes, namely Sgr A* and M87*. These
observations, documented in studies such as [71, 72, 73, 74], revealed that
the size of the black hole shadow agrees with predictions based on the Kerr
metric, with an accuracy of around 10%. This discovery offers an additional
tool to investigate the nature of strong-field gravity and potentially
constrain deviations from the Kerr metric, such as Kerr-like black holes that
arise in theories like $f(R,T)$ gravity or other alternative gravity theories.
Thus, the observations from the EHT can test fundamental theories of gravity,
including $f(R,T)$ in the strong field regime, where a rotating black hole
plays a vital role. However, the scarcity of rotating black hole models in
modified gravity significantly impedes progress in testing such theories
through observational means. To address this limitation, we have tackled the
problem by utilizing the revised Newman-Janis algorithm, a viable method for
generating rotating solutions from a spherical Kiselev seed metric (13). The
$f(R,T)$ gravity changes the structure of the Kerr black hole, resulting in
the presence of an additional hair term in the metric (15). This metric
describes FKRBH, which is asymptotically flat and encompasses various well-
known black hole solutions, including Kerr ($K=0$), Kerr-Newman
($K=Q^{2},\;d=2$), Reissner-Nordstrom ($K=Q^{2},\;d=2,\;a=0$), and
Schwarzschild ($K=0,\;a=0$) black holes. Like the Kerr black hole, the FKRBH
retains the Cauchy and event horizons and the stationary limit surface (SLS).
However, the radii of these horizons and the SLS are affected by the parameter
$\gamma$, resulting in their decrease. This has exciting implications for the
ergosphere and can lead to significant consequences in the astrophysical
Penrose process. From the parameter space defined by $a$ and $\gamma$ for
FKRBH, we can observe that when $a_{1}<a<a_{2}$, there is a critical value
$\gamma=\gamma_{E}$ which corresponds to extreme value black holes
characterized by degenerate horizons. When $a<a_{1}$ ($a>a_{2}$), we encounter
two distinct critical values $\gamma=\gamma_{E1},\;\gamma_{E2}$ with
$\gamma_{E1}>\gamma_{E2}$ (or $\gamma=\gamma_{E3},\;\gamma_{E4}$ with
$\gamma_{E3}>\gamma_{E4}$ (cf. Fig 1-4).

Despite the complexity of the FKRBH metric (15), we have analytically derived
exact expressions for the conserved mass $M_{\text{eff}}$ and angular momentum
$J_{\text{eff}}$ using the Komar prescription. These expressions hold at any
radial distance. Notably, the $f(R,T)$ gravity introduces notable
modifications to these conserved quantities compared to those of the Kerr
black hole. However, in the limit $K=0$, the conserved mass $M_{\text{eff}}$
and angular momentum $J_{\text{eff}}$ converge to their values for the Kerr
black hole. It is worth mentioning that the influence of the $f(R,T)$ gravity
diminishes at large radial distances from the horizon, as $M_{\text{eff}}$ and
$J_{\text{eff}}$ approach the values of the Kerr black hole in the asymptotic
region ($r\to\infty$).

In conclusion, the resulting FKRBH spacetime (15) possesses mathematical
properties reminiscent of the Kerr metric and several other intriguing
features, demonstrating a rich spacetime structure.

Numerous avenues exist for future research, particularly in analyzing the
accretion process onto black holes. Notably, the structure of Kerr black holes
is significantly influenced by $f(R,T)$ gravity, which can have various
astrophysical implications, such as the impact on wormholes, gravitational
lensing properties, and black holes in AdS/CFT. Exploring the possibility of
extending these findings to the other forms of $f(R,T)$ gravity presents an
exciting avenue for future investigations. In line with the no-hair
conjectures, the role of FKRBH black hole deserves consideration and can
potentially provide valuable insights. Moreover, studying the shadow of FKRBHs
in light of results obtained from the observations made by the Event Horizon
Telescope collaboration is essential.

##  VII Acknowledgments

S.U.I and S.G.G. would like to thank SERB-DST for project No. CRG/2021/005771.
S.D.M acknowledges that this work is based upon research supported by the
South African Research Chair Initiative of the Department of Science and
Technology and the National Research Foundation.

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