# Tunneling and entropy analysis of parameterized black hole with rotating
case

Muhammad Asgher<EMAIL_ADDRESS>Department of Mathematics, The Islamia
University of Bahawalpur, Bahawalpur-63100, Pakistan  Anosha Karamat
<EMAIL_ADDRESS>Department of Mathematics, Minhaj University,
Lahore-54000, Pakistan  Rimsha Babar<EMAIL_ADDRESS>Division of
Science and Technology, University of Education, Township, Lahore-54590,
Pakistan  Riasat Ali∗<EMAIL_ADDRESS>Department of Mathematics, GC,
University Faisalabad Layyah Campus, Layyah-31200, Pakistan

###### Abstract

In this work, we study the parameterized black hole solution by applying the
Newman-Janis approach and also examine the Hawking temperature. We consider a
Lagrangian field equation associated with the generalized uncertainty
principle to study the motion of boson particles. By using semi-classical
phenomenon, we analyze the modified Hawking temperature and graphically check
the effects of deformation, rotation and correction parameter on black hole
geometry. Furthermore, we investigate the logarithmic corrected entropy and
also analyze the graphical behavior of deformation and quantum gravity
parameter on the logarithmic corrected entropy of black hole.

Black Holes; Newman-Janis algorithmic rule; Lagrangian field equation; Hawking
Temperature; Logarithmic corrected entropy.

##  I Introduction

Black Holes (BHs) are a very dominating objects in the field of quantum
gravity (QG). Many types of BH solutions in different theories have been
investigated. Alternately, to find the specific BHs solutions as
representation in gravitational theory, it is interesting to use a different
approach with parametric solutions 1 . Many parameterized BH solutions were
presented and examined in the recent research with various inspirations 2 -5 .
Chrusciel and his colleagues 6  have investigated the Konoplya–Zhidenko
parameterization with a static deformation that allows the resulting metric to
violates the no-hair theorem. Cardoso et. al., 4  have studied the
parameterized BH in quasi-normal modes and analyzed decoupled equations for
non-rotating BHs. Magalhães and his fellows 8  have studied the parameterized
BH with a single deformation parameter and investigated the impacts of
deformation parameter on null geodesics. Konoplya and Zhidenko 9  have
computed the general parameterization of a BH metric in various gravity
theories with many parameters i.e., parameters of shadow, electromagnetic
radiation, quasi-normal modes and accretion matter.

The interesting class of charged rotating BHs are depicted by Kerr-Newman
metrics, that can be created by a procedure known as the Newman-Janis strategy
7 . Whereas this strategy is especially effective for inferring the Kerr
metric and its electromagnetic speculation, the Kerr-Newman spacetime, it has
frequently been criticized 10  because the system is definitely not an overall
strategy for creating vacuum from vacuum metrics as well as there is a sure
discretion in the decision of complexification of terms in the unique basic
metrics (Schwarzschild or Reissner Nordstr$\ddot{o}$m). The Kerr space-time
portrays the metric of a rotating BH, it is actually of interest whether it
depicts the outside of a modified axisymmetric rotating gigantic object.
Whereas Birkhoff’s theorem 11  states that the metric outside to a spherically
symmetric gigantic object is uniquely expressed by the Schwarzschild metric
there’s shockingly no reason to expect that the metric outside to an arbitrary
stationary axisymmetric perfect fluid ought to be the Kerr metric. Although,
stays an open problem whether the Kerr arrangement can performs the outside of
any perfect fluid by any means. The metrics for different types of BHs and
black rings for rotating and non-rotating cases have been widely discussed in
literature 13 ; 13a ; 15a .

Moreover, to discuss the thermodynamics of BHs is always a very interesting
feature for researchers. In this regard, many approaches has been used and the
quantum tunneling method is also a very popular technique to analyze the
thermodynamics like Hawking temperature of BHs. It is also a well known fact
that quantum gravity influences the physical properties of gigantic objects.
In order to reduce the gravitational effects, researchers take help of
generalized uncertainty principle (GUP). The GUP is a sufficient tool for
depicting BHs and their thermodynamic properties. The researchers calculate
the corrected temperature and entropy with the help of GUP. The GUP is
associated to the Heisenberg principle 17 ; 18 . Ali 19  has discussed the
formation of mini BHs via GUP. Faizal and Khalil 20  have introduced the
universality law of entropy related to its area and discussed the quantum
corrections of thermodynamics of black objects. Gangopadhyay and Dutta 21
have investigated the corrections to the thermodynamics of a Schwarzschild BH
for higher dimensions incorporating GUP. Gecim, and Sucu 22  have studied the
GUP effects for fermions and scalar particles by utilizing the Hamilton–Jacobi
technique for Martinez–Zanelli BH and calculated the tunneling probability and
temperature for the corresponding BH. They have also investigated the modified
temperature of BTZ BH by using the quantum tunneling approach for scalar
particles 23 . The quantum corrected temperature for different kind of BHs
through tunneling method based on GUP has been discussed in detail 28 .
Moreover, by using the quantum tunneling phenomenon for vector particles, the
Hawking temperature for BHs have been anaylzed 29 ; 30 ; 31 ; 32 ; 33 ; c4 .
Hassanabadi and his colleagues 34  have analyzed a detailed thermodynamics
analysis for a charged BH via GUP.

The important motivation of this work is to introduce the rotating case of
parameterized BH through Newman-Jannis approach as well as to derive the
thermodynamics of parameterized BH that add the spin parameter in the metric.
Moreover, to discuss the effects of rotation parameter in the presence of
quantum gravity.

We have formed our paper as follows; In Sec. II, we derive a Parameterized BH
solution in the context of Newman-Janis algorithmic rule and also investigate
the Hawking temperature for the corresponding BH. In Sec. III, we investigate
the graphical importance of temperature under the effects of deformation
parameter as well as rotation parameter. In Sec. IV, we examine the modified
tunneling and modified temperature. Sec V, gives the analysis of Hawking
temperature with respect to horizon and check the stabile state of
parameterized BH with rotation parameter under quantum gravity effects. Sec
VI, studies the logarithmic entropy corrections for parameterized BH in the
rotating case. In Sec. VII, we analyze the physical significance of corrected
entropy plots under the effects of deformation and correction parameter for
rotating parameterized BH. Sec VIII, we have analyzed our results.

##  II Parameterized Black Hole With Rotation Parameter

We consider the parametrization studied in Ref. P1 , which the BH solution
modifies into extra parameters incorporating in the mass like term known as
deformation parameters. The non-spinning form’s parametrized like BH solution
is provided P2  by

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

with $A_{KZ}(r)=1-\frac{2M(r)}{r}$. In order to include more parameters, the
mass-term $M(r)$ is used as

$2M(r)=\sum_{i=0}^{\infty}\frac{\eta_{i}}{r_{i}}+2M,$ (2)

with $\eta_{i}$ represent the deformation parameters. When $\eta_{i}=0$, we
get Schwarzschild BH solution. In the BH family of parametrized solutions, a
group of solutions with the same Schwarzschild post-Newtonian asymptotics can
be chosen. In this section, we analyze the parameterized BH solution in the
frame of Newman-Janis approach. Additionally, we calculate the Hawking
temperature for the parameterized BH metric. To do so, we choose a static
spherically symmetric BH metric 8

$ds^{2}=-A(r)dt^{2}+\frac{1}{A(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (3)

where

$A(r)=1-\frac{2M}{r}-\frac{\eta}{r^{3}},$

here, $M$ and $\eta$ represents the mass and deformation parameters of BH,
respectively.

To study the solution of parameterized BH in the frame of Newman-Janis
approach. Firstly, we consider a transformation in the background of
parameterized BH in Eq. (3) from the coordinates $(t,r,\theta,\phi)$ to new
coordinates $(w,r,\theta,\phi)$ as

$\displaystyle dw=dt-\frac{dr}{A(r)},$ (4)

the metric Eq. (3) can be rewritten in the given form under the defined
transformation

$ds^{2}=-A(r)dw^{2}-2dwdr+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}.$ (5)

The inverse metric elements are defined in the form

$g^{01}=g^{10}=-1,~{}~{}g^{11}=A,~{}~{}g^{22}=\frac{1}{r^{2}},~{}~{}g^{33}=\frac{1}{r^{2}\sin^{2}\theta}.$
(6)

The inverse components of the metric in the form of null tetrad are
represented as

$\displaystyle
g^{ij}=-l^{j}n^{i}-l^{i}n^{j}+m^{i}\bar{m}^{j}+m^{j}\bar{m}^{i}.$ (7)

The elements related to null tetrad can be given as

$\displaystyle l^{i}$ $\displaystyle=$
$\displaystyle\delta_{r}^{i},~{}~{}~{}n^{i}=\delta_{w}^{i}-\frac{1}{2}A\delta_{r}^{i},$
$\displaystyle m^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}r}\delta_{\theta}^{i}+\frac{\iota}{\sqrt{2}r\sin\theta}\delta_{\phi}^{i},$
$\displaystyle\bar{m}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}r}\delta_{\theta}^{i}-\frac{\iota}{\sqrt{2}r\sin\theta}\delta_{\phi}^{i}.$

Although, at the some points in the parameterized BH metric, the relationship
of null tetrad with vectors is defined as

$l_{i}l^{i}=n_{i}n^{i}=m_{i}m^{i}=l_{i}m^{i}=m_{i}m^{i}=0,$

and

$l_{i}n^{i}=-m_{i}\bar{m}^{i}=1.$

The transformation of coordinates in the $(w,r)$ plane can be expressed as

$\displaystyle w$ $\displaystyle\rightarrow$ $\displaystyle w-\iota
a\cos\theta,$ $\displaystyle r$ $\displaystyle\rightarrow$ $\displaystyle
r+\iota a\cos\theta.$

Furthermore, we compute the transformations in the following form

$A(r)\rightarrow\tilde{A}(r,a,\theta),$ (8)

and

$r^{2}+a^{2}\cos^{2}\theta=\sigma^{2}.$ (9)

The null vectors in the $(w,r)$ plane obtain the following form

$\displaystyle l^{i}$ $\displaystyle=$
$\displaystyle\delta_{r}^{i},~{}~{}~{}n^{i}=\delta_{u}^{i}-\frac{1}{2}\tilde{A}\delta_{r}^{i},$
$\displaystyle m^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}r}\left(\delta_{\theta}^{i}+\iota
a\sin\theta(\delta_{u}^{i}-\delta_{r}^{i})+\frac{\iota}{\sin\theta}\delta_{\phi}^{\iota}\right),$
$\displaystyle\bar{m}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}r}\left(\delta_{\theta}^{i}-\iota
a\sin\theta(\delta_{u}^{i}-\delta_{r}^{i})-\frac{\iota}{\sin\theta}\delta_{\phi}^{i}\right).$
(10)

By the definition of null tetrad, the significant inverse elements of the
metric $g^{ij}$ in the coordinates $(w,r,\theta,\phi)$ can be computed in the
form

$\displaystyle g^{00}$ $\displaystyle=$
$\displaystyle\frac{a^{2}\sin^{2}\theta}{\sigma^{2}},~{}~{}~{}g^{01}=g^{10}=-1-\frac{a^{2}\sin^{2}\theta}{\sigma^{2}},~{}~{}~{}g^{11}=\tilde{A}(r,\theta)+\frac{a^{2}\sin^{2}\theta}{\sigma^{2}},~{}~{}~{}$
$\displaystyle g^{33}$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma^{2}\sin^{2}\theta},~{}~{}~{}g^{03}=g^{30}=\frac{a}{\sigma^{2}},~{}~{}~{}g^{13}=g^{31}=-\frac{a}{\sigma^{2}},~{}~{}~{}g^{22}=\frac{1}{\sigma^{2}},$

here

$\tilde{A}(r,\theta)=\frac{r^{2}A+a^{2}\cos^{2}\theta}{\sigma^{2}}.$ (11)

Moreover, we derive a transformation of coordinates from $(w,r,\theta,\phi)$
to the coordinates $(t,r,\theta,\phi)$ in the following state

$dw=dt+\zeta(r)dr,~{}~{}~{}d\phi=d\phi+\xi(r)dr,$ (12)

here

$\displaystyle\zeta(r)$ $\displaystyle=-$
$\displaystyle\frac{r^{2}+a^{2}}{r^{2}A+a^{2}},$ $\displaystyle\xi(r)$
$\displaystyle=$ $\displaystyle-\frac{a}{r^{2}A+a^{2}}.$

At last, we obtain the BH solution in the given condition

$ds^{2}=-\tilde{A}(r)dt^{2}+\sigma^{2}d\theta^{2}+\frac{\sigma^{2}}{\tilde{A}(r)}dr^{2}+2a\sin^{2}\theta\Big{[}\tilde{A}(r)-1\Big{]}dtd\phi+\sin^{2}\theta\Big{[}a^{2}\sin^{2}\theta\Big{(}2-\tilde{A}(r)\Big{)}+\sigma^{2}\Big{]}d\phi^{2}.$
(13)

The above equation can be derived in the given form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\left(1-\frac{2Mr+\eta
r^{-1}}{\sigma^{2}}\right)dt^{2}+\frac{\sigma^{2}}{\Delta_{r}}dr^{2}-2a\sin^{2}\theta\left(\frac{2Mr+\eta
r^{-1}}{\sigma^{2}}\right)dtd\phi+\sigma^{2}d\theta^{2}$ (14) $\displaystyle+$
$\displaystyle\sin^{2}\theta\Big{[}a^{2}\sin^{2}\theta\Big{(}\frac{2Mr+\eta
r^{-1}}{\sigma^{2}}\Big{)}+\sigma^{2}\Big{]}d\phi^{2}.$

here

$\Delta_{r}=r^{2}-2Mr+a^{2}-\eta r^{-1}.$ (15)

In order to study the thermodynamics of the corresponding BH, we derive the
temperature in the presence of rotation parameter with the help of surface
gravity $\kappa$ by using the following formula

$T_{H}=\frac{\kappa}{2\pi},$ (16)

where

$\kappa=\frac{\Delta^{\prime}_{r}}{2(r^{2}_{+}+a^{2})},~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}\Delta^{\prime}_{r}=\frac{d}{dr}(\Delta_{r}).$
(17)

The Hawking temperature for parameterized BH with the rotation parameter can
be derived as

$T_{H}=\frac{2Mr_{+}^{4}+3\eta r_{+}^{2}+a^{2}(\eta-2Mr_{+}^{2})}{4\pi
r_{+}^{2}\Big{(}r_{+}^{2}+a^{2}\Big{)}^{2}}.$ (18)

The temperature $T_{H}$ depends on spin parameter $a$, BH mass $M$ and
deformation parameter $\eta$. It is very important to express here that in the
absence of spin parameter i.e., $a=0$, we get the Hawking temperature which is
independent of Newman-Janis algorithmic rule.

The temperature in its original form can be obtained in the following form

$T_{H}=\frac{2Mr_{+}^{4}+3\eta r_{+}^{2}}{4\pi r_{+}^{6}}.$ (19)

The above expression indicates the temperature of parameterized BH in the
absence of rotation parameter.

##  III Graphical interpretation of Temperature versus horizon

This section studies the graphical conduct of $T_{H}$ via horizon. We analyze
the physical understanding of the plots with the impacts of deformation and
rotation parameters. We discuss the stable conditions of parameterized BH in
the rotating case.

Fig. 1: Temperature versus horizon for $M=1$. Left $a=0.5$, $\eta=0.05$
(black), $\eta=0.06$ (blue), $\eta=0.07$ (red), and $\eta=0.08$ (purple).
Right $\eta=0.06$, $a=0.1$ (black), $a=0.2$ (blue), $a=0.3$ (red), and $a=0.4$
(purple).

The left plot in Fig. 1 shows that the $T_{H}$ constantly decreases and after
a minima it rises to reach at a maxima and then again gradually decrease to
shows an asymptotically flat description until $r\rightarrow\infty$. There can
be seen two critical values $r_{+}^{min}$ and $r_{+}^{max}$ in the plot where
the BH is stable for $r_{+}^{min}\leq r_{+}\leq r_{+}^{max}$. The $T_{H}$
rises with the rising values of deformation parameter $\eta$.

The right plot gives that the $T_{H}$ constantly decreases and then approaches
to an asymptotically flat behavior upto $r_{+}\rightarrow\infty$ in the range
$0\leq r_{+}\leq 8$. Moreover, for the rising values of $a$, the $T_{H}$ shows
a decreasing conduct.

##  IV Modified Tunneling of Parameterized Black Hole with Rotation Parameter

In this chapter, we examine the quantum mechanical process of Hawking
temperature as a tunneling process for the parameterized BH with rotation
parameter. We compute the tunneling rate of the charged boson and compute the
corrected temperature for parameterized BH. From the Eq. (14), we get

$ds^{2}=-\Delta
dt^{2}+\frac{1}{\Delta}dr^{2}+\sigma^{2}d\theta^{2}+Ud\phi^{2}+2Vdtd\phi,$
(20)

with $\Delta=\frac{\sigma^{2}r}{r^{2}-2Mr+a^{2}-\eta
r^{-1}},~{}~{}U=\sin^{2}\theta\Big{[}\sigma^{2}+a^{2}\sin^{2}\theta\Big{(}\frac{2Mr+\eta
r^{-1}}{\sigma^{2}}\Big{)}\Big{]}$ and
$V=-a\sin^{2}\theta\left(\frac{2Mr^{2}+\eta}{\sigma^{2}r}\right).$

Now, we choose the emission of boson particles from parameterized BH with
rotation parameter in the background of tunneling process around the horizon.
This is done by solving charged Lagrangian equation with GUP parameter for
vector field $\psi$, which is given by 15a

$\partial_{i}(\sqrt{-g}\psi^{ji})+\sqrt{-g}\frac{m^{2}}{\hbar^{2}}\psi^{j}+\sqrt{-g}\frac{\iota}{\hbar}A_{i}\psi^{ji}+\sqrt{-g}\frac{\iota}{\hbar}eF^{ji}\psi_{i}+\hbar^{2}\beta\partial_{0}\partial_{0}\partial_{0}(\sqrt{-g}g^{00}\psi^{0j})-\hbar^{2}\beta\partial_{i}\partial_{i}\partial_{i}(\sqrt{-g}g^{ii}\psi^{ij})=0,$
(21)

here $\beta$ is the GUP or quantum gravity parameter, $g$ denotes the
determinant of the metric, $\psi^{ji}$ denotes the anti-symmetric tensor as
well as $m$ stands for mass of particle, since

$\psi_{ji}=(1-\beta{\hbar^{2}\partial_{j}^{2}})\partial_{j}\psi_{i}-(1-\beta{\hbar^{2}\partial_{i}^{2}})\partial_{i}\psi_{j}+(1-\beta{\hbar^{2}\partial_{j}^{2}})\frac{\iota}{\hbar}eA_{j}\psi_{i}-(1-\beta{\hbar^{2}}\partial_{\nu}^{2})\frac{\iota}{\hbar}eA_{i}\psi_{j},~{}~{}~{}F_{ji}=\nabla_{j}A_{i}-\nabla_{i}A_{j},$

here $\beta,~{}A_{i},~{}e~{}$ and $\nabla_{i}$ shows the dimensionless
positive parameter, vector potential of parameterized BH, the particle’s
charge and covariant derivatives, respectively. The non zero values of
$\psi^{i}$ and $\psi^{ji}$ are computed as

$\displaystyle\psi^{0}=\frac{-U\psi_{0}+V\psi_{3}}{U\Delta+V^{2}},~{}~{}~{}\psi^{1}=\Delta\psi_{1},~{}~{}~{}\psi^{2}=\frac{\psi_{2}}{\sigma^{2}},~{}~{}~{}\psi^{3}=\frac{V\psi_{0}+\Delta\psi_{3}}{U\Delta+V^{2}},~{}~{}\psi^{12}=\frac{\Delta\psi_{12}}{\sigma^{2}},~{}~{}\psi^{13}=\frac{\Delta\psi_{13}}{U\Delta+V^{2}},$
$\displaystyle\psi^{01}=\frac{-U\psi_{01}+V\psi_{13}}{(U\Delta+V^{2})},~{}~{}~{}\psi^{02}=\frac{-U\psi_{02}}{\sigma^{2}(U\Delta+V^{2})},~{}~{}~{}\psi^{03}=\frac{(U\Delta+\Delta^{2})\psi_{03}}{(U\Delta+V^{2})^{2}},~{}\psi^{23}=\frac{V\psi_{02}+\Delta\psi_{23}}{\sigma^{2}(U\Delta+V^{2})}.$
(22)

The WKB method 13  for vector particles can be defined in the way

$\psi_{j}=c_{j}\exp\Big{[}\frac{\iota}{\hbar}I_{0}(t,r,\theta,\phi,)+\Sigma\hbar^{n}I_{n}(t,r,\theta,\phi,)\Big{]}.$
(23)

By putting the Eqs. (20) and (23) into Eq. (21), we collect a set of field
equations in APPENDIX A and after applying the technique of separation
variables 15a , we can get

$I_{0}=-(E-j\omega)t+W(r)+J\phi+\nu(\theta),$ (24)

where $E$ denotes the particle’s energy, $J$ represents the particle’s angular
momentum associated with angle $\phi$. After substituting the separation of
variables into set of field equations, we obtain

$Z(c_{0},c_{1},c_{2},c_{3})^{T}=0,$

which implies a $4\times 4$ matrix labeled as ”$Z$” in APPENDIX B, whose
components are given as follows

$\displaystyle Z_{00}$ $\displaystyle=$
$\displaystyle\frac{{-U\Delta}}{(U\Delta+V^{2})}[W_{1}^{2}+\beta
W_{1}^{4}]-\frac{U}{\sigma^{2}(U\Delta+V^{2})}[J^{2}+\beta
J^{4}]-\frac{U\Delta}{(U\Delta+V^{2})^{2}}[\nu_{1}^{2}+\beta\nu_{1}^{4}]-\frac{m^{2}U}{(U\Delta+V^{2})},$
$\displaystyle Z_{01}$ $\displaystyle=$
$\displaystyle\frac{-D\Delta}{(U\Delta+V^{2})}[(E-j\omega)+\beta(E-j\omega)^{3}+eA_{0}+\beta
eA_{0}(E-j\omega)^{2}]W_{1}+\frac{V\Delta}{(U\Delta+V^{2})}+[\nu_{1}+\beta\nu_{1}^{3}],$
$\displaystyle Z_{02}$ $\displaystyle=$
$\displaystyle\frac{-U\Delta}{\sigma^{2}(U\Delta+V^{2})}[(E-j\omega)+\beta(E-j\omega)^{3}-eA_{0}-\beta
eA_{0}(E-j\omega)^{2}]J,$ $\displaystyle Z_{03}$ $\displaystyle=$
$\displaystyle\frac{-V\Delta}{(U\Delta+V^{2})}[W_{1}^{2}+\beta
W_{1}^{4}]-\frac{U\Delta}{\sigma^{2}(U\Delta+V^{2})^{2}}[(E-j\omega)+\beta(E-j\omega)^{3}-eA_{0}-\beta
eA_{0}(E-j\omega)^{2}]\nu_{1}+\frac{m^{2}V}{(U\Delta+V^{2})^{2}},$
$\displaystyle Z_{10}$ $\displaystyle=$
$\displaystyle\frac{-U\Delta}{(U\Delta+V^{2})}[(E-j\omega)W_{1}+\beta(E-j\omega)W_{1}^{3}]-m^{2}\Delta-\frac{eA_{0}U\Delta}{(U\Delta+V^{2})}[W_{1}+\beta
W_{1}^{3}],$ $\displaystyle Z_{11}$ $\displaystyle=$
$\displaystyle\frac{-U\Delta}{(U\Delta+V^{2})}[(E-j\omega)^{2}+\beta(E-j\omega)^{4}-eA_{0}(E-j\omega)-\beta
eA_{0}(E-j\omega)W_{1}^{2}]+\frac{V\Delta}{(U\Delta+V^{2})}+[\nu_{1}+\beta\nu_{1}^{3}](E-j\omega)$
$\displaystyle-$ $\displaystyle\frac{\Delta}{\sigma^{2}}[J^{2}+\beta
J^{4}]-\frac{\Delta}{(U\Delta+V^{2})}[\nu_{1}+\beta\nu_{1}^{3}]-m^{2}\Delta-\frac{eA_{0}U\Delta}{(U\Delta+V^{2})}[(E-j\omega)+\beta(E-j\omega)^{3}-eA_{0}-\beta
eA_{0}(E-j\omega)^{2}]$ $\displaystyle+$
$\displaystyle\frac{eA_{0}V\Delta}{(U\Delta+V^{2})}[\nu_{1}+\beta\nu_{1}^{3}],$
$\displaystyle Z_{12}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{\sigma^{2}}[W_{1}+\beta W_{1}^{3}]J,$
$\displaystyle Z_{13}$ $\displaystyle=$
$\displaystyle\frac{-V\Delta}{(U\Delta+V^{2})}[W_{1}+\beta
W_{1}^{3}](E-j\omega)+\frac{\Delta}{(U\Delta+V^{2})^{2}}[W_{1}+\beta
W_{1}^{3}]\nu_{1}+\frac{V\Delta eA_{0}}{(U\Delta+V^{2})}[W_{1}+\beta
W_{1}^{3}],$ $\displaystyle Z_{20}$ $\displaystyle=$
$\displaystyle\frac{U}{\sigma^{2}(U\Delta+V^{2})}[(E-j\omega)J+\beta(E-j\omega)J^{3}]+\frac{V}{\sigma^{2}(U\Delta+V^{2})}[(E-j\omega)+\beta(E-j\omega)^{3}\nu_{1}^{2}]-\frac{UeA_{0}}{\sigma^{2}(U\Delta+V^{2})}[J+\beta
J^{3}],$ $\displaystyle Z_{21}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{\sigma^{2}}[J+\beta J^{3}]W_{1},$ $\displaystyle
Z_{22}$ $\displaystyle=$
$\displaystyle\frac{U}{\sigma^{2}(U\Delta+V^{2})}[(E-j\omega)^{2}+\beta(E-j\omega)^{4}-eA_{0}(E-j\omega)-\beta
eA_{0}(E-j\omega)]-\frac{\Delta}{\sigma^{2}}+\frac{V}{\sigma^{2}(U\Delta+V^{2})}[(E-j\omega)$
$\displaystyle+$ $\displaystyle\beta(E-j\omega)^{3}-eA_{0}-\beta
eA_{0}(E-j\omega)^{2}]\nu_{1}-\frac{\Delta}{\sigma^{2}(U\Delta+V^{2})}[\nu_{1}^{2}+\beta\nu_{1}^{4}]-\frac{m^{2}}{\sigma^{2}}-\frac{eA_{0}U}{\sigma^{2}(U\Delta+V^{2})}[(E-j\omega)$
$\displaystyle+$ $\displaystyle\beta(E-j\omega)^{3}-eA_{0}-\beta
eA_{0}(E-j\omega)^{2}],$ $\displaystyle Z_{23}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{\sigma^{2}(U\Delta+V^{2})}[J+\beta J^{3}]\nu_{1},$
$\displaystyle Z_{30}$ $\displaystyle=$
$\displaystyle\frac{(U\Delta-\Delta^{2})}{(U\Delta+V^{2})^{2}}[\nu_{1}+\beta\nu_{1}^{3}](E-j\omega)+\frac{V}{\sigma^{2}(U\Delta+V^{2})}[J^{2}+\beta
J^{4}]-\frac{m^{2}V}{(U\Delta+V^{2})}-\frac{eA_{0}(U\Delta-\Delta^{2})}{(U\Delta+V^{2})^{2}}[\nu_{1}+\beta\nu_{1}^{3}],$
$\displaystyle Z_{31}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{(U\Delta+V^{2})}[\nu_{1}+\beta\nu_{1}^{3}]W_{1},$
$\displaystyle Z_{32}$ $\displaystyle=$
$\displaystyle\frac{V}{\sigma^{2}(U\Delta+V^{2})}[J+\beta
J^{3}](E-j\omega)+\frac{\Delta}{\sigma^{2}(U\Delta+V^{2})}[\nu_{1}+\beta\nu_{1}^{3}]J,$
$\displaystyle Z_{33}$ $\displaystyle=$
$\displaystyle\frac{(U\Delta-\Delta^{2})}{(U\Delta+V^{2})}[(E-j\omega)^{2}+\beta(E-j\omega)^{4}-eA_{0}(E-j\omega)-\beta
eA_{0}(E-j\omega)^{3}]-\frac{\Delta}{(U\Delta+V^{2})}[W_{1}^{2}+\beta
W_{1}^{4}]$ $\displaystyle-$
$\displaystyle\frac{\Delta}{\sigma^{2}(U\Delta+V^{2})}[J^{2}+\beta
J^{4}]-\frac{m^{2}\Delta}{(U\Delta+V^{2})}-\frac{eA_{0}(U\Delta-\Delta^{2})}{(U\Delta+V^{2})}[(E-j\omega)+\beta(E-j\omega)^{3}-eA_{0}(E-j\omega)^{2}],$

whereas $J=\partial_{\phi}S_{0}$, $W_{1}=\partial_{r}{S_{0}}$,
$\nu_{1}=\partial_{\theta}{S_{0}}$. For the purpose of non-trivial solution,
we set the determinant of Z equals to the and attain the solution in the form

$\displaystyle ImW^{\pm}$ $\displaystyle=$
$\displaystyle\pm\int\sqrt{\frac{(E-j\omega-
eA_{0})^{2}+X_{1}\Big{[}1+\beta\frac{X_{2}}{X_{1}}\Big{]}}{\frac{(U\Delta+V^{2})\Delta}{U}}}dr,$
(25) $\displaystyle=$ $\displaystyle\pm i\pi\frac{(E-j\omega-
eA_{0})+[1+\beta\Xi]}{2\kappa(r_{+})},$

where, $\Xi$ is a arbitrary parameter and $X_{1}$, $X_{2}$ are defined as

$\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle\frac{V}{(U\Delta+V^{2})\Delta}[(E-j\omega)-eA_{0}]\nu_{1}+\frac{\nu_{1}^{2}}{(U\Delta+V^{2})}-\frac{m^{2}}{\Delta},$
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle\frac{U}{\Delta(U\Delta+V^{2})}[(E-j\omega)^{4}-2eA_{0}(E-j\omega)^{3}+(eA_{0})^{2}(E-j\omega)^{2}]-\frac{\nu_{1}^{4}}{(U\Delta+V^{2})}-W_{1}^{4}$
$\displaystyle+$
$\displaystyle\frac{V}{\Delta\sigma^{2}(U\Delta+V^{2})}[(E-j\omega)^{3}-eA_{0}(E-j\omega)^{2}]\nu_{1}.$

The tunneling probability tp  of charged boson particles is derived as

$\Gamma\simeq\exp\Big{[}{-2\pi}\frac{(E-j\omega-
eA_{0})}{\kappa(r_{+})}\Big{]}\Big{[}1+\beta\Xi\Big{]}.$ (26)

The corrected temperature of parameterized BH with rotation parameter can be
computed in the form

$T{{}^{\prime}}_{H}=\frac{2Mr_{+}^{4}+3\eta
r_{+}^{2}+a^{2}(\eta-2Mr_{+}^{2})}{4\pi
r_{+}^{2}\Big{(}r_{+}^{2}+a^{2}\Big{)}^{2}}\Big{[}1-\beta\Xi\Big{]}.$ (27)

The modified Hawking temperature $T^{\prime}_{H}$ of BH depends upon rotation
parameter $a$, deformation parameter $\eta$, mass $M$, radius horizon $r_{+}$
as well as quantum gravity parameter $\beta$. When the correction parameter
$\beta=0$, we get the original temperature of Eq. (18) as well as by putting
$\eta=0=a$, we obtain the Hawking temperature of Schwarzschild BH
$T_{Sch}=\frac{1}{8\pi M}$ at $r_{+}\thickapprox 2M$.

##  V Graphical Explanation of Corrected Temperature

This section demonstrates the graphical description of corrected temperature
$T^{\prime}_{H}$ versus $r_{+}$. We analyze the physical value of the plots
with the impacts of deformation parameter $\eta$, spin parameter $a$ and GUP
parameter $\beta$.

Fig. 2: Corrected Temperature via horizon for $M=1=\Xi$. Left
$a=0.1,~{}\eta=0.1$, $\beta=0.1$ (black), $0.2$ (blue), $0.3$ (red), and $0.4$
(purple). Right $a=0.5,~{}\eta=0$, $\beta=0.5$ (black), $0.6$ (blue), $0.7$
(red), and $0.8$ (purple).

The left plot in Fig. 2 represents that the $T^{\prime}_{H}$ exponentially
decreases for rising horizon in the region $0\leq r_{+}\leq 8$ for different
variations of $\beta$. For rising values of $\beta$ the $T^{\prime}_{H}$
decreases.

The right plot depicts that afterwards gaining a peak height the
$T^{\prime}_{H}$ suddenly falls down in a decreasing way and reaches an
asymptotically flat state until $r_{+}\rightarrow\infty$. In the absence of
deformation parameter, the $T^{\prime}_{H}$ also decreases within the
increasing variations of $\beta$ in $0\leq r_{+}\leq 10$.

As stated by Hawking’s scenario with the increasing temperature more radiation
emit and the radius of BH decreases. In Fig. 2, one can examine that the
temperature is maximum at minimum horizon. This physical strategy demonstrates
the stable state of BH.

Fig. 3: Corrected Temperature w. r. t horizon for $M=1=\Xi$. Left
$a=0.5=\beta$, $\eta=0.05$ (black), $\eta=0.06$ (blue), $\eta=0.07$ (red), and
$\eta=0.08$ (purple). Right $\beta=0.5=\eta$, $a=0.1$ (black), $a=0.2$ (blue),
$a=0.3$ (red), and $a=0.4$ (purple).

The left plot in Fig. 3 indicates, the corrected temperature slowly decreases
with distinct variations of $\eta$ in $0\leq r_{+}\leq 8$ and after a minima
it rises to reach at a maxima and then again gradually decrease to shows an
asymptotically flat description until $r\rightarrow\infty$. The plot
represents the two critical values $r^{min}_{+}$ and $r^{max}_{+}$. The BH
assures its stability for $r^{min}_{+}\leq r_{+}\leq r^{max}_{+}$ by getting
an asymptotically flat state upon $r_{+}\rightarrow\infty$. In the existence
of correction parameter with rising $\eta$, the $T^{\prime}_{H}$ also rises.

The right plot demonstrate the conduct of $T^{\prime}_{H}$ for distinct values
of $a$ in $0\leq r_{+}\leq 8$.

Moreover, it is an admitted fact in quantum gravity that the quantum
corrections are a cause of reduction in rising temperature. So, this
phenomenon can be clearly observed in plots with the presence/absence of
gravity parameter.

##  VI Entropy Corrections for Parameterized Black Hole within Spin Parameter

This section studies the Bekenstein-Hawking entropy associated with correction
factor in the theory of quantum loop expansion. The logarithmic expression
appears as an additional loop into the standard entropy of BH. The entropy of
BH has a very significant role in primordial BHs A1 . It established a least
mass of the primordial BHs so that the small BHs have sufficient opportunity
to vanish completely in the current form. We investigate the entropy
corrections for parameterized BH with rotation parameter. The entropy
corrections via null geodesic technique has been studied by Banerjee and Majhi
A2 -A4 . We investigate the entropy corrections for parameterized BH with
rotation parameter by utilizing the formula of Bekenstein-Hawking entropy for
$1^{st}$ order corrections A5 . By using the expressions of $T^{\prime}_{H}$
and standard entropy $\mathbb{S}_{T}$, we calculate the logarithmic entropy
corrections for parameterized BH with rotation parameter through the given
formula

$\mathbb{S}_{P}=\mathbb{S}_{T}-\frac{1}{2}\ln\Big{|}T_{H}^{2},\mathbb{S}_{T}\Big{|}+...~{}.$
(28)

The standard entropy for parameterized BH with rotation parameter can be
derived through the given formula

$\mathbb{S}_{T}=\frac{A_{P}}{4},$ (29)

where

$\displaystyle A_{P}$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\int_{0}^{\pi}\sqrt{g_{\theta\theta}g_{\phi\phi}}d\theta
d\phi,$ (30) $\displaystyle=$
$\displaystyle\pi\left[\frac{2(r_{+}^{2}+a^{2}\Big{)}^{2}+a^{2}\Big{(}2Mr_{+}+\eta
r^{-1}\Big{)}}{\Big{(}r_{+}^{2}+a^{2}\Big{)}}\right].$

The standard entropy for parameterized BH with rotation parameter can be
calculated in the form

$\mathbb{S}_{T}=\pi\left[\frac{2(r_{+}^{2}+a^{2}\Big{)}^{2}+a^{2}\Big{(}2Mr_{+}+\eta
r^{-1}\Big{)}}{4\Big{(}r_{+}^{2}+a^{2}\Big{)}}\right].$ (31)

After substituting the values from (31) into Eq. (28), we obtain the entropy
corrections as

$\displaystyle\mathbb{S}_{P}$ $\displaystyle=$
$\displaystyle\pi\left[\frac{2(r_{+}^{2}+a^{2}\Big{)}^{2}+a^{2}\Big{(}2Mr_{+}+\eta
r^{-1}\Big{)}}{4\Big{(}r_{+}^{2}+a^{2}\Big{)}}\right]$ (32) $\displaystyle-$
$\displaystyle\frac{1}{2}\ln\left|\frac{\left[\Big{\\{}2Mr_{+}^{4}+3\eta
r_{+}^{2}+a^{2}(\eta-2Mr_{+}^{2})\Big{\\}}\Big{\\{}1-\beta\Xi\Big{\\}}\right]^{2}\Big{[}2\Big{(}r_{+}^{2}+a^{2}\Big{)}^{2}+a^{2}\Big{(}2Mr_{+}+\eta
r^{-1}\Big{)}\Big{]}}{64\pi
r^{4}_{+}\left(r^{2}_{+}+a^{2}\right)^{5}}\right|+...,$

The Eq. (32) shows the corrected entropy for parameterized BH with rotation
parameter.

##  VII Corrected Entropy Analysis

The graphical analysis of entropy corrections $S_{P}$ versus horizon $r_{+}$
for parameterized BH with rotation parameter has been analyzed in this
section. We analyze the impacts of $\beta$ and $\eta$ on corrected entropy
$S_{P}$ via graphs.

Figure 4: Corrected Entropy $S_{P}$ via horizon $r_{+}$ for $\Xi=1$. Left
$a=0.3,\eta=0.1$, $M=10$, $\beta=0.1$ (black), $0.3$ (red), $0.6$ (blue) and
$0.8$ (purple). Right $a=0.5,\beta=0.5,M=1$, $\eta=0.01$ (black), $0.02$
(red), $0.03$ (blue) and $0.04$ (purple).

The left plot in Fig. 4 depicts the graphical conduct of $S_{P}$ for changing
values of $\beta$ in the region $0\leq r_{+}\leq 5$. The corrected entropy
slowly decreases and after showing some variations the $S_{P}$ continuously
increases for rising values of horizon. This conduct with positive entropy
shows the stable form of parameterized BH. The $S_{P}$ increases with the
rising values of correction parameter.

The right plot in Fig. 4 gives the behavior of $S_{P}$ versus $r_{+}$ for
different values of deformation parameter $\eta$ in the region $0\leq
r_{+}\leq 5$. One can see from the plot that with the rising values of
deformation parameter $\eta$ the $S_{P}$ also increases. It is also observable
that at most of the points sharp edge appears for varying values of
deformation parameter that reduces the stability of entropy.

It can be conclude from the observation of both plots, the BH entropy is more
stable for varying values of correction parameter as compared to deformation
parameter.

##  VIII Results and Discussion

In this research work, we have investigated the solution of parameterized BH
with spin parameter by using the Newman-Janis strategy. By setting the
rotation coupling $(a\rightarrow 0)$ in the Eq. (14), we obtained the metric
of parameterized BH free off rotation parameter. The solution of parameterized
BH within Newman-Janis approach is actually in contrast from the BH solution
in general relativity. Furthermore, we have studied the thermodynamics for
parameterized BH with rotation parameter. The temperature is dependent on the
mass $M$, deformation parameter $\eta$, radius of BH $r_{+}$ as well as spin
parameter $a$. We recover the temperature for parameterized BH without
rotation parameter for $a=0$. We have analyzed the physical importance of
temperature versus horizon under the effects of deformation and rotation
parameter. The $T_{H}$ rises with the rising values of deformation parameter
$\eta$.

The $T_{H}$ constantly decreases and after a minima it rises to reach at a
maxima and then again gradually decrease to shows an asymptotically flat
description until $r\rightarrow\infty$ for changing values of deformation
parameter. There can be seen two critical values $r_{+}^{min}$ and
$r_{+}^{max}$ in the plot where the BH is stable for $r_{+}^{min}\leq
r_{+}\leq r_{+}^{max}$. For the rising values of $a$, the $T_{H}$ shows a
decreasing conduct. The stability of BH can be observed by Hawking’s scenario
with the increasing temperature more radiation emit and the radius of BH
decreases. Furthermore, we have studied the corrected temperature for
parameterized BH with rotation parameter accompanying by GUP effects. When the
correction parameter $\beta=0$, we get the temperature of Eq. (18) as well as
by putting $\eta=0=a$, we obtain the Hawking temperature of Schwarzschild BH
$T_{Sch}=\frac{1}{8\pi M}$ at $r_{+}\thickapprox 2M$. We have also studied the
graphical evaluation of corrected temperature with respect to horizon and
graphically discussed the stable condition of parameterized BH with rotation
parameter under the impact of quantum gravity. The $T^{\prime}_{H}$
exponentially decreases for rising horizon in the region $0\leq r_{+}\leq 8$
for different variations of $\beta$. For rising values of $\beta$ the
$T^{\prime}_{H}$ decreases. In the absence of deformation parameter, the
$T^{\prime}_{H}$ also decreases with the rising variations of $\beta$ in the
region $0\leq r_{+}\leq 10$. In Fig. 2, one can examined that the temperature
is maximum at minimum horizon. This physical strategy demonstrates the stable
state of BH. In the existence of gravity parameter with rising $\eta$, the
$T^{\prime}_{H}$ also rises.

The conduct of $T^{\prime}_{H}$ for distinct values of $\eta$ in $0\leq
r_{+}\leq 8$ demonstrates the two critical values $r^{min}_{+}$ and
$r^{max}_{+}$. The BH assures its stability for $r^{min}_{+}\leq r_{+}\leq
r^{max}_{+}$ by getting an asymptotically flat state upon
$r_{+}\rightarrow\infty$.

Moreover, it is an admitted fact in quantum gravity that the quantum
corrections are a cause of reduction in rising temperature. We have concluded
that the temperature in Fig. 1 (absence of quantum gravity parameter) is
higher than in Fig. 3 (presence of quantum gravity parameter), so, this
phenomenon can be clearly observed in plots with the presence/absence of
gravity parameter.

At last, the corrected entropy for parameterized BH with rotation parameter is
calculated. By neglecting the gravity parameter effects, we attained the
standard entropy of parameterized BH with rotation parameter. The corrected
entropy slowly decreases and after some variations the $S_{P}$ continuously
increases for rising values of horizon. This conduct shows the stable form of
parameterized BH. The $S_{P}$ increases with the increasing values of
correction parameter.

For different values of deformation parameter $\eta$ in the region $0\leq
r_{+}\leq 5$, one can observed that with the rising values of deformation
parameter $\eta$ the $S_{P}$ also increases. It is also observable that at
most of the points sharp edge appears for varying values of deformation
parameter that reduces the stability of entropy.

It can be concluded from the observation of both plots of corrected entropy,
the BH entropy is more stable for varying values of correction parameter as
compared to deformation parameter.

##  IX Appendix A

We apply the WKB approximation in Lagrangian field equation and collect a set
of field equations in the form

$\displaystyle\frac{U\Delta}{(U\Delta+V^{2})}[c_{1}(\partial_{0}I_{0})(\partial_{1}I_{0})+\beta
c_{1}(\partial_{0}I_{0})^{3}(\partial_{1}I_{0})-c_{0}(\partial_{1}I_{0})^{2}-\beta
c_{0}(\partial_{1}I_{0})^{4}$ $\displaystyle+$ $\displaystyle
c_{1}eA_{0}(\partial_{1}I_{0})+c_{1}\beta
eA_{0}(\partial_{0}I_{0})^{2}(\partial_{1}I_{0})]-\frac{V\Delta}{(U\Delta+V^{2})}[c_{3}(\partial_{1}I_{0})^{2}+\beta
c_{3}(\partial_{1}I_{0})^{4}$ $\displaystyle-
c_{1}(\partial_{1}I_{0})(\partial_{3}I_{0})-\beta
c_{1}(\partial_{1}I_{0})(\partial_{3}I_{0})^{2}]+\frac{U}{\sigma^{2}(U\Delta+V^{2})}[c_{2}(\partial_{0}I_{0})(\partial_{2}I_{0})+\beta
c_{2}$
$\displaystyle(\partial_{0}I_{0})^{3}(\partial_{2}I_{0})-c_{0}(\partial_{2}I_{0})^{2}-\beta
c_{0}(\partial_{2}I_{0})^{4}+c_{2}eA_{0}(\partial_{2}I_{0})+c_{2}eA_{0}\beta(\partial_{0}I_{0})^{2}(\partial_{1}I_{0})]$
$\displaystyle+$
$\displaystyle\frac{U\Delta}{(U\Delta+V^{2})^{2}}[c_{3}(\partial_{0}I_{0})(\partial_{3}S_{0})+\beta
c_{3}(\partial_{0}I_{0})^{3}(\partial_{3}I_{0})-c_{0}(\partial_{3}I_{0})^{2}-\beta
c_{0}(\partial_{3}I_{0})^{4}$ $\displaystyle+$ $\displaystyle
c_{3}eA_{0}(\partial_{3}I_{0})+c_{3}eA_{0}(\partial_{0}I_{0})^{2}(\partial_{3}I_{0})]-m^{2}\frac{Uc_{0}-{Vc_{3}}}{(U\Delta+V^{2})}=0,$
$\displaystyle-$
$\displaystyle\frac{U\Delta}{(U\Delta+V^{2})}[c_{1}(\partial_{0}I_{0})^{2}+\beta
c_{1}(\partial_{0}I_{0})^{4}-c_{0}(\partial_{0}I_{0})(\partial_{1}I_{0})-\beta
c_{0}(\partial_{0}I_{0})(\partial_{1}I_{0})^{3}$ $\displaystyle+$
$\displaystyle c_{1}eA_{0}(\partial_{0}I_{0})+\beta
c_{1}eA_{0}(\partial_{0}I_{0})^{3}]+\frac{V\Delta}{(U\Delta+V^{2})}[c_{3}(\partial_{0}I_{0})(\partial_{1}I_{0})+\beta
c_{3}(\partial_{0}I_{0})$
$\displaystyle(\partial_{1}I_{0})^{3}-c_{1}(\partial_{0}I_{0})(\partial_{3}I_{0})-\beta
c_{1}(\partial_{0}I_{0})(\partial_{3}I_{0})^{3}]+\frac{\Delta}{\sigma^{2}}[c_{2}(\partial_{1}I_{0})(\partial_{2}I_{0})+\beta
c_{2}$
$\displaystyle(\partial_{1}I_{0})(\partial_{2}I_{0})^{3}-c_{1}(\partial_{2}I_{0})^{2}-\beta
c_{1}(\partial_{2}I_{0})^{4}]+\frac{\Delta}{(U\Delta+V^{2})}[c_{3}(\partial_{1}I_{0})(\partial_{3}I_{0})$
$\displaystyle+$ $\displaystyle\beta
c_{3}(\partial_{1}I_{0})(\partial_{3}I_{0})^{3}-c_{1}(\partial_{3}I_{0})^{2}-\beta
c_{1}(\partial_{3}I_{0})^{4}]-m^{2}\Delta
c_{1}+\frac{eA_{0}U\Delta}{(U\Delta+V^{2})}[c_{1}$
$\displaystyle(\partial_{0}I_{0})+\beta
c_{1}(\partial_{0}I_{0})^{3}-c_{0}(\partial_{1}I_{0})-\beta
c_{0}(\partial_{1}I_{0})^{3}+eA_{0}c_{1}+\beta
c_{1}eA_{0}(\partial_{0}I_{0})^{2})]$ $\displaystyle+$
$\displaystyle\frac{eA_{0}V\Delta}{(U\Delta+V^{2})}[c_{3}(\partial_{1}I_{0})+\beta
c_{3}(\partial_{1}I_{0})^{3}-c_{1}(\partial_{3}I_{0})-\beta
c_{1}(\partial_{1}I_{0})^{3}]=0,$
$\displaystyle\frac{U}{\sigma^{2}(U\Delta+V^{2})}[c_{2}(\partial_{0}I_{0})^{2}+\beta
c_{2}(\partial_{0}I_{0})^{4}-c_{0}(\partial_{0}I_{0})(\partial_{2}I_{0})-\beta
c_{0}(\partial_{0}I_{0})(\partial_{2}I_{0})^{3}$ $\displaystyle+$
$\displaystyle c_{2}eA_{0}(\partial_{0}I_{0})+\beta
c_{2}eA_{0}(\partial_{0}I_{0})^{3}]+\frac{\Delta}{\sigma^{2}}[c_{2}(\partial_{1}I_{0})^{2}+\beta
c_{2}(\partial_{1}I_{0})^{4}-c_{1}(\partial_{1}I_{0})$
$\displaystyle(\partial_{2}I_{0})-\beta
c_{1}(\partial_{1}I_{0})(\partial_{2}I_{0})^{3}]-\frac{V}{\sigma^{2}(U\Delta+V^{2})}[c_{2}(\partial_{0}I_{0})(\partial_{3}I_{0})+\beta
c_{2}(\partial_{0}I_{0})^{3}$
$\displaystyle(\partial_{3}I_{0})-c_{0}(\partial_{0}I_{0})(\partial_{3}I_{0})-\beta
c_{0}(\partial_{0}I_{0})^{3}(\partial_{3}I_{0})+c_{2}eA_{0}(\partial_{3}I_{0})+\beta
c_{2}eA_{0}$
$\displaystyle(\partial_{3}I_{0})^{3}]+\frac{\Delta}{\sigma^{2}(U\Delta+V^{2})}[c_{3}(\partial_{2}I_{0})(\partial_{3}I_{0})+\beta
c_{3}(\partial_{2}I_{0})^{3}(\partial_{3}I_{0})-c_{2}(\partial_{3}I_{0})^{2}$
$\displaystyle-$ $\displaystyle\beta
c_{2}(\partial_{3}I_{0})^{4}]-\frac{m^{2}c_{2}}{\sigma^{2}}+\frac{eA_{0}U}{\sigma^{2}(U\Delta+V^{2})}[c_{2}(\partial_{0}I_{0})+\beta
c_{2}(\partial_{0}I_{0})^{3}-c_{0}(\partial_{2}I_{0})$ $\displaystyle-$
$\displaystyle\beta c_{0}(\partial_{2}I_{0})^{3}+c_{2}eA_{0}+c_{2}\beta
eA_{0}(\partial_{0}I_{0})^{2}]=0,$ (35)
$\displaystyle\frac{(U\Delta)-\Delta^{2}}{(U\Delta+V^{2})^{2}}[c_{3}(\partial_{0}I_{0})^{2}+\beta
c_{3}(\partial_{0}I_{0})^{4}-c_{0}(\partial_{0}I_{0})(\partial_{3}I_{0})-\beta
c_{0}(\partial_{0}I_{0})(\partial_{3}I_{0})^{3}$ (36) $\displaystyle+$
$\displaystyle{eA_{0}c_{3}}(\partial_{0}I_{0})+\beta
c_{3}eA_{0}(\partial_{0}I_{0})^{3}]-\frac{U}{\sigma^{2}(U\Delta+V^{2})}[c_{3}(\partial_{1}I_{0})^{2}+\beta
c_{3}(\partial_{1}I_{0})^{4}$ $\displaystyle-$ $\displaystyle
c_{1}(\partial_{1}I_{0})(\partial_{3}I_{0})-\beta
c_{1}(\partial_{1}I_{0})(\partial_{3}I_{0})^{3}]-\frac{V}{\sigma^{2}(U\Delta+V^{2})}[c_{2}(\partial_{0}I_{0})(\partial_{2}I_{0})$
$\displaystyle+$ $\displaystyle\beta
c_{2}(\partial_{0}I_{0})^{3}(\partial_{2}I_{0})-c_{0}(\partial_{2}I_{0})^{2}+\beta
c_{0}(\partial_{2}I_{0})^{4}+{eA_{0}c_{2}}(\partial_{2}I_{0})+\beta
c_{2}eA_{0}$
$\displaystyle(\partial_{0}I_{0})^{2}(\partial_{2}I_{0})]-\frac{eA_{0}\Delta}{\sigma^{2}(U\Delta+V^{2})}[c_{3}(\partial_{2}I_{0})^{2}+\beta
c_{3}(\partial_{2}I_{0})^{4}-c_{2}(\partial_{2}I_{0})(\partial_{3}I_{0})$
$\displaystyle-$ $\displaystyle\beta
c_{2}(\partial_{0}I_{0})(\partial_{3}I_{0})^{3}]-\frac{m^{2}(Vc_{0}-\Delta
c_{3})}{(U\Delta+V^{2})}+\frac{U\Delta
eA_{0}-\Delta^{2}}{(U\Delta+V^{2})^{2}}[c_{3}(\partial_{0}I_{0})+\beta c_{3}$
$\displaystyle(\partial_{0}I_{0})^{3}-c_{0}(\partial_{3}I_{0})-\beta
c_{0}(\partial_{3}I_{0})^{3}+c_{3}eA_{0}+\beta
eA_{0}(\partial_{0}I_{0})^{2}]=0.$

## Appendix B

We have made the assumption that the Lagrangian gravity equation exists in
order to see the bosonic tunneling. Additionally, we generated a set of
equations using the WKB approximation to the Lagrangian gravity equation. So,
we have used the variable separation action to obtain the solutions in matrix
form. The equations (IX)-(36) of general matrix form can be expressed as

$\begin{pmatrix}0\\\ 0\\\ 0\\\
0\end{pmatrix}=\begin{pmatrix}Z_{00}&Z_{01}&Z_{02}&Z_{03}\\\
Z_{10}&Z_{11}&Z_{12}&Z_{13}\\\ Z_{20}&Z_{21}&Z_{22}&Z_{23}\\\
Z_{30}&Z_{31}&Z_{32}&Z_{33}\\\ \end{pmatrix}\begin{pmatrix}c_{0}\\\ c_{1}\\\
c_{2}\\\ c_{3}\\\ \end{pmatrix}$

## References

  * (1) Johannsen, T.; Psaltis, D.; Metric for rapidly spinning black holes suitable for strong-field tests of the no-hair theorem. Phys. Rev. D 2011, 83, 124015.
  * (2) Cardoso, V.; Pani, P.; Rico, J.; On generic parametrizations of spinning black-hole geometries. Phys. Rev. D 2014, 89, 064007.
  * (3) Rezzolla, L.; Zhidenko, A.; New parametrization for spherically symmetric black holes in metric theories of gravity. Phys. Rev. D 2014, 90, 084009.
  * (4) McManus, R.; Berti, E.; Macedo, C. F. B.; Kimura, M.; Maselli, A.; Cardoso, V.;  Parametrized black hole quasinormal ringdown. II. Coupled equations and quadratic corrections for nonrotating black holes. Phys. Rev. D 2019, 100, 044061.
  * (5) Chrusciel, P. T.; Costa, J. L.; Heusler, M.; Stationary Black Holes: Uniqueness and Beyond. Living Rev. Relativ. 2012, 15, 7.
  * (6) Cardoso, V.; Kimura, M.; Maselli, A.; Berti, E.; Macedo, C. F. B.; McManus, R.; Parametrized black hole quasinormal ringdown. I. Decoupled equations for nonrotating black holes. Rev. D 2019,99, 104077.
  * (7) Magalhães, R. B.; Leite, L. C. S.; Crispino, L. C. B.; Parametrized black holes: scattering investigation. Eur. Phys. J. C 2022, 82, 698.
  * (8) Konoplya, R. A.; Zhidenko, A.; General parametrization of black holes: The only parameters that matter. Phys. Rev. D 2020 ,101, 124004.
  * (9) Newman, E. T.; Couch, E.; Chinnapared, K.; Exton, A.; Prakash A.; Torrence, R.; Metric of a rotating, charged mass. J. Math. Phys. 1965, 6, 918.
  * (10) Flaherty, E. J.; Hermitian and K¨ahlerian geometry in relativity. Lecture notes in Physics, 1975, 46 (Springer-Verlag, Berlin 976).
  * (11) Misner, C. W.; Thorne, K. S.; Wheeler, J. A.; Gravitation (W.H. Freeman and Co., San Francisco, 1973).
  * (12) Ali, R.; Bamba, K.; Shah, S. A. A.; Effect of quantum gravity on the stability of black holes. Symmetry 2019, 631, 11.
  * (13) Ali, R.; Asgher, M.; Tunneling Analysis Under the Influences of Einstein-Gauss-Bonnet Black Holes Gravity Theory. New Astronomy 2022, 93, 101759.
  * (14) Ali, R.; Babar, R.; Sahoo, P. K.; Quantum gravity evolution in the Hawking radiation of a rotating regular Hayward black hole. Physics of the Dark Universe 2022, 35, 100948.
  * (15) Konishi, K.; Pafiuti, G.; Provero, P.; Minimum physical length and the generalized uncertainty principle in string theory. Phys. Lett. B 1990, 234, 276.
  * (16) Maggiore, M.; A generalized uncertainty principle in quantum gravity. Phys. Lett. B 1993, 304, 65.
  * (17) Ali, A. F.; No existence of black holes at LHC due to minimal length in quantum gravity. JHEP 2012, 1209, 067.
  * (18) Faizal, M.; Khalil, M. M.; GUP-corrected thermodynamics for all black objects and the existence of remnants. Int. J. Mod. Phys. A 2015, 30, 1550144.
  * (19) Gangopadhyay, S.; Dutta, A.; Faizal, M.; Constraints on the generalized uncertainty principle from black-hole thermodynamics. Eur. Phys. Lett. 2015, 112, 20006.
  * (20) Gecim, G.; Sucu, Y.; The GUP effect on Hawking radiation of the 2+ 1 dimensional black hole. Phys. Lett. B 2017, 773, 391.
  * (21) Gecim, G.; Sucu, Y.; Quantum gravity effect on the Hawking radiation of charged rotating BTZ black hole. Gen. Relativ. Gravit. 2018, 50, 152.
  * (22) Javed, W.; Babar, R.; $\ddot{O}$vg$\ddot{u}$n, A.; Effect of the GUP on the Hawking radiation of black hole in 2+1 dimensions with quintessence and charged BTZ-like magnetic black hole. Mod. Phys. Lett. A 2020, 35, 2050104.
  * (23) Sakalli, I.; $\ddot{O}$vg$\ddot{u}$n, A.; Hawking radiation and deflection of light from rindler modified schwarzschild black hole. EPL 2015, 110, 10008.
  * (24) Sakalli, I.; $\ddot{O}$vg$\ddot{u}$n, A.; Hawking radiation of spin-1 particles from a three-dimensional rotating hairy black hole. Exp. Theor. Phys. 2015, 121, 404.
  * (25) Javed, W.; Ali, R.; Babar, R.; $\ddot{O}$vg$\ddot{u}$n, A.; Tunneling of massive vector particles from types of BTZ-like black holes. Eur. Phys. J. Plus 2019, 134, 511.
  * (26) $\ddot{O}$vg$\ddot{u}$n, A.; Jusufi, K.; Massive vector particles tunneling from noncommutative charged black holes and their GUP-corrected thermodynamics. The European Physical Journal Plus 2016, 131, 177.
  * (27) $\ddot{O}$vg$\ddot{u}$n, A.; Jusufi, K.; The effect of the GUP on massive vector and scalar particles tunneling from a warped DGP gravity black hole. Eur. Phys. J. Plus 2017, 132, 298.
  * (28) Javed, W.; Ali, R.; Abbas, G.; Charged vector particles tunneling from black ring and 5D black hole. Can. J. Phys. 2018, 97, 176.
  * (29) Hassanabadi, H.; Maghsoodi, E.; Chung, W. S.; Analysis of black hole thermodynamics with a new higher order generalized uncertainty principle. Eur. Phys. J. C 2019, 79, 358.
  * (30) Konoplya, R.; Zhidenko, A.; Detection of gravitational waves from black holes: is there a window for alternative theories? Phys. Lett. B 2016, 756, 350.
  * (31) Magalhaes, R. B.; Leite, L. C. S.; Crispino, L. C. B.; Absorption by deformed black holes. Phys. Lett. B 2020, 805, 135418.
  * (32) Shahraeini, S.; Nozari, K.; Saghaf, S.; Journal of Holography Applications in Physics 2022, 2, 55.
  * (33) Solodukhin, S. N.; Evidence of rescattering effect in Pb–Pb collisions at the LHC through production of K (892)o and $\phi$(1020) mesons. Phys. Lett. B 2020,802, 135225.
  * (34) Banerjee, R.; Majhi, B. R.; Quantum Tunneling and Back Reaction. Phys. Lett. B 2018, 662, 62.
  * (35) Banerjee, R.; Majhi, B. R.; Quantum tunneling beyond semiclassical approximation.JHEP 2008, 2008, 095.
  * (36) Banerjee, R.; Majhi, B. R.; Samanta, S.; Noncommutative black hole thermodynamics. Phys. Rev. D 2008, 77, 124035.
  * (37) Pradhan, P.; CFT and logarithmic corrections to the black hole entropy product formula,. Adv. High Energy Phys. 2017, 2017, 2367387.