# Study of tunneling radiation and thermal fluctuations of a gauge super
gravity like black hole

Riasat Ali<EMAIL_ADDRESS>Department of Mathematics, GC University
Faisalabad Layyah Campus, Layyah-31200, Pakistan  Muhammad Asgher
<EMAIL_ADDRESS>Department of Mathematics, The Islamia University of
Bahawalpur, Bahawalpur-63100, Pakistan  P.K. Sahoo<EMAIL_ADDRESS>pilani.ac.in Department of Mathematics, Birla Institute of Technology and
Science-Pilani, Hyderabad Campus, Hyderabad-500078, India

###### Abstract

The Newman-Janis technique and the semi-classical Hamilton-Jacobi approach are
two distinct phenomena that we apply in this paper to examine the Hawking
temperature $(T_{H})$ for $4$-dimensional gauge super gravity like black hole
with rotation parameters. First, using the Newman-Janis algorithmic approach,
we compute the gauge super gravity like black holes solution. We use surface
gravity to derive the $T_{H}$ for a gauge super gravity like black hole. In
order to do this, we use the Lagrangian field equation in the background of
generalized uncertainty principle, semi-classical Hamilton-Jacobi approach and
the WKB approximation technique. Furthermore, we investigate the stability of
considered geometry with the help of corrected entropy and well known
thermodynamic quantities. It is concluded that the gauge super gravity like
black holes is stable under first-order corrections.

Black hole; Gauge super gravity; Newman-Janis technique; Tunneling radiation;
Thermal fluctuations; Corrected entropy; phase transition.

##  I Introduction

The Janis-Newman method is presented in its most general form of BH geometry.
With the help of this extension, configurations with all bosonic fields having
spin of one can be created complex and real scalar fields, gauge fields and
metric fields. The Newman-Penrose tetrad formalism has been used in the
original prescription, which seems to be exceedingly time-consuming because it
calls for inverting the metric, looking for a basis of null tetrad where the
transformation may be used, and then inverting into metric 1 ; 2 ; 2a ; 3 ; 4
; 4a . One of these solution generating methods has the Janis-Newman
phenomenon, which, in its original form, has derived rotational metrics from
static ones. It was discovered by Janis and Newman as an alternate Kerr metric
formulation 5 , and soon after, it was utilised once more to find the Kerr-
Newman metric 6 . The Kerr-Newman metric is the only rotational solution the
Janis-Newman algorithm has produced for non-fluid configurations (excluding
radiated and internal solutions) over the course of its long history, and only
few known cases have been created 7 ; 8 ; 9 . In general, applying the Janis-
Newman phenomenon to interior and emitting systems 10 ; 11 ; 12 ; 13 ; 14
entails deriving a configuration that does not immediately solve the motion
equations and interpreting the fluid (whose features can be analysed);
however, these types of applications are not the focus of this review.

The generalized uncertainty principle (GUP) is the most appropriate technique
to investigate this minimum length GUP1 . A definition of the modified
commutation relationship is

$[O_{u},p_{v}]=i\hbar\left[1+\alpha(p^{2})\right]\delta_{uv},$ (1)

with $p_{v}$ and $O_{u}$ are generalized momentum and position operators,
respectively. Moreover, a connection to GUP can be made in the following
manner

$\Delta O\Delta p\geq\frac{\hbar}{2}\left[1+\alpha(\Delta p^{2})\right].$ (2)

The Planck mass is denoted by $M_{p}$, and the correction parameter $\alpha$
can be represented in terms of a dimensionless parameter as
$\alpha=\frac{\alpha_{0}}{M_{p^{2}}}$. The quantum effects have a significant
impact close to a BH’s horizon and the GUP relationship is highly helpful in
understanding the BH physics.

Hawking investigated the theory that black holes (BHs) emit radiation around
the horizon as a result of the effects of the associated quantum vacuum
fluctuation 15 . The quantum tunneling method can be used to define Hawking
radiation near to the BHs horizon. It was discovered that the black body
radiation was the Hawking radiation. This scenario suggests that the BH
totally 16  disappears. Different inferences based on relativity quantum
mechanics and quantum theory in curved space-time 17  demonstrate the accuracy
of the Hawking formula. The semi-classical tunnelling strategy 18 ; 19 ; 20 ;
21 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28 ; 28a  is a useful model for
investigating the Hawking radiation. One can calculate the corrected $T_{H}$
using this method by accounting for the different space-time. All sorts of
particles, including fermions, bosons, and scalar particles, are radiated by
the generalised BH. For these kinds of particles in the geometry of various
BHs metrics, the standard $T_{H}$ as well as the corrected $T_{H}$ have been
examined 29 ; 30 ; 31 . The fact that the corrected $T_{H}$ rate is higher
than the standard $T_{H}$ indicates that the evaporation process is
accelerated by the various background geometries. There are mentions to a
number of commutation relations generalisations 32 ; 33 ; 34 . These
alterations are widely used in order to discover more about the quantum
features of gravity. Black holes can be used to examine the effects of quantum
gravity. The introduction of gravity influences into physics of BH through the
use of GUP leads to several interesting outcomes and discoveries 30 .
Additionally, the analysis included the remnant mass and modifications to the
entropy and temperature. The effects of GUP to spin-$0$ and spin-$1$ particles
in the context of warped DGP gravity BH have been studied 35  by Övgün and
Jusufi.

Thermal fluctuations are caused by statistical perturbations in compact
objects. Because Hawking radiations are emitted from BHs, they reduce the size
of the BH and, as a result, raise the temperature of the considered BH. Faizal
and Khalil z3  looked on the thermodynamics and thermal fluctuations of
charged AdS BHs. This type of BH shows the stability under the first-order
corrections. Pourhassan and Faizal z4  investigated the thermodynamics of
spinning Kerr-AdS BH and they studied the corrected entropy and analyzed that
the corrections are extremely effective for the small BHs. Additionally,
utilizing heat capacity and the Hessian matrix stability test, the non-minimal
regular BHs of thermodynamics are studied z5 . They came to the conclusion
that the these BHs are locally and globally stable for increasing levels of
the correction parameter. Zhang and Pradhan z6  investigated that how
corrected entropy affected the Kerr-Newman-AdS and RN-AdS BHs, they found that
there was a quadratic phase transition occurring because of the thermal
fluctuations. Moreover, the thermodynamics of various types of BHs under the
influence of correction parameter is analyzed, they checked the relation of
other thermodynamics quantities with the corrected entropy and for stability
reasons z13 ; z14a ; z14b ; z14c ; z15 ; z16 ; z17 ; z18 ; RB1 ; RB2 .

In this article, we examine the Newman-Janis method that generates rotating
metrics by performing a complex transformation. In section II, we develop the
rotating metric for a spherically symmetric metric that will be used to
analyse the $T_{H}$. In section. III generates both corrected tunnelling
solutions and corrected $T_{H}$. Graphics are used in Sec. IV to describe the
solution’s physical behaviour, viability, and stability. In section V, we
investigate the behaviour of thermodynamic quantities under the effect of
thermal fluctuations. We finalise our results in section VI.

##  II Gauged Super Gravity Like Black Hole with Rotation Parameter

The theory of representation for the extended algebra of supersymmetry studied
by Wess and Bagger S1  in chapter 2. Gibbons S2  studied the Reissner-
Nordstrom BH as a supersymmetric solition. The supergravity models of BH
solutions have been studied S3  and also discussed the general BHs solution in
supersymmetric theories and general relativity. Black holes are significant
objects in any gravity theory for the insight BHs implies into quantum
gravity. For this reason, obtaining all potential BH solutions is an important
stage in any theory. Since it is thought that the majority of astrophysical
BHs are revolving, rotating BHs are the most pertinent sub-cases for
astrophysics. These systems might also include revolving star external
metrics. This algorithm implies an approach to generating an axi-symmetric
solution from a seed solution that is spherically symmetric by first
complexifying the radial and null time coordinates, then performing a complex
coordinate transformation. To write the outcome in Boyer-Lindquist(BL)
coordinates, one frequently changes the spherical coordinates. The gauge super
gravity like BH is given as 24 ; 36 ,

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

where $F(r)=\frac{f}{(H_{1}H_{2}H_{3}H_{4})^{\frac{1}{2}}}$ and
$G(r)=(H_{1}H_{2}H_{3}H_{4})^{\frac{1}{2}}$, with
$f=k-\frac{\mu}{r^{2}}+g^{2}r^{2}H_{1}H_{2}H_{3}H_{4},~{}H_{i}=\frac{q_{i}}{r^{2}}+1$,
for $(i=1,2,3,4).$ The $g=1/L$ and $L$ is related to the cosmological constant
$\Lambda=-3g^{2}=-3/L^{2}$ and the $\mu$ represent the non-extremality
parameter 37  for radius $k=1$ and $k=0,$ then represents the metrics on
$S_{2}$, and $R_{2}$ respectively. The four electric potentials $A_{\mu}^{i}$
are defined as; $A_{0}^{i}=\frac{\tilde{q_{i}}}{r^{2}+q_{i}}$ $(i=1,2,3,4),$
where $q_{i}$ and $\tilde{q_{i}}$ represent charges and physical charges of a
BH. Through the transformation of $(t,r,\theta,\phi)$ to $(u,r,\theta,\phi)$
coordinates

$\displaystyle du=dt-\frac{dr}{F(r)},$ (4)

After using the above transformation the metric (3) can be expressed as
follows

$ds^{2}=-F(r)du^{2}-2dudr+r^{2}G(r)(d\theta^{2}+sin^{2}d\phi^{2}).$ (5)

The non-zero component of the above metric is

$\displaystyle g_{uu}$ $\displaystyle=$
$\displaystyle-F(r),~{}g_{ur}=g_{ru}=-1,~{}~{}g_{\theta\theta}=G(r)r^{2},$
$\displaystyle g_{\phi\phi}$ $\displaystyle=$ $\displaystyle
G(r)r^{2}\sin^{2}\theta.$ (6)

Now, we take the inverse metric coefficient

$\displaystyle g^{rr}$ $\displaystyle=$
$\displaystyle-F(r),~{}~{}g^{\theta\theta}=\frac{1}{G(r)r^{2}},~{}~{}g^{\phi\phi}=\frac{1}{G(r)r^{2}sin^{2}\theta},$
$\displaystyle g^{ur}$ $\displaystyle=$ $\displaystyle g^{ru}=-1.$ (7)

In the null tetrad framework the metric can be written as

$\displaystyle
g^{\mu\nu}=-l^{\nu}n^{\mu}-l^{\mu}n^{\nu}+m^{\mu}\bar{m}^{\nu}+m^{\nu}\bar{m}^{\mu}.$
(8)

Here the corresponding components are

$\displaystyle l^{\mu}$ $\displaystyle=$
$\displaystyle\delta_{r}^{\mu},~{}~{}~{}n^{\nu}=\delta_{u}^{\mu}-\frac{1}{2}F\delta_{r}^{\mu},$
$\displaystyle m^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}rG(r)}(\delta_{\theta}^{\mu}+\frac{i}{sin\theta}\delta_{\phi}^{\mu}),$
$\displaystyle\bar{m}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}rG(r)}(\delta_{\theta}^{\mu}-\frac{i}{sin\theta}\delta_{\phi}^{\mu}).$

For any point in the BH spacetime the null vectors of the null tetrad satisfy
the relations of
$l_{\mu}l^{\mu}=n_{\mu}n^{\mu}=m_{\mu}m^{\mu}=l_{\mu}m^{\mu}=m_{\mu}m^{\mu}=0$
and $l_{\mu}n^{\nu}=-m_{\mu}\bar{m}^{\mu}=1$ and $(u,r)$ place coordinate
transformation are $u\rightarrow u-iacos\theta$, $r\rightarrow r+iacos\theta$,
then we perform the transformation $F(r)\rightarrow\tilde{F}(r,a,\theta)$ and
$\sigma^{2}=r^{2}+a^{2}cos^{2}\theta$. In the $(u,r)$ space the vectors become

$\displaystyle l^{\mu}$ $\displaystyle=$
$\displaystyle\delta_{r}^{\mu},~{}~{}~{}n^{\nu}=\delta_{u}^{\mu}-\frac{1}{2}\tilde{F}\delta_{r}^{\mu},$
$\displaystyle m^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}rG(r)}(\delta_{\theta}^{\mu}+iasin\theta(\delta_{u}^{\mu}-\delta_{r}^{\mu})+\frac{i}{sin\theta}\delta_{\phi}^{\mu}),$
$\displaystyle\bar{m}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}rG(r)}(\delta_{\theta}^{\mu}-iasin\theta(\delta_{u}^{\mu}-\delta_{r}^{\mu})-\frac{i}{sin\theta}\delta_{\phi}^{\mu}),$
(9)

From the definition of the null tetrad, using Eq. (9) in Eq. (8), we get a
matrix equation.

By using the coefficient of above matrix in Eq. (8) to obtained $g^{\mu\nu}$
in the EF coordinates are given by

From the definition of the null tetrad the metric tensor $g^{\mu r}$ in the EF
coordinate are computed. Furthermore, the matrix’s lower index components in
the EF coordinates can be expressed as

$\displaystyle g_{uu}$ $\displaystyle=$
$\displaystyle-\frac{a^{2}G^{2}\sigma^{2}sin^{2}\theta+\tilde{F}G^{4}\sigma^{4}}{a^{4}sin^{2}\theta},~{}~{}~{}g_{ur}=-\frac{G^{2}\sigma^{2}}{a^{2}sin^{2}\theta},~{}~{}~{}g_{\theta\theta}=\sigma^{2}G^{2},$
$\displaystyle~{}~{}~{}g_{\phi\phi}$ $\displaystyle=$
$\displaystyle-\frac{\tilde{F}G^{4}\sigma^{4}}{a^{2}},~{}~{}~{}g_{u\phi}=\frac{G^{2}\sigma^{2}(a^{2}sin^{2}\theta+\tilde{F}G^{2}\sigma^{2})}{a^{3}sin^{2}\theta},~{}~{}~{}g_{r\phi}=\frac{G^{2}\sigma^{2}}{a},$

Finally, we perform the coordinate transformation from the EF to BL
coordinates as

$du=dt+\lambda(r)dr,~{}~{}~{}d\phi=d\phi+h(r)dr$ (10)

where

$\displaystyle\lambda(r)$ $\displaystyle=$
$\displaystyle-\frac{r^{2}+a^{2}}{r^{2}F+a^{2}},$ $\displaystyle h(r)$
$\displaystyle=$ $\displaystyle-\frac{a}{r^{2}F+a^{2}}$
$\displaystyle\tilde{F}(r,a,\theta)$ $\displaystyle=$
$\displaystyle\frac{r^{2}F(r)+a^{2}cos^{2}\theta}{\sigma^{2}}.$ (11)

By using the Eq.(10) and Eq.(11) in Eq. (5) we get the final metric

$ds^{2}=-F(dt-\lambda(r)dr)^{2}-2(dt-\lambda(r)dr)dr+\sigma^{2}d\phi^{2}sin^{2}\theta+\sigma^{2}sin^{2}\theta(d\phi+h(r)dr)^{2}.$
(12)

From the Eq. (12), we are neglecting non-diagonal values expect $dtd\phi$ as
like in these Refs(R6 -R9 ). The gauged super gravity like BH metric with new
spin (rotation) parameter in $(t,r,\theta,\phi)$ can be derived in the
following form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\sigma^{2}(a^{2}r^{2}q_{r}sin^{2}\theta+k+\frac{\mu}{r^{2}}g^{2}\sigma^{2})}{q_{r}}dt^{2}+\frac{q_{r}}{\sigma^{2}(a^{2}r^{2}q_{r}sin^{2}\theta+k+\frac{\mu}{r^{2}}g^{2}\sigma^{2})}dr^{2}$
(13) $\displaystyle+$
$\displaystyle\sigma^{2}(\frac{q_{r}}{r^{2}})d\theta^{2}+\frac{r^{4}k+\mu
r^{2}g^{2}\sigma^{4}}{q_{r}}d\phi^{2}+2\sigma^{2}(a^{2}r^{2}sin^{2}\theta(\frac{q_{r}}{r^{2}})+k+\frac{\mu}{r^{2}}g^{2}\sigma^{2})dtd\phi,$

with $q_{r}=q_{i}+r^{2}$ and $q_{i}=q_{1}q_{2}q_{3}q_{4}$. Xu et al.
calculated R6  the Kerr-Newman-AdS metric in the Rastall gravity by using the
Newman-Janis approach and the rotating metric must satisfy the Einstein-
Maxwell field equation. Drake and Turolla discovered 12  that the Newman-Janis
method can produce a new Kerr metric interior solution that is suitable. The
Kerr metric can be smoothly matched to these new solutions. Since all the
necessary supergravity components are now available in equation (13), (gauged)
supergravity is a major playground for this modified Janis-Newman algorithm.
If the transformation is infinitesimal, it is important to confirm that it is
a perfectly valid diffeomorphism and is integrable 1 . Additionally,
significant solutions to higher-dimensional Einstein-Maxwell problems remain
unsolved (particularly the black rings solution), and it is hoped that an
understanding of the Janis-Newman algorithm in these dimensions would help
solve this issue. To make the metric simpler, modify the coordinate system
(for particular, to the BL system). The $T_{H}$ is given as

$T_{H}=\frac{r^{4}q_{i}(2r^{2}+a^{2}cos^{4}\theta)\\{k+g^{2}\mu-a^{2}r^{2}cos^{2}\theta(k+2g^{2}\mu)-a^{4}g^{2}\mu
cos^{4}\theta(q_{i}+2r^{2})+r^{4}a^{2}sin^{2}\theta(q_{i}+2r^{2})\\}}{2\pi
r^{3}(q_{i}+2r^{2})}.$ (14)

##  III Corrected tunneling

The gauge theory that the super-partner of the particle, the gauge boson
charged particle, is defined by the super-gravity theory. Additionally,
because this theory is more significant than the ungauged case, it contains a
negative cosmological constant, where cosmological constant is expressed in an
anti de-sitter BH. Now, let’s examine the tunnelling of a boson particle from
a BH. In the theory of gauged supergravity in four dimensions, we determine
the Hawking temperature of BH bosonic tunneling phenomena at BH horizon. We
investigate the stability of charged black holes’ thermodynamic properties in
gauged supergravity theories with $D=4$. The metric Eq. (13) can be re-written
as

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
Ldt^{2}+\frac{1}{L}dr^{2}+Md\theta^{2}+Nd\phi^{2}+2Zdtd\phi,$

where

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}(a^{2}r^{2}sin^{2}\theta(q_{i}+r^{2})+k+\frac{\mu}{r^{2}}g^{2}\sigma^{2})}{q_{i}+r^{2}},$
$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{q_{i}\sigma^{2}+r^{2}\sigma^{2}}{r^{2}},$ $\displaystyle
N$ $\displaystyle=$ $\displaystyle\frac{r^{4}k+\mu
r^{2}g^{2}\sigma^{4}}{q_{i}+r^{2}},$

and

$Z=2\sigma^{2}(a^{2}r^{2}sin^{2}\theta(\frac{q_{i}+r^{2}}{r^{2}})+k+\frac{\mu}{r^{2}}g^{2}\sigma^{2}).$

There are other kinds of solutions, but this review will only focus on four-
dimensional BH like solutions, which are described as particle-like boson
particles that carry charges like mass. The Lagrangian gravity equation can be
defined 4 ; 22  as

$\displaystyle\partial_{\mu}(\sqrt{-g}\phi^{\nu\mu})+\sqrt{-g}\frac{m^{2}}{\hbar^{2}}\phi^{\nu}+\sqrt{-g}\frac{i}{\hbar}A_{\mu}\phi^{\nu\mu}+\sqrt{-g}\frac{i}{\hbar}eF^{\nu\mu}\phi_{\mu}+\alpha\hbar^{2}\partial_{0}\partial_{0}\partial_{0}(\sqrt{-g}g^{00}\phi^{0\nu})-$
$\displaystyle\alpha\hbar^{2}\partial_{i}\partial_{i}\partial_{i}(\sqrt{-g}g^{ii}\phi^{i\nu})=0,$
(16)

here $g$ is determinant coefficient matrix, $\phi^{\nu\mu}$ is anti-symmetric
tensor and $m$ is particle mass, since

$\displaystyle\phi_{\nu\mu}$ $\displaystyle=$
$\displaystyle(1-\alpha{\hbar^{2}\partial_{\nu}^{2}})\partial_{\nu}\phi_{\mu}-(1-\alpha{\hbar^{2}\partial_{\mu}^{2}})\partial_{\mu}\phi_{\nu}+(1-\alpha{\hbar^{2}\partial_{\nu}^{2}})\frac{i}{\hbar}eA_{\nu}\phi_{\mu}-(1-\alpha{\hbar^{2}}\partial_{\nu}^{2})\frac{i}{\hbar}eA_{\mu}\phi_{\nu},$
$\displaystyle~{}~{}and~{}~{}F_{\nu\mu}=\nabla_{\nu}A_{\mu}-\nabla_{\mu}A_{\nu},$

where $\alpha,~{}A_{\mu},~{}e~{}$ and $\nabla_{\mu}$ are the positive
dimensionless parameter(GUP parameter), vector potential of gauge super
gravity like BH, the charge of particle and covariant derivatives
respectively.

$\displaystyle\phi^{0}=\frac{-N\phi_{0}+Z\phi_{3}}{LN+Z^{2}},~{}~{}~{}\phi^{1}=L\phi_{1},~{}~{}~{}\phi^{2}=\frac{1}{M}\phi_{2},~{}~{}~{}\phi^{3}=\frac{Z\phi_{0}+L\phi_{3}}{LN+Z^{2}},$
$\displaystyle\phi^{01}=\frac{(-N\phi_{01}+Z\phi_{13})L}{LN+Z^{2}},~{}~{}~{}\phi^{02}=\frac{-N\phi_{02}}{M(LN+Z^{2})},~{}~{}~{}\phi^{03}=\frac{(L^{2}-LN)\phi_{03}}{(LN+Z^{2})^{2}},$
$\displaystyle\phi^{12}=\frac{L}{M}\phi_{12},~{}\phi^{13}=\frac{L}{LN+Z^{2}}\phi_{13},~{}~{}\phi^{23}=\frac{Z\phi_{02}+L\phi_{23}}{M(LN+Z^{2})},$
(17)

The WKB approximation is

$\phi_{\nu}=c_{\nu}\exp[\frac{i}{\hbar}I_{0}(t,r,\theta,\phi)+\Sigma\hbar^{n}I_{n}(t,r,\theta,\phi)].$
(18)

By computing the Lagrangian equation using the WKB approximation, the set of
field equations in Appendix A is generated. Using separation of variables
technique, we can choose

$I_{0}=-(E-J\Omega)t+W(r)+J\phi+\nu(\theta),$ (19)

where $E$ is the energy of the particle, $J$ represents the particle’s angular
momentum corresponding to angles $\phi$.

$\displaystyle U_{00}$ $\displaystyle=$ $\displaystyle-
LN\tilde{Z}[W_{1}^{2}+\alpha W_{1}^{4}]-\frac{N\tilde{Z}}{M}[J^{2}+\alpha
J^{4}]-LN\tilde{Z}^{2}[\nu_{1}^{2}+\alpha\nu_{1}^{4}]-m^{2}N\tilde{Z},$
$\displaystyle U_{01}$ $\displaystyle=$
$\displaystyle\\-LN\tilde{Z}[\tilde{\Omega}+\alpha\tilde{\Omega}^{3}+eA_{0}+\alpha
eA_{0}\tilde{\Omega}W_{1}+LZ\tilde{Z}+[\nu_{1}+\alpha\nu_{1}^{3}],$
$\displaystyle U_{02}$ $\displaystyle=$
$\displaystyle\frac{-N\tilde{Z}}{M}[\tilde{\Omega}+\alpha\tilde{\Omega}^{3}-eA_{0}-\alpha
eA_{0}\tilde{\Omega}^{2}]J,$ $\displaystyle U_{03}$ $\displaystyle=$
$\displaystyle-L\tilde{\Omega}\tilde{Z}[W_{1}^{2}+\alpha
W_{1}^{4}]-\frac{LN\tilde{Z}^{2}}{M}[\tilde{\Omega}+\alpha\tilde{\Omega}^{3}-eA_{0}-\alpha
eA_{0}\tilde{\Omega}^{2}]\nu_{1}+m^{2}Z\tilde{Z},$ $\displaystyle U_{10}$
$\displaystyle=$ $\displaystyle-
LN\tilde{Z}[\tilde{\Omega}W_{1}+\alpha\tilde{\Omega}W_{1}^{3}]-m^{2}L-eA_{0}LN\tilde{Z}[W_{1}+\alpha
W_{1}^{3}],$ $\displaystyle U_{11}$ $\displaystyle=$ $\displaystyle-
LN\tilde{Z}[\tilde{\Omega}^{2}+\alpha\tilde{\Omega}^{4}-eA_{0}\tilde{\Omega}-\alpha
eA_{0}\tilde{\Omega}W_{1}^{2}]+LZ\tilde{Z}+[\nu_{1}+\alpha\nu_{1}^{3}]\tilde{\Omega}-\frac{L}{M}[J^{2}+\alpha
J^{4}]$
$\displaystyle-L\tilde{Z}[\nu_{1}+\alpha\nu_{1}^{3}]-m^{2}L-eA_{0}LN\tilde{Z}[\tilde{\Omega}+\alpha\tilde{\Omega}^{3}-eA_{0}-\alpha
eA_{0}\tilde{\Omega}^{2}]+eA_{0}LZ\tilde{Z}[\nu_{1}+\alpha\nu_{1}^{3}],$
$\displaystyle U_{12}$ $\displaystyle=$ $\displaystyle\frac{L}{M}[W_{1}+\alpha
W_{1}^{3}]J,$ $\displaystyle U_{13}$ $\displaystyle=$
$\displaystyle-L\tilde{\Omega}\tilde{Z}[W_{1}+\alpha
W_{1}^{3}]\tilde{\Omega}+L\tilde{Z}^{2}[W_{1}+\alpha
W_{1}^{3}]\nu_{1}+LZeA_{0}\tilde{Z}[W_{1}+\alpha W_{1}^{3}],$ $\displaystyle
U_{20}$ $\displaystyle=$
$\displaystyle\frac{N\tilde{Z}}{M}[\tilde{\Omega}J+\alpha\tilde{\Omega}J^{3}]+\frac{Z\tilde{Z}}{M}[\tilde{\Omega}+\alpha\tilde{\Omega}^{3}\nu_{1}^{2}]-\frac{NeA_{0}\tilde{Z}}{M}[J+\alpha
J^{3}],$ $\displaystyle U_{21}$ $\displaystyle=$
$\displaystyle\frac{L}{M}[J+\alpha J^{3}]W_{1},$ $\displaystyle U_{22}$
$\displaystyle=$
$\displaystyle\frac{N\tilde{Z}}{M}[\tilde{\Omega}^{2}+\alpha\tilde{\Omega}^{4}-eA_{0}\tilde{\Omega}-\alpha
eA_{0}\tilde{\Omega}]-\frac{L}{M}+\frac{Z\tilde{Z}}{M}[\tilde{\Omega}+\alpha\tilde{\Omega}^{3}-eA_{0}-\alpha
eA_{0}\tilde{\Omega}^{2}]\nu_{1}$
$\displaystyle-\frac{L\tilde{Z}}{M}[\nu_{1}^{2}+\alpha\nu_{1}^{4}]-\frac{m^{2}}{M}-\frac{eA_{0}N\tilde{Z}}{M}[\tilde{\Omega}+\alpha\tilde{\Omega}^{3}-eA_{0}-\alpha
eA_{0}\tilde{\Omega}^{2}],$ $\displaystyle U_{23}$ $\displaystyle=$
$\displaystyle\frac{L\tilde{Z}}{M}[J+\alpha J^{3}]\nu_{1},$ $\displaystyle
U_{30}$ $\displaystyle=$
$\displaystyle(LN-L^{2})\tilde{Z}^{2}[\nu_{1}+\alpha\nu_{1}^{3}]\tilde{\Omega}+\frac{Z\tilde{Z}}{M}[J^{2}+\alpha
J^{4}]-m^{2}Z\tilde{Z}-eA_{0}(LN-L^{2})\tilde{Z}^{2}[\nu_{1}+\alpha\nu_{1}^{3}],$
$\displaystyle U_{31}$ $\displaystyle=$ $\displaystyle
L\tilde{Z}[\nu_{1}+\alpha\nu_{1}^{3}]W_{1},$ $\displaystyle U_{32}$
$\displaystyle=$ $\displaystyle\frac{Z\tilde{Z}}{M}[J+\alpha
J^{3}]\tilde{\Omega}+\frac{L\tilde{Z}}{M}[\nu_{1}+\alpha\nu_{1}^{3}]J,$
$\displaystyle U_{33}$ $\displaystyle=$
$\displaystyle(LN-L^{2})\tilde{Z}[\tilde{\Omega}^{2}+\alpha\tilde{\Omega}^{4}-eA_{0}\tilde{\Omega}-\alpha
eA_{0}\tilde{\Omega}^{3}]-L\tilde{Z}[W_{1}^{2}+\alpha W_{1}^{4}]$
$\displaystyle-\frac{L\tilde{Z}}{M}[J^{2}+\alpha
J^{4}]-m^{2}L\tilde{Z}-eA_{0}(LN-L^{2})\tilde{Z}[\tilde{\Omega}+\alpha\tilde{\Omega}^{3}-eA_{0}\tilde{\Omega}^{2}],$

where $W_{1}=\partial_{r}{I_{0}}$, $J=\partial_{\phi}I_{0}$,
$\nu_{1}=\partial_{\theta}{I_{0}}$, $\tilde{\Omega}=E-J\Omega$ and
$\tilde{Z}=\frac{1}{LN+Z^{2}}$. For the non-trivial solution, the determinant
U is equal to zero, so we get

$\displaystyle ImW^{\pm}_{1}$ $\displaystyle=$
$\displaystyle\pm\int\sqrt{\frac{(\tilde{\Omega}-eA_{0})^{2}+X_{1}[1+\alpha\frac{X_{2}}{X_{1}}]}{\frac{L(LN+Z^{2})}{N}}}dr$
(20) $\displaystyle=$ $\displaystyle\pm
i\pi\frac{(\tilde{\Omega}-eA_{0})+[1+\alpha\Xi]}{2\kappa(r_{+})},$

with $\Xi$ is a arbitrary parameter and

$\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle\frac{Z\tilde{Z}}{L)}[\tilde{\Omega}-eA_{0}]\nu_{1}+\tilde{Z}[\nu_{1}^{2}]-\frac{m^{2}}{L},$
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle\frac{N\tilde{Z}}{L}[\tilde{\Omega}^{4}-2eA_{0}\tilde{\Omega}^{3}+(eA_{0})^{2}\tilde{\Omega}^{2}]$
$\displaystyle+$
$\displaystyle\frac{Z\tilde{Z}}{LM}[\tilde{\Omega}^{3}-eA_{0}\tilde{\Omega}^{2}]\nu_{1}-\nu_{1}^{4}\tilde{Z}-W_{1}^{4}.$

The modified tunneling probability as of charge bosonic particles as given as

$\Gamma=exp[{-2\pi}\frac{(E-J\Omega-eA_{0})}{\kappa(r_{+})}][1+\alpha\Xi].$
(21)

The corrected $\acute{T_{H}}$ of BH is given as

$\displaystyle\acute{T_{H}}$ $\displaystyle=$
$\displaystyle\frac{[r^{4}q_{i}(2r^{2}+a^{2}cos^{4}\theta)\\{k+g^{2}\mu-a^{2}r^{2}cos^{2}\theta(k+2g^{2}\mu)-a^{4}g^{2}\mu
cos^{4}\theta(q_{i}+2r^{2})+r^{4}a^{2}sin^{2}\theta(q_{i}+2r^{2})\\}]}{2\pi
r^{3}(q_{i}+2r^{2})}$ (22) $\displaystyle[1-\alpha\Xi],$

The modified $\acute{T_{H}}$ of BH depend on metric parameter, quantum gravity
parameter as well as rotation parameter. Moreover the Bekenstein entropy for
this geometry is defined as

$\displaystyle S=2\pi r_{+}^{2}\sqrt{q_{i}+r_{+}^{2}}.$ (23)

##  IV Graphical analysis

In this section, We examine the graphical behaviour of temperature
$\acute{T_{H}}$ versus horizon $r_{+}$ over a range of values for the charge
$q_{i}$ and GUP parameter $\alpha$. Only the physical example accurately
depicts the stable condition of gauge super gravity like BH. The
$\acute{T_{H}}$ strongly increases with the decreasing outer horizon $r+$ and
a small parameter value $\Xi=1$ can generate a small difference in
temperature. Three distinct conditions are expressed by the starting mass,
which really is greater than the remnant mass, depending on different values
of the horizon radius, charge, and GUP parameter. We note that the temperature
of the gauge super gravity like BH quantum gravity of 4 dimensions reduces and
implies with decreasing GUP parameter.

Figure 1: $\acute{T_{H}}$ versus $r_{+}$ for
$\mu=-1,~{}a=0.3,~{}k=0.2,~{}q_{i}=10,~{}g=0.4,~{}\Xi=1$ and
$\alpha=50~{}to~{}150.$ ($i.e:~{}i=1~{}to~{}4$)

##  V Thermal Fluctuations

This section investigates the effects of thermal fluctuations on thermodynamic
quantities of gauge super gravity such as BH. We compute the equations of
thermodynamics quantities under the logarithmic corrections and graph them for
the parameters under consideration. To compute corrected entropy, we first
define the partition function as follows.

$R(\varrho)=\int_{0}^{\infty}\exp(-\varrho\lambda)\rho(\lambda)d\lambda,$ (24)

where $\lambda$ and $\rho(\lambda)$ represent state’s average energy and
density, respectively. We use the inverse Laplace transform of the partition
function defined as to compute $\rho(\lambda)$.

$\rho(\lambda)=\frac{1}{2i\pi}\int_{-i\infty+\varrho_{0}}^{i\infty+\varrho_{0}}\exp(\varrho\lambda)R(\varrho)d\varrho=\frac{1}{2i\pi}\int_{-i\infty+\varrho_{0}}^{i\infty+\varrho_{0}}\exp(\tilde{S}(\varrho))d\varrho,$
(25)

where $\tilde{S}(\varrho)=\beta\lambda+\ln Z(\varrho)$ with $\varrho>0$
depicts the corrected entropy of the BH, which depends on Hawking temperature.
The expression of Taylor series for corrected entropy described as

$\tilde{S}(\varrho)=S+\frac{1}{2}(\varrho-\varrho_{0})^{2}\frac{\partial^{2}\tilde{S}(\varrho)}{\partial\varrho^{2}}\Big{|}_{\varrho=\varrho_{0}}+O(\varrho-\varrho_{o})^{2},$
(26)

putting the Eq.(26) in (25), the equation leads to

$\rho(\lambda)=\frac{1}{\sqrt{2\pi}}\exp(S)\Big{(}\Big{(}\frac{\partial^{2}\tilde{S}(\varrho)}{\partial\varrho^{2}}\Big{)}\Big{|}_{\varrho=\varrho_{0}}\Big{)}^{-\frac{1}{2}},$
(27)

it can be simplified as

$\tilde{S}=S-\frac{1}{2}(\ln S+\ln T^{2})+\frac{\zeta}{S}.$ (28)

The number $\frac{1}{2}$ might be used in place of $\omega$ to enhance the the
influence of corrections terms. So, the expression of corrected entropy is
given as

$\tilde{S}=S-\omega(\ln S+\ln T^{2})+\frac{\zeta}{S}.$ (29)

Under first-order corrections, the corrected entropy relation is modified. The
entropy expression is expressed as by ignoring higher order terms.

$\tilde{S}=S-\omega(\ln S+\ln T^{2}).$ (30)

We use Eqs.(22) and (23) into (30), this leads to

$\displaystyle\tilde{S}$ $\displaystyle=$ $\displaystyle 2\pi
r_{+}^{2}\sqrt{q_{i}+r_{+}^{2}}-\omega\log\Big{(}\Big{(}r^{3}q_{i}\sqrt{q_{i}+r^{2}}\Big{(}a^{2}\cos^{4}\theta+2r_{+}^{2}\Big{)}\Big{(}a^{2}\Big{(}\Big{(}q+2r^{2}\Big{)}\Big{(}r_{+}^{4}\sin^{2}\theta-a^{2}g^{2}\mu\cos^{4}\theta\Big{)}$
(31) $\displaystyle-$ $\displaystyle
r_{+}^{2}\cos^{2}\theta\Big{(}2g^{2}\mu+k\Big{)}\Big{)}+g^{2}\mu+k\Big{)}\Big{)}\Big{(}q+2r_{+}^{2}\Big{)}^{-1}\Big{)}.$

Figure 3: $\tilde{S}$ versus $r_{+}$ for $\mu$=0.4, g=0.4 and $q_{i}$=0.2.

The graphical analysis of corrected entropy for various values of $\omega$ is
shown in figure 3. It is noted that corrected entropy grows with increasing
horizon radius and has a positive trend for all parameter values. It should be
emphasized that the second law of thermodynamics is obeyed for the parameter
values under consideration, and that both parameters have a minor impact on
corrected entropy. Now, using the corrected entropy expression, examine the
other quantities of thermodynamic in the presence of thermal fluctuations. So,
the energy of Helmholtz ($F=-\int\tilde{S}dT$) goes to the form.

$\displaystyle F$ $\displaystyle=$
$\displaystyle-\Big{(}r_{+}^{2}q_{i}\sqrt{q_{i}+r_{+}^{2}}\Big{(}a^{2}\Big{(}r_{+}^{4}\Big{(}q+2r_{+}^{2}\Big{)}{}^{2}\Big{(}-\sin^{2}\theta\Big{)}\Big{(}5a^{2}\cos^{4}\theta+14r_{+}^{2}\Big{)}+r_{+}^{2}\cos^{2}\theta\Big{(}2g^{2}\mu+k\Big{)}\Big{(}a^{2}\cos^{4}\theta\Big{(}3q+2r_{+}^{2}\Big{)}$
(32) $\displaystyle+$ $\displaystyle
2r_{+}^{2}\Big{(}5q+6r_{+}^{2}\Big{)}\Big{)}+\cos^{4}\theta\Big{(}a^{4}g^{2}\mu\cos^{4}\theta\Big{(}q+2r_{+}^{2}\Big{)}^{2}+g^{2}\mu\Big{(}2r_{+}^{2}\Big{(}3a^{2}q^{2}+1\Big{)}+24a^{2}qr_{+}^{4}+24a^{2}r_{+}^{6}-q\Big{)}-k$
$\displaystyle\times$
$\displaystyle\Big{(}q-2r_{+}^{2}\Big{)}\Big{)}\Big{)}-2r_{+}^{2}\Big{(}3q+2r_{+}^{2}\Big{)}\Big{(}g^{2}\mu+k\Big{)}\Big{)}\Big{)}\Big{(}\Big{(}q+2r_{+}^{2}\Big{)}{}^{2}\Big{)}^{-1}.$

Figure 3: $F$ versus $r_{+}$ for $\mu$=0.4, g=0.4 and $q_{i}$=0.2.

The curve of free energy of Helmholtz w.r.t horizon radius is shown in Fig. 3.
The behavior of energy is seen to steadily decrease for the various correction
parameter $\omega$ values. While the uncorrected entropy graph exhibits the
opposite graph increases behavior. This action indicates that the system under
consideration is shifting toward equilibrium, making it unable to extract any
more work from it. For the considered system, the formula for internal energy
($E=F+T\tilde{S}$) is calculated as z7

$\displaystyle E$ $\displaystyle=$
$\displaystyle\Big{(}r_{+}^{2}q_{i}\sqrt{q_{i}+r_{+}^{2}}\Big{(}a^{2}\Big{(}-a^{4}g^{2}\mu\Big{(}r_{+}+1\Big{)}\cos^{4}\theta\Big{(}q+2r_{+}^{2}\Big{)}{}^{2}+\cos^{4}\theta\Big{(}g^{2}\mu\Big{(}-2a^{2}q^{2}\Big{(}r_{+}+3\Big{)}r_{+}^{2}+q$
(33) $\displaystyle\times$
$\displaystyle\Big{(}-8a^{2}r_{+}^{5}-24a^{2}r_{+}^{4}+r_{+}+1\Big{)}-2r_{+}^{2}\Big{(}r_{+}\Big{(}4a^{2}r_{+}^{3}\Big{(}r_{+}+3\Big{)}-1\Big{)}+1\Big{)}\Big{)}+a^{2}(r+5)r_{+}^{4}\Big{(}q+2r_{+}^{2}\Big{)}{}^{2}\sin^{2}\theta$
$\displaystyle+$ $\displaystyle
k\Big{(}q\Big{(}r_{+}+1\Big{)}+2\Big{(}r_{+}-1\Big{)}r_{+}^{2}\Big{)}\Big{)}+r_{+}^{2}\cos^{2}\theta\Big{(}2g^{2}\mu+k\Big{)}\Big{(}a^{2}\Big{(}-\cos^{4}\theta\Big{)}\Big{(}q\Big{(}r_{+}+3\Big{)}+2\Big{(}r_{+}+1\Big{)}r_{+}^{2}\Big{)}$
$\displaystyle-$ $\displaystyle
2r_{+}^{2}\Big{(}q\Big{(}r_{+}+5\Big{)}+2\Big{(}r_{+}+3\Big{)}r_{+}^{2}\Big{)}\Big{)}+2\Big{(}r_{+}+7\Big{)}r_{+}^{6}\Big{(}q+2r_{+}^{2}\Big{)}{}^{2}\sin^{2}\theta\Big{)}+2r_{+}^{2}\Big{(}q\Big{(}r_{+}+3\Big{)}+2\Big{(}r_{+}+1\Big{)}r_{+}^{2}\Big{)}$
$\displaystyle\times$
$\displaystyle\Big{(}g^{2}\mu+k\Big{)}\Big{)}\Big{)}\Big{(}\Big{(}q+2r_{+}^{2}\Big{)}{}^{2}\Big{)}^{-1}.$

Figure 4: $E$ versus $r_{+}$ for $\mu$=0.4, g=0.4 and $q_{i}$=0.2.

In Fig. 4, the internal energy behavior for the various horizon radius options
is depicted graphically. It should be noticed that the graph rapidly lowers
and even shifts to the negative side for tiny values of radii, but the $E$
shows physical behavior. This implies that in order to retain its state, the
considered BH must absorb an increasing amount of heat from its surroundings.
Since BHs are thought to as thermodynamic systems, pressure is another crucial
thermodynamic quantity. The form of the BH pressure ($P=-\frac{dF}{dV}$)
expression when thermal fluctuations are present is

$\displaystyle P$ $\displaystyle=$
$\displaystyle\Big{(}r_{+}q_{i}\Big{(}a^{6}g^{2}\mu\cos^{4}\theta\Big{(}q+2r_{+}^{2}\Big{)}{}^{3}\Big{(}2q_{i}+3r_{+}^{2}\Big{)}+a^{2}\cos^{4}\theta\Big{(}2q_{i}\Big{(}g^{2}\mu\Big{(}12a^{2}q^{3}r_{+}^{2}+q^{2}\Big{(}72a^{2}r_{+}^{4}-1\Big{)}+6q$
(34) $\displaystyle\times$
$\displaystyle\Big{(}24a^{2}r_{+}^{6}+r_{+}^{2}\Big{)}+96a^{2}r_{+}^{8}\Big{)}-kq\Big{(}q-6r_{+}^{2}\Big{)}\Big{)}+r_{+}^{2}\Big{(}g^{2}\mu\Big{(}4r_{+}^{4}\Big{(}45a^{2}q^{2}+1\Big{)}+6qr_{+}^{2}\Big{(}5a^{2}q^{2}+2\Big{)}+360a^{2}qr_{+}^{6}$
$\displaystyle+$ $\displaystyle
240a^{2}r_{+}^{8}-3q^{2}\Big{)}+k\Big{(}-3q^{2}+12qr_{+}^{2}+4r_{+}^{4}\Big{)}\Big{)}-5a^{2}r_{+}^{4}\Big{(}q+2r_{+}^{2}\Big{)}{}^{3}\sin^{2}\theta\Big{(}6q_{i}+7r_{+}^{2}\Big{)}\Big{)}+a^{2}r_{+}^{2}\cos^{2}\theta\Big{(}2g^{2}\mu+k\Big{)}$
$\displaystyle\times$
$\displaystyle\Big{(}a^{2}\cos^{2}\theta\Big{(}4q_{i}\Big{(}3q^{2}+3qr_{+}^{2}+2r_{+}^{4}\Big{)}+15q^{2}r_{+}^{2}+20qr_{+}^{4}+12r_{+}^{6}\Big{)}+2\Big{(}q_{i}\Big{(}30q^{2}r_{+}^{2}+68qr_{+}^{4}+48r_{+}^{6}\Big{)}+35q^{2}r_{+}^{4}+84qr_{+}^{6}$
$\displaystyle+$ $\displaystyle
60r_{+}^{8}\Big{)}\Big{)}-2r_{+}^{2}\Big{(}7a^{2}r_{+}^{4}\Big{(}q+2r_{+}^{2}\Big{)}{}^{3}\sin^{2}\theta\Big{(}8q_{i}+9r_{+}^{2}\Big{)}+\Big{(}g^{2}\mu+k\Big{)}\Big{(}4q_{i}\Big{(}3q^{2}+3qr_{+}^{2}+2r_{+}^{4}\Big{)}+15q^{2}r_{+}^{2}+20qr_{+}^{4}$
$\displaystyle+$ $\displaystyle
12r_{+}^{6}\Big{)}\Big{)}\Big{)}\Big{)}\Big{(}\Big{(}q+2r_{+}^{2}\Big{)}{}^{3}\sqrt{q_{i}+r_{+}^{2}}\Big{)}^{-1}.$

Figure 5: $P$ versus $r_{+}$ for $\mu$=0.4, g=0.4 and $q_{i}$=0.2.

The pressure graph in Fig. 5 is coincides with the point of equilibrium. For
various correction parameter values, the pressure dramatically rises for the
system under consideration. Enthalpy ($H=E+PV$) is a further significant
thermodynamic quantity that is given in Appendix B.

Figure 6: $H$ versus $r_{+}$ for $\mu$=0.4, g=0.4 and $q_{i}$=0.2.

Fig. 6 shows that the corrected enthalpy views with the corrected one plot and
abruptly shifts to the side of negative. This implies that reactions of
exothermic exist and that a significant amount of energy will be released into
the environment. The Gibbs free energy ($G=H-T\tilde{S}$) under the effect of
thermal fluctuations is stated in Appendix B.

Figure 7: $G$ versus $r_{+}$ for $\mu$=0.4, g=0.4 and $q_{i}$=0.2.

From Fig. 7 illustrates the $G$ w.r.t the horizon radius. Positive energy
indicates the non-spontaneous presence processes, which is the system to reach
equilibrium needs more energy. The stability of the system, which is verified
by specific heat ($C_{\tilde{S}}=\frac{dE}{dT}$) is given in Appendix B.

Figure 8: $C_{\tilde{S}}$ versus $r_{+}$ for $\mu$=0.4, g=0.4 and $q_{i}$=0.2.

In the Fig. 8, the behavior of $C_{\tilde{S}}$ is seen in relation to the
horizon radius under the various $\omega$ selections. It is clear that whereas
the quantity of uncorrected represents negative presentation, indicating that
the unstable system, the $C_{\tilde{S}}$ exhibits positive presentation across
the whole examined area. This plot’s positivity is a sign of a stable area. It
is evident that the thermal fluctuations under the correction terms arrives at
the stable of system.

##  VI Discussion

In this paper, the quantum tunneling mechanism of massive boson particles from
a gauge super gravity like BH have examined. First, the particles motion deal
field equation with GUP parameter by using the quantum tunneling approach and
a $(t,r,\theta,\phi)$ kind of coordinate system. To determine the modified
tunneling rate of the bosonic particles, we examined the semi-classical action
in Planck’s constant power series. Then, using this methodology appropriately,
the surface gravity has estimated while neglecting the back reaction
influences of space-time. We reached the realization that the modified
tunneling probability depends not only on the features of BHs and GUP
parameter but also on the properties of the emitted vector.

We investigated the thermodynamic quantities heat capacity, Hawking
temperature, and corrected entropy in gauge super gravity systems. For
stability reasons, we first derived the expressions for entropy, Hawking
temperature, and heat capacity, and then calculated the state’s density using
the inverse Laplace transformation. We investigated various thermodynamic
quantities such as Helmholtz free energy, internal energy, Gibbs free energy,
enthalpy, pressure, and heat capacity using corrected entropy and also checked
stability of the system by two different techniques such as heat capacity and
Hessian matrix trace. Under the influence of first-order corrections, we plot
a corrected entropy graph for $\omega$ parameter values. Throughout the domain
under consideration, the graph of corrected entropy increases monotonically.
The graph (black) of usual entropy increases only for small values of horizon
radius, whereas the corrected expression of energy increases smoothly. As a
result, for small BHs, these correction terms are more effective. Now, using
the corrected entropy expression, examine the other thermodynamic quantities
in the presence of thermal fluctuations.

Furthermore, for different correction parameter $\omega$ values, the behavior
of Helmholtz energy gradually decreases. The entropy, on the other hand,
exhibits the graph increases in the other direction. This observation
indicates that the studied system is approaching equilibrium and no further
work can be produced from it. The graphical behavior of internal energy is
also examined for various horizon radius choices. It is worth noting that for
small radii values, while the studied internal energy displays positive
behaviour, the graph steadily declines and even shifts to the negative side.
The plots of the corrected enthalpy graph match up with the standard enthalpy
graph, which then sharply declines and shifts to the negative side. As a
result, there will be an exothermic reaction, and a significant quantity of
energy will be released into the environment. Furthermore, Fig. 7 depicts the
Gibbs free energy in terms of the horizon radius. Non-spontaneous reactions
suggest that more energy is needed for this system to reach equilibrium. Fig.
8 illustrates how specific heat behaves in relation to horizon radius and
various correction parameter options.

Acknowledgement
PKS acknowledges the National Board for Higher Mathematics (NBHM) under the
Department of Atomic Energy (DAE), Govt. of India for financial support to
carry out the Research project No.: 02011/3/2022 NBHM(R.P.)/R & D II/2152
Dt.14.02.2022. We are very much grateful to the honorable referees and to the
editor for the illuminating suggestions that have significantly improved our
work in terms of research quality, and presentation.

## Appendix A

By computing the Lagrangian equation using the WKB approximation, the set of
field equations is generated as

$\displaystyle\frac{LN}{N+Z^{2}}[c_{1}(\partial_{0}I_{0})(\partial_{1}I_{0})+\alpha
c_{1}(\partial_{0}I_{0})^{3}(\partial_{1}I_{0})-c_{0}(\partial_{1}I_{0})^{2}-c_{0}(\partial_{1}I_{0})^{4}\alpha+eA_{0}(\partial_{1}I_{0})c_{1}+\alpha
eA_{0}(\partial_{0}I_{0})^{2}(\partial_{1}I_{0})c_{1}]$
$\displaystyle-\frac{LZ}{LN+Z^{2}}[c_{3}(\partial_{1}I_{0})^{2}+\alpha
c_{3}(\partial_{1}I_{0})^{4}-c_{1}(\partial_{1}I_{0})(\partial_{3}I_{0})-\alpha
c_{1}(\partial_{1}I_{0})(\partial_{3}I_{0})^{2}]+\frac{N}{M(LN+Z^{2})}[c_{2}(\partial_{0}I_{0})(\partial_{2}I_{0})+\alpha
c_{2}$
$\displaystyle(\partial_{0}I_{0})^{3}(\partial_{2}I_{0})-c_{0}(\partial_{2}I_{0})^{2}-\alpha
c_{0}(\partial_{2}I_{0})^{4}+c_{2}eA_{0}(\partial_{2}I_{0})+c_{2}eA_{0}\alpha(\partial_{0}I_{0})^{2}(\partial_{1}I_{0})]+\frac{LN}{(LN+Z^{2})^{2}}[c_{3}(\partial_{0}I_{0})(\partial_{3}I_{0})$
$\displaystyle+\alpha
c_{3}(\partial_{0}I_{0})^{3}(\partial_{3}I_{0})-c_{0}(\partial_{3}I_{0})^{2}-\alpha
c_{0}(\partial_{3}I_{0})^{4}+c_{3}eA_{0}(\partial_{3}I_{0})+c_{3}eA_{0}(\partial_{0}I_{0})^{2}(\partial_{3}I_{0})]-m^{2}\frac{Nc_{0}-Zc_{3}}{LN+Z^{2}}=0,$
(35) $\displaystyle\frac{-LN}{LN+Z^{2}}[c_{1}(\partial_{0}I_{0})^{2}+\alpha
c_{1}(\partial_{0}I_{0})^{4}-c_{0}(\partial_{0}I_{0})(\partial_{1}I_{0})-\alpha
c_{0}(\partial_{0}I_{0})(\partial_{1}I_{0})^{3}+c_{1}eA_{0}(\partial_{0}I_{0})+\alpha
c_{1}eA_{0}(\partial_{0}I_{0})^{3}]+$
$\displaystyle\frac{LZ}{LN+Z^{2}}[c_{3}(\partial_{0}I_{0})(\partial_{1}I_{0})+\alpha
c_{3}(\partial_{0}I_{0})(\partial_{1}I_{0})^{3}-c_{1}(\partial_{0}I_{0})(\partial_{3}I_{0})-\alpha
c_{1}(\partial_{0}I_{0})(\partial_{3}I_{0})^{3}]+\frac{L}{M}[c_{2}(\partial_{1}I_{0})(\partial_{2}I_{0})+\alpha
c_{2}$
$\displaystyle(\partial_{1}I_{0})(\partial_{2}I_{0})^{3}-c_{1}(\partial_{2}I_{0})^{2}-\alpha
c_{1}(\partial_{2}I_{0})^{4}]+\frac{L}{LN+Z^{2}}[c_{3}(\partial_{1}I_{0})(\partial_{3}I_{0})+\alpha
c_{3}(\partial_{1}I_{0})(\partial_{3}I_{0})^{3}-c_{1}(\partial_{3}I_{0})^{2}-\alpha
c_{1}(\partial_{3}I_{0})^{4}]$
$\displaystyle-m^{2}Lc_{1}+\frac{eA_{0}LN}{LN+Z^{2}}[c_{1}(\partial_{0}I_{0})+\alpha
c_{1}(\partial_{0}I_{0})^{3}-c_{0}(\partial_{1}I_{0})-\alpha
c_{0}(\partial_{1}I_{0})^{3}+eA_{0}c_{1}+\alpha
c_{1}eA_{0}(\partial_{0}I_{0})^{2})]+$
$\displaystyle\frac{eA_{0}LZ}{LN+Z^{2}}[c_{3}(\partial_{1}I_{0})+\alpha
c_{3}(\partial_{1}I_{0})^{3}-c_{1}(\partial_{3}I_{0})-\alpha
c_{1}(\partial_{1}I_{0})^{3}]=0,$ (36)
$\displaystyle\frac{N}{M(LN+Z^{2})}[c_{2}(\partial_{0}I_{0})^{2}+\alpha
c_{2}(\partial_{0}I_{0})^{4}-c_{0}(\partial_{0}I_{0})(\partial_{2}I_{0})-\alpha
c_{0}(\partial_{0}I_{0})(\partial_{2}I_{0})^{3}+c_{2}eA_{0}(\partial_{0}I_{0})+\alpha
c_{2}eA_{0}(\partial_{0}I_{0})^{3}]$
$\displaystyle+\frac{L}{M}[c_{2}(\partial_{1}I_{0})^{2}+\alpha
c_{2}(\partial_{1}I_{0})^{4}-c_{1}(\partial_{1}I_{0})(\partial_{2}I_{0})-\alpha
c_{1}(\partial_{1}I_{0})(\partial_{2}I_{0})^{3}]-\frac{Z}{M(LN+Z^{2})}[c_{2}(\partial_{0}I_{0})(\partial_{3}I_{0})+\alpha
c_{2}(\partial_{0}I_{0})^{3}$
$\displaystyle(\partial_{3}I_{0})-c_{0}(\partial_{0}I_{0})(\partial_{3}I_{0})-\alpha
c_{0}(\partial_{0}I_{0})^{3}(\partial_{3}I_{0})+c_{2}eA_{0}(\partial_{3}I_{0})+\alpha
c_{2}eA_{0}(\partial_{3}I_{0})^{3}]+\frac{L}{M(LN+Z^{2})}[c_{3}(\partial_{2}I_{0})(\partial_{3}I_{0})$
$\displaystyle+\alpha
c_{3}(\partial_{2}I_{0})^{3}(\partial_{3}I_{0})-c_{2}(\partial_{3}I_{0})^{2}-\alpha
c_{2}(\partial_{3}I_{0})^{4}]-\frac{m^{2}c_{2}}{M}+\frac{eA_{0}N}{M(LN+Z^{2})}[c_{2}(\partial_{0}I_{0})+\alpha
c_{2}(\partial_{0}I_{0})^{3}-(\partial_{2}I_{0})c_{0}$
$\displaystyle-(\partial_{2}I_{0})^{3}c_{0}\alpha+eA_{0}c_{2}+eA_{0}\alpha(\partial_{0}I_{0})^{2}c_{2}]=0,$
(37)
$\displaystyle\frac{LN-L^{2}}{(LN+Z^{2})^{2}}[(\partial_{0}I_{0})^{2}c_{3}+\alpha(\partial_{0}I_{0})^{4}c_{3}-(\partial_{0}I_{0})(\partial_{3}I_{0})c_{0}-\alpha(\partial_{0}I_{0})(\partial_{3}I_{0})^{3}c_{0}+{eA_{0}c_{3}}(\partial_{0}I_{0})+\alpha
c_{3}eA_{0}(\partial_{0}I_{0})^{3}]$
$\displaystyle-\frac{N}{M(LN+Z^{2})}[c_{3}(\partial_{1}I_{0})^{2}+\alpha
c_{3}(\partial_{1}I_{0})^{4}-c_{1}(\partial_{1}I_{0})(\partial_{3}I_{0})-\alpha
c_{1}(\partial_{1}I_{0})(\partial_{3}I_{0})^{3}]-\frac{Z}{M(LN+Z^{2})}[c_{2}(\partial_{0}I_{0})(\partial_{2}I_{0})$
$\displaystyle+\alpha
c_{2}(\partial_{0}I_{0})^{3}(\partial_{2}I_{0})-c_{0}(\partial_{2}I_{0})^{2}+\alpha
c_{0}(\partial_{2}I_{0})^{4}+{eA_{0}c_{2}}(\partial_{2}I_{0})+\alpha
c_{2}eA_{0}(\partial_{0}I_{0})^{2}(\partial_{2}I_{0})]-\frac{eA_{0}L}{M(LN+Z^{2})}$
$\displaystyle[c_{3}(\partial_{2}I_{0})^{2}+\alpha
c_{3}(\partial_{2}I_{0})^{4}-c_{2}(\partial_{2}I_{0})(\partial_{3}I_{0})-\alpha
c_{2}(\partial_{0}I_{0})(\partial_{3}I_{0})^{3}]-\frac{m^{2}(Zc_{0}-Lc_{3})}{LN+Z^{2}}+\frac{eA_{0}(LN-L^{2})}{(LN+Z^{2})^{2}}[c_{3}(\partial_{0}I_{0})+\alpha
c_{3}$
$\displaystyle(\partial_{0}I_{0})^{3}-(\partial_{3}I_{0})c_{0}-\alpha(\partial_{3}I_{0})^{3}c_{0}+eA_{0}c_{3}+\alpha
eA_{0}(\partial_{0}I_{0})^{2}c_{3}]=0.$ (38)

## Appendix B

Enthalpy is computed as

$\displaystyle H$ $\displaystyle=$
$\displaystyle(r_{+}q_{i}(4r_{+}(2r_{+}^{2}+q)((2\sin^{2}\theta(r_{+}+7)(2r_{+}^{2}+q)^{2}r_{+}^{6}+(2\mu
g^{2}+k)\cos^{2}\theta(-\cos^{4}\theta(2(r_{+}+1)r_{+}^{2}$ (39)
$\displaystyle+q(r_{+}+3))a^{2}-2r_{+}^{2}(2(r_{+}+3)r_{+}^{2}+q(r_{+}+5)))r_{+}^{2}-a^{4}g^{2}\mu\cos^{4}\theta(r_{+}+1)(2r_{+}^{2}+q){}^{2}+$
$\displaystyle\cos^{4}\theta(a^{2}\sin^{2}\theta(r_{+}+5)(2r_{+}^{2}+q)^{2}r_{+}^{4}+k(2(r_{+}-1)r_{+}^{2}+q(r_{+}+1))+g^{2}\mu(-2a^{2}q^{2}(r_{+}+3)r_{+}^{2}$
$\displaystyle-2(r_{+}(4a^{2}r_{+}^{3}(r_{+}+3))+1)r_{+}^{2}+q(-8a^{2}r_{+}^{5}-24a^{2}r_{+}^{4}+r_{+}+1))))a^{2}+2(\mu
g^{2}+k)r_{+}^{2}$
$\displaystyle(2(r_{+}+1)r_{+}^{2}+q(r_{+}+3)))(r_{+}^{2}+q_{i})-(3g^{2}(g^{2}\mu\cos^{4}\theta(2r_{+}^{2}+q)^{3}(3r_{+}^{2}+2q_{i})a^{6}+\cos^{4}\theta$
$\displaystyle(-5a^{2}\sin^{2}\theta(2r_{+}^{2}+q)^{3}(7r_{+}^{2}+6q_{i})r_{+}^{4}+(\mu(240a^{2}r_{+}^{8}+360a^{2}qr_{+}^{6}+4(45a^{2}q^{2}+1)r_{+}^{4}+6q(5a^{2}q^{2}+2)$
$\displaystyle
r_{+}^{2}-3q^{2})g^{2}+k(-3q^{2}+4r_{+}^{2}))r_{+}^{2}+2(g^{2}\mu(96a^{2}r_{+}^{8}+12a^{2}q^{3}r_{+}^{2}+q^{2}(72a^{2}r_{+}^{4}-1)+6q(24a^{2}r_{+}^{6}+r_{+}^{2}))$
$\displaystyle-kq(q-6r_{+}^{2}))q_{i})a^{2}+(2\mu g^{2}+k)\cos^{2}\theta
r_{+}^{2}(\cos^{4}\theta(12r_{+}^{6}+20qr_{+}^{4}+15q^{2}r_{+}^{2}+4(2r_{+}^{4}+3qr_{+}^{2}+3q^{2})q_{i})$
$\displaystyle
a^{2}+2(60r_{+}^{8}+84qr_{+}^{6}+35q^{2}r_{+}^{4}+(48r_{+}^{6}+68qr_{+}^{4}+30q^{2}r_{+}^{2})q_{i}))a^{2}-2r_{+}^{2}(7a^{2}\sin^{2}\theta(2r_{+}^{2}+q)^{3}$
$\displaystyle(9r_{+}^{2}+8q_{i})r_{+}^{4}+(\mu
g^{2}+k)(12r_{+}^{6}+20qr_{+}^{4}+15q^{2}r_{+}^{2}+4(2r_{+}^{4}+3qr_{+}^{2}+3q^{2})q_{i}))))(\pi)^{-1}))$
$\displaystyle(4(q+2r_{+}^{2})^{3}\sqrt{r_{+}^{2}+q_{i}})^{-1}.$

The Gibbs free energy is stated as

$\displaystyle G$ $\displaystyle=$ $\displaystyle(r_{+}^{4}(\cos^{4}\theta
a^{2}+2r_{+}^{2})(((2r_{+}^{2}+q)(\sin^{2}\theta
r_{+}^{4}-a^{2}g^{2}\mu\cos^{4}\theta)-r^{2}(2\mu
g^{2}+k)\cos^{2}\theta)a^{2}+k+g^{2}\mu)$ (40) $\displaystyle
q_{i}^{2}(4r_{+}(2r_{+}^{2}+q)((2\cos^{2}\theta(r_{+}+7)(2r_{+}^{2}+q)^{2}r_{+}^{6}+(2\mu
g^{2}+k)\cos^{2}\theta(-\cos^{4}\theta(2(r_{+}+1)r_{+}^{2}+$ $\displaystyle
q(r_{+}+3))a^{2}-2r_{+}^{2}(2(r_{+}+3)r_{+}^{2}+q(r_{+}+5)))r_{+}^{2}-a^{4}g^{2}\mu\cos^{2}\theta(r_{+}+1)(2r_{+}^{2}+q){}^{2}+\cos^{4}\theta$
$\displaystyle(a^{2}\sin^{2}\theta(r_{+}+5)(2r_{+}^{2}+q)^{2}+k(2(r_{+}-1)r_{+}^{2}+q(r_{+}+1))+g^{2}\mu(-2a^{2}q^{2}(r_{+}+3)r_{+}^{2}-2$
$\displaystyle(r_{+}(4a^{2}r_{+}^{3}(r_{+}+3)-1)+1)r_{+}^{2}+q(-8a^{2}r_{+}^{5}-24a^{2}r_{+}^{4}+r_{+}+1))))a^{2}+2(\mu
g^{2}+k)r_{+}^{2}(2(r_{+}+1)$ $\displaystyle
r_{+}^{2}+q(r_{+}+3)))(r_{+}^{2}+q_{i})-(3g^{2}(g^{2}\mu\cos^{4}\theta(2r_{+}^{2}+q)^{3}(3r_{+}^{2}+2q_{i})a^{6}+\cos^{4}\theta(-5a^{2}\sin^{2}\theta$
$\displaystyle(2r_{+}^{2}+q)^{3}(7r_{+}^{2}+6q_{i})r_{+}^{4}+(\mu(240a^{2}r_{+}^{8}+360a^{2}qr_{+}^{6}+4(45a^{2}q^{2}+1)r_{+}^{4}+6q(5a^{2}q^{2}+2)r_{+}^{2}-3q^{2})g^{2}$
$\displaystyle+k(4r_{+}^{4}+12qr_{+}^{2}-3q^{2}))r_{+}^{2}+2(g^{2}\mu(96a^{2}r_{+}^{8}+12a^{2}q^{3}r_{+}^{2}+q^{2}(72a^{2}r_{+}^{4}-1)+6q(24a^{2}r_{+}^{6}+r_{+}^{2}))-kq$
$\displaystyle(q-6r_{+}^{2}))q_{i})a^{2}+(2\mu g^{2}+k)\cos^{2}\theta
r_{+}^{2}(\cos^{4}\theta(12r_{+}^{6}+20qr_{+}^{4}+15q^{2}r_{+}^{2}+4(2r_{+}^{4}+3qr_{+}^{2}+3q^{2})q_{i})a^{2}$
$\displaystyle+2(60r_{+}^{8}+84qr_{+}^{6}+35q^{2}r_{+}^{4}+(48r_{+}^{6}+68qr_{+}^{4}+30q^{2}r_{+}^{2})q_{i}))a^{2}-2r_{+}^{2}(7a^{2}\sin^{2}\theta(2r_{+}^{2}+q)^{3}$
$\displaystyle(9r_{+}^{2}+8q_{i})r_{+}^{4}+(\mu
g^{2}+k)(12r_{+}^{6}+20qr_{+}^{4}+15q^{2}r_{+}^{2}+4(2r_{+}^{4}+3qr_{+}^{2}+3q^{2})q_{i}))))(\pi)^{-1}))$
$\displaystyle(4(q)^{4})^{-1}.$

The specific heat is given as

$\displaystyle C_{\tilde{S}}$ $\displaystyle=$
$\displaystyle-(2\text{$\pi$r}_{+}(-g^{2}\mu\cos^{4}\theta(2r_{+}^{2}+q)^{3}((4r_{+}+3)r_{+}^{2}+(3r_{+}+2)q_{i})a^{6}+\cos^{4}\theta(a^{2}\sin^{2}\theta(2r_{+}^{2}+q)^{3}$
(41)
$\displaystyle((8r_{+}+35)r_{+}^{2}+(7r_{+}+30)q_{i})r_{+}^{4}+(k(4(2r_{+}-1)r_{+}^{4}+12q(r_{+}-1)r_{+}^{2}+q^{2}(4r_{+}+3))-g^{2}\mu$
$\displaystyle(4(24a^{2}r_{+}^{5}+60a^{2}r_{+}^{4}-2r_{+}+1)r_{+}^{4}+6a^{2}q^{3}(2r_{+}+5)r_{+}^{2}+12q(r_{+}(6a^{2}r_{+}^{3}(2r_{+}+5)-1)+1)r_{+}^{2}$
$\displaystyle+q^{2}(4r_{+}(9a^{2}r_{+}^{3}(2r_{+}+5)-1)-3)))r_{+}^{2}+(k(4r_{+}^{5}+4q(2r_{+}-3)r_{+}^{2}+q^{2}(3r_{+}+2))-g^{2}\mu$
$\displaystyle(4(4a^{2}r_{+}^{3}(5r_{+}+12)-1)r_{+}^{5}+2a^{2}q^{3}(5r_{+}+12)r_{+}^{2}+4q(30a^{2}r_{+}^{5}+72a^{2}r_{+}^{4}-2r_{+}^{2}+3)r_{+}^{2}+q^{2}$
$\displaystyle(3r_{+}(4a^{2}r_{+}^{3}(5r_{+}+12)-1)-2)))q_{i})-(2\mu
g^{2}+k)\cos^{2}\theta r_{+}^{2}(2(12(2r_{+}^{2}+5)r_{+}^{4}+28q(r_{+}+3)$
$\displaystyle
r_{+}^{2}+q^{2}(8r_{+}+35))r_{+}^{4}+2(4(5r_{+}+12)r_{+}^{4}+4q(6r_{+}+17)r_{+}^{2}+q^{2}(7r_{+}+30))q_{i}r_{+}^{2}+a^{2}\cos^{4}\theta$
$\displaystyle((4(4r_{+}+3)r_{+}^{4}+20q(r_{+}+1)r_{+}^{2}+3q^{2}(2r_{+}+5))r_{+}^{2}+(4(3r_{+}+2)r_{+}^{4}+(4r_{+}+3)+q^{2}$
$\displaystyle(5r_{+}+12))q_{i}))a^{2}+2r_{+}^{2}(a^{2}\sin^{2}\theta(2r_{+}^{2}+q)^{3}((10r_{+}+63)r_{+}^{2}+(9r_{+}^{2}+56))r_{+}^{4}+(\mu
g^{2}+k)$
$\displaystyle((4(4r_{+}+3)r_{+}^{4}+20q(r_{+}+1)r_{+}^{2}+3q^{2}(2r_{+}^{2}+5))r_{+}^{2}+(4(3r_{+}+2)r_{+}^{4}+4q(4r_{+}+3)r_{+}^{2}$
$\displaystyle+q^{2}(5r_{+}+12))q_{i}))))((2r_{+}^{2}+q)(a^{2}(-\sin^{2}\theta(2r_{+}^{2}+q)^{2}\Big{(}5\cos^{4}\theta
a^{2}+14r_{+}^{2})r_{+}^{4}+(2\mu g^{2}+k)$
$\displaystyle\cos^{2}\theta(\cos^{4}\theta(2r_{+}^{2}+3q)a^{2}+2r_{+}^{2}(6r_{+}^{2}+5q))r_{+}^{2}+\
\cos^{4}\theta(g^{2}\mu\cos^{4}\theta(2r_{+}^{2}+q){}^{2}a^{4}-k(q-2r_{+}^{2})$
$\displaystyle+g^{2}\mu(24a^{2}r_{+}^{6}+24a^{2}qr_{+}^{4}+2(3a^{2}q^{2}+1)r_{+}^{2}-q)))-2(\mu
g^{2}+k)r_{+}^{2}(2r_{+}^{2}+3q))\sqrt{r_{+}^{2}+q_{i}})^{-1}.$

## References

  * (1) H. Erbin, Universe 3 (2017) 19.
  * (2) M. Azreg-Aïnou, Phys. Lett. B 730 (2014) 95.
  * (3) R. Ali, R. Babar, M. Asgher and T. C. Xia, Int. J. Mod. Phys. A 37 (2022) 2250108.
  * (4) R. Ali, R. Babar and M. Asgher, Annalen der Physik, 2200074 (2022) 12.
  * (5) R. Ali, R. Babar and P. K. Sahoo, Phys. Dark Universe, 35 (2022) 100948.
  * (6) R. Ali, R. Babar, M. Asgher and G. Mustafa, Int. J. Mod. Phys. A 37 (2022) 2250134.
  * (7) E.T. Newman and A.I. Janis, J. Math. Phys. 6 (1965) 915.
  * (8) E.T. Newman, et al, J. Math. Phys. 6 (1965) 918.
  * (9) D.Y. Xu, Class. Quantum Gravity 5 (1988) 871.
  * (10) H. Kim, Phys. Rev. D 59 (1999) 064002.
  * (11) S. Yazadjiev, Gen. Relativ. Gravit. 32 (2000) 2345.
  * (12) M. Azreg-Aïnou, Eur. Phys. J. C 74 (2014) 2865.
  * (13) L. Herrera and J. Jimenez, J. Math. Phys.23 (1982) 2339.
  * (14) S.P. Drake and R. Turolla, Class. Quantum Gravity 14 (1997) 1883.
  * (15) N. Ibohal, Gen. Relativ. Gravit. bf 37 (2005) 19.
  * (16) M. Azreg-Aïnou, Phys. Rev. D 90 (2014) 064041.
  * (17) A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52 (1995) 1108.
  * (18) S.W. Hawking, Commun. Math. Phys. 43 (1975) 199.
  * (19) M. Sharif and W. Javed, Eur. Phys. J. C 72 (2012) 1997.
  * (20) T. Damoar and R. Ruffini, Phys. Rev. D 14 (1976) 332.
  * (21) W. Javed, G. Abbas and R. Ali, Eur. Phys. J. C 77 (2017) 296.
  * (22) A. Övgün, W. Javed and R. Ali, Advance in High Energy Physics. 2018 (2018) 11.
  * (23) W. Javed, R. Ali and G. Abbas, Can. J. Phys. 97 (2018) 176.
  * (24) W. Javed, R. Ali, R. Babar, A. Övgün, Eur. Phys. J. Plus 134 (2019) 511.
  * (25) W. Javed, R. Ali, R. Babar, A. Övgün, Chinese Physics C 144 (2020) 015104.
  * (26) G. Johnson, JHEP 03 (2020) 038.
  * (27) R. Ali, K. Bamba and S.A.A. Shah, Symmetry. 631 (2019) 11.
  * (28) R. Ali, K. Bamba, M. Asgher, M.F. Malik and S.A.A. Shah, Symmetry. 1165 (2020) 12.
  * (29) R. Ali, M. Asgher and M.F. Malik, Mod. Phys. Lett. A 35 (2020) 2050225.
  * (30) M. Rizwan, M.Z. Ali and A. Övgün, Mod. Phys. Lett. A 34 (2019) 1950184.
  * (31) R. Ali, K. Bamba, M. Asgher and S.A.A. Shah, Int. J. Mod. Phys. D 30 (2021) 2150002.
  * (32) R. Ali, K. Bamba, S.A.A. Shah and M.J. Saleem,, Int. J. Mod. Phys. D 31 (2022) 2250069.
  * (33) W. Javed, R. Babar and A. Övgün, Mod. Phys. Lett. A 34 (2019) 1950057.
  * (34) W. Javed and R. Babar, Chin. J. Phys. 61 (2019) 138.
  * (35) R. Babar, W. Javed and A. Övgün, Mod. Phys. Lett. A 35 (2020) 2050104.
  * (36) A. Kempf, J. Phys. A 30 (1997) 2093.
  * (37) F. Brau, J. Phys. A 32 (1999) 7691.
  * (38) A.F. Ali, S. Das and E.C. Vagenas, Phys. Lett. B 678 (2009) 497.
  * (39) A. Övgün and K. Jusufi, Eur. Phys. J. Plus 132 (2017) 298.
  * (40) M. Faizal and M.M. Khalil, Int. J. Mod. Phys. A 30 (2015) 1550144.
  * (41) B. Pourhassan and M. Faizal, Nucl. Phys. B 913 (2016) 834.
  * (42) A. Jawad and M.U. Shahzad, Eur. Phys. J. C 77 (2017) 349.
  * (43) M. Zhang, Nucl. Phys. B 935 (2018) 170.
  * (44) K. Ghaderi and B. Malakolkalami, Nucl. Phys. B 903 (2016) 10.
  * (45) R. Ali, R. Babar, M. Asgher and S.A.A. Shah, Annals of Physics, 432 (2021) 168572.
  * (46) R. Ali and M. Asgher, New Astronomy 93 (2022) 101759.
  * (47) R. Ali, R. Babar, M. Asgher and S.A.A. Shah, Int. J. Geom. Methods Mod. Phys. 19 (2022) 2250017.
  * (48) M. Sharif and Z. Akhtar, Phys. Dark Universe 29 (2020) 100589.
  * (49) M. Sharif and Z. Akhtar, Chin. J. Phys 71 (2021) 669.
  * (50) W. Javed, et al, Mod. Phys. Lett. A 33 (2018) 1850089.
  * (51) Z. Yousaf, et al, Int. J. Geom. Methods Mod. 19 (2022) 2250102.
  * (52) R. Babar, M. Asgher and R. Ali, Phys. Scr. 97(2022)125201.
  * (53) Z. Akhtar, A. Khan, Z. Ahmad, R. Ali, New Astronomy 98(2023)101936.
  * (54) J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, 1992.
  * (55) G.W. Gibbons, Supersymmetric Soliton States in Extended Supergravity Theories, in: P. Breitenlohner and H.P. Dürr (eds.), Unified Theories of Elementary Particles, Lecture Notes in Physics 160, Springer, 1982.
  * (56) A. Gallerati, Int. J. Mod. Phys. A 34(2019) 1930017.
  * (57) M. Cvetic and S.S. Gubser, JHEP 04 (1999) 024.
  * (58) K. Behrndt, M. Cvetic and W.A. Sabra, Nucl. Phys. B 553 (1999) 317.
  * (59) Z. Xu, X. Hou, X Gong and J. Wang Eur. Phys. J. C 78(2018) 513.
  * (60) H.C.D.L. Junior, L.C.B. Crispino, P.V.P. Cunha and C.A.R. Herdeiro, Eur. Phys. J. C 80(2020) 1036.
  * (61) C.Y. CHEN, Int. J. Geom. Methods Mod. 19(2022) 2250176.
  * (62) R.C. Pantig and E.T. Rodulfo, Chin. J. Phys. 68(2020) 236.
  * (63) P. Pradhan, Universe 5 (2019) 57.