aainstitutetext: Department of Mathematics,
Manipur University, Canchipur, Imphal, Manipur, India

# Massive vector particle tunneling from Kerr-Newman-de Sitter black hole
under generalized uncertainty principle

Yenshembam Priyobarta Singh,111Corresponding author. a,1 Telem Ibungochouba
Singh<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The quantum tunneling of charged massive vector boson particles across the
event horizon of Kerr-Newman-de Sitter black hole is investigated under the
influence of quantum gravity effects.  The modified Hawking temperatures and
heat capacities across the event horizon of KNdS black hole are derived in
3-dimensional and 4-dimensional frame dragging coordinates. It is found that
due to quantum gravity effects the modified Hawking temperatures and heat
capacities depend on the mass and angular momentum of the emitted vector boson
particles. For 3-dimensional KNdS black hole, the modified Hawking temperature
is lower than the original Hawking temperature but the modified heat capacity
is higher than the original heat capacity due to quantum gravity effects. In
the case of 4-dimensional KNdS black hole, the modified Hawking temperature
and heat capacity are lower or greater than the original Hawking temperature
and heat capacity depending upon the choices of black hole parameters due to
quantum gravity effects. We also discuss the remnant and graphical analysis of
the modified Hawking temperatures and heat capacities.

###### Keywords:

Quantum tunneling; quantum gravity effects; Kerr-Newman-de Sitter black hole;
Hawking temperatures and heat capacities.

##  1 Introduction

In the early 1970s, Hawking proposed a black hole radiation called Hawking
radiation using quantum field theory techniques on a curve space-time
background hawking1 ; hawking2 . The discovery of Hawking radiation is one of
the significant developments in gravitational physics. It shows that black
holes have a thermodynamical features. Refs. bekenstein1 ; bekenstein2
determined that the horizon area and the black hole’s entropy are
proportional. More precisely, the black hole’s entropy is equal to one-fourth
of its horizon area. Since then, theoretical physicists have paid a lot of
attention to Hawking radiation and a number of approaches have been used to
determine Hawking radiation.

Generally, there are two approaches to investigate the imaginary part of the
action. They are the radial null geodesic method and the Hamilton-Jacobi
method. The radial null geodesic method was put forth by Parikh and Wilczek
kraus1 ; kraus2 ; kraus3 ; parikh1 . In this method, the emitted particles act
as the potential barrier and the semiclassical WKB approximation is used to
determine the imaginary part of the radial action. Later, Zhang and Zhao
zhang1 ; zhang2 ; zhang3 have made significant progress by extending Parikh-
Wilczek method to non-spherical symmetric stationary black holes and the
radiation of charged massive particle. Moreover, using the Parikh-Wilczek
method, extensive investigations of many types of black holes have been
conducted vagenas2 ; vagenas3 ; v4 ; vagenas4 ; a3 . The second is the
Hamilton-Jacobi method, used by Angheben et al. angheben1 and Vagenas vagenas1
as an extension of the complex path integral method introduced by Padmanabhan
et al. padmanabhan1 ; padmanabhan2 ; padmanabhan3 . In both methods, the WKB
approximation is used to investigate the tunneling probability for classically
forbidden trajectory from inside to outside the horizon and it is given by
$\Gamma=\exp\left[-\frac{2}{\hbar}\text{Im}~{}I\right]$, where $I$ and $\hbar$
denote the semiclassical action of the outgoing particle and Planck’s
constant. Kerner and Mann kerner1 ; kerner2 investigated the tunneling of
spin-1/2 fermions particle from rotating and non-rotating black holes using
Dirac equation, Pauli sigma matrix and Fenyman prescription. By the fermions
tunneling method the Hawking radiation from the black ring is also derived
jiang1 . Using their method, many interesting results have been obtained in
ren1 ; wang1 ; rahaman1 ; ibungochouba1 . Damour and Ruffini damour1 used a
new approach called tortoise coordinate transformation to study Hawking
radiation. Later, Sannan sannan1 extended the work of Damour and Ruffini by
deriving the probability distributions for boson and fermions emission from a
black hole. Moreover, topological approach has been used to derive Hawking
temperature of black holes robson1 ; robson2 ; ali11 ; ali12 ; ali13 .

The existence of a minimum measurable length that can be identified with the
order of the Planck scale is predicted by numerous quantum gravity theories,
such as string theory, loop quantum gravity, non-commutative geometry and
Gedanken experiments townsend1 ; yoneya1 ; konishi1 ; maggiore1 ; garay1 ;
scardigli1 ; amelino1 . Constructing new theoretical models is one of the
fields of study for quantum gravity. The Generalized Uncertainty Principle
(GUP) is a modified theory that realizes this minimum length. It is the
generalization of the Heisenberg uncertainty principle (HUP). The modified
fundamental commutation relation proposed by Kempf et al. kempf1 is of the
form

$\displaystyle\bigl{[}x_{i},p_{j}\bigr{]}=i\hbar\delta_{i,j}\bigl{[}1+\beta
p^{2}\bigr{]},$ (1)

where the positions $x_{i}$ and momentum operators $p_{j}$ are defined by

$\displaystyle x_{i}=x_{0i},~{}~{}~{}~{}~{}p_{j}=p_{0j}\bigl{(}1+\beta
p_{0}^{2}\bigr{)}.$ (2)

Then $x_{0i}$ and $p_{0j}$ satisfy the canonical commutation relations as
$\bigl{[}x_{0i},p_{0j}\bigr{]}=i\hbar~{}\delta_{ij}$. The corresponding GUP
takes the form

$\displaystyle\Delta x\Delta p\geq\frac{\hbar}{2}\bigl{[}1+(\Delta
p)^{2}\beta\bigr{]},$ (3)

where
$\beta=\frac{\beta_{0}l^{2}_{p}}{\hbar^{2}}=\frac{\beta_{0}}{M_{p}^{2}c^{2}}$.
Here $\beta_{0}(\leq 10^{34})$, $l_{p}$ and $M_{p}$ represent the
dimensionless parameter of order unity, Planck length and Planck mass
respectively.

The implications of the aspects of GUP have been investigated in many contexts
such as modifications of the quantum Hall effect, unruh effect, neutrino
oscillations, Landau levels, Newton’s law, cosmology, and the weak equivalence
principle (WEP) das1 ; ali1 ; zhu1 ; sprenger1 ; tawfik1 ; ali2 ; vagnozzi ;
s2 ; s3 . Adler et al. adler1 studied the influence of GUP on the
thermodynamics of the Schwarzschild black hole. Later, Ref. ovgunR
investigated the tunneling of scalar and fermion particles from a
Schwarzschild black hole immersed in an electromagnetic universe under the
effect of GUP . Further, using the Parikh-Wilczek tunnelling approach, Nozari
and Saghaf nozari1 studied the Hawking radiation for massless scalar particles
in the background geometry of Schwarzschild black hole and retrieved the
tunnelling rate as well as the corrected Hawking temperature by taking GUP
into account. It should be noted that the GUP has also influenced the
thermodynamics of black holes. As a result, the GUP concept has been applied
to various black holes in order to study their thermodynamics properties a2 .
Applying the WKB approximation to the Dirac equation, the tunnelling of
fermions from the Kerr and Kerr-Newman black holes are studied in li1 ; jian1
. Yale yale1 investigated the tunneling of arbitrary scalars, fermions and
spin-1 bosons by ignoring back-reaction effects. Banerjee and Majhi banerjee1
; banerjee2 discussed the tunneling of fermions and bosons at the event
horizon of black holes beyond the semiclassical approximation. They derived
the modified Hawking temperature and change in black hole entropy.
Ibungochouba et al. ibungochouba2 studied fermions quantum tunneling from the
BTZ black hole. Li li2 studied the tunneling of massive spin-1 particle from
Reissner-Nordstrom and Kerr black hole under the effect of quantum gravity.
Övgün et al. ovgun1 investigated the charged massive bosons tunneling from
noncommutative charged black holes.

GUP prevents a black hole from complete evaporation as the black hole’s mass
approaches the Planck scale. This late stage of a black hole under Hawking
evaporation is referred as black hole remnant. A black hole remnant consists
of a black hole phase that evaporates under the Hawking radiation, which is
either stable or long-lived/meta-stable. Many researchers have investigated
the black hole remnant in adler ; adler2 ; chen1 ; chen2 ; myung1 ;
gangopadhyay1 ; feng1 ; eslamR . In the context of the information loss
paradox, the study of black hole remnant plays a crucial role unruh ; chenR .

This article’s layout is constructed in the following manner: In section 2, we
revisit the generalized field equation for massive vector boson particles. The
quantum tunneling of massive vector boson from KNdS black hole under quantum
gravity effects is presented in section 3. Further, section 4 provides the
analysis of the remnant of 3-dimensional KNdS black hole under quantum gravity
effects. The graphical analysis of quantum corrected Hawking temperatures and
heat capacities are presented in section 5. Lastly, some concluding remarks
are presented in section 6.

##  2 Generalized field equations for massive vector bosons

The GUP-corrected Lagrangian of massive vector field $\Psi_{\mu}$ is given by
li2

$\displaystyle\mathfrak{L}_{GUP}=$
$\displaystyle-\frac{1}{2}\left(\mathfrak{D}^{+}_{\mu}\Psi^{+}_{\nu}-\mathfrak{D}^{+}_{\nu}\Psi^{+}_{\mu}\right)\left(\mathfrak{D}^{-\mu}\Psi^{-\nu}-\mathfrak{D}^{-\nu}\Psi^{-\mu}\right)-\dfrac{m^{2}}{\hbar^{2}}W^{+}_{\mu}W^{-\mu}$
$\displaystyle-\dfrac{i}{\hbar}eF^{\mu\nu}W^{+}_{\mu}W^{-}_{\nu},$ (4)

where $\mathcal{D}_{0}^{\pm}=\left(1+\beta\hbar^{2}g^{00}D_{0}^{\pm
2}\right)D_{0}^{\pm}$,
$\mathcal{D}_{i}^{\pm}=\left(1-\beta\hbar^{2}g^{ii}D_{i}^{\pm
2}\right)D_{i}^{\pm}$,
$F_{\mu\nu}=\widehat{\nabla}_{\mu}A_{\nu}-\widehat{\nabla}_{\nu}A_{\mu}$ with
$D_{\mu}^{\pm}=\nabla_{\mu}\pm\frac{i}{\hbar}eA_{\mu}$,
$\widehat{\nabla}_{0}=\left(1+\beta\hbar^{2}g^{00}\nabla_{0}^{2}\right)\nabla_{0}$
and
$\widehat{\nabla}_{i}=\left(1-\beta\hbar^{2}g^{ii}\nabla_{i}^{2}\right)\nabla_{i}$.
Here $e$, $m$ and $A_{\mu}$ denote the charge of $W^{+}$ boson, mass of the
vector particle and electromagnetic potential of the black hole respectively.

The generalized action of the massive vector bosons particles takes the form

$\displaystyle\mathcal{S}_{GUP}=\displaystyle\int
dx^{4}\sqrt{-g}~{}\mathfrak{L}_{GUP}\left(\Psi_{\mu}^{\pm},\partial_{\mu}\Psi_{\nu}^{\pm},\partial_{\mu}\partial_{\rho}\Psi_{\nu}^{\pm},\partial_{\mu}\partial_{\rho}\partial_{\lambda}\Psi_{\nu}^{\pm}\right).$
(5)

Following from eq. (5), the modified wave equation for massive vector bosons
is obtained as

$\displaystyle\partial_{\mu}$
$\displaystyle(\sqrt{-g}\Psi^{\mu\nu})-3\beta\partial_{0}\left[\sqrt{-g}{}{}g^{00}\left(e^{2}A_{0}^{2}+i\hbar
e\nabla_{0}A_{0}\right)\Psi^{0v}\right]$
$\displaystyle+3\beta~{}\partial_{i}\left[\sqrt{-g}{}{}g^{ii}\left(e^{2}A_{i}^{2}+i\hbar
e\nabla_{i}A_{i}\right)\Psi^{iv}\right]+3\beta\partial_{0}\partial_{0}\left[\sqrt{-g}{}{}g^{00}i\hbar
eA_{0}\Psi^{0v}\right]$
$\displaystyle-3\beta\partial_{i}\partial_{i}\left[\sqrt{-g}{}{}g^{ii}i\hbar
eA_{i}\Psi^{iv}\right]+\beta\hbar^{2}\partial_{0}\partial_{0}\partial_{0}\left[\sqrt{-g}{}{}g^{00}\Psi^{0v}\right]$
$\displaystyle-\beta\hbar^{2}\partial_{i}\partial_{i}\partial_{i}\left[\sqrt{-g}{}{}g^{ii}\Psi^{iv}\right]+\sqrt{-g}\frac{i}{\hbar}eA_{\mu}\Psi^{\mu\nu}-\sqrt{-g}\frac{m^{2}}{\hbar^{2}}\Psi^{\nu}$
$\displaystyle-\sqrt{-g}\frac{i}{\hbar}eF^{\mu\nu}\Psi_{\mu}+\beta\sqrt{-g}g^{00}\left[i\hbar
e\nabla_{0}\nabla_{0}A_{0}+3e^{2}A_{0}\nabla_{0}A_{0}-\frac{i}{\hbar}e^{3}A_{0}^{3}\right]\Psi^{0\nu}$
$\displaystyle-\beta\sqrt{-g}g^{ii}\left[i\hbar
e\nabla_{i}\nabla_{i}A_{i}+3e^{2}A_{i}\nabla_{i}A_{i}-\frac{i}{\hbar}e^{3}A_{i}^{3}\right]\Psi^{i\nu}~{}=~{}0,$
(6)

where GUP modified anti-symmetric tensor $\Psi_{\mu\nu}$ is given by
$\Psi_{\mu\nu}=\mathcal{D}_{\mu}\Psi_{\nu}-\mathcal{D}_{\nu}\Psi_{\mu}$.

##  3 Quantum tunneling from KNdS black hole

The line element of KNdS black hole in well known Boyer-Lindquist coordinates
$(t,r,\theta,\varphi)$ carter1 is expressed as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(\dfrac{\Delta-\Delta_{\theta}~{}a^{2}\sin^{2}\theta}{\Sigma^{2}\rho^{2}}\right)dt^{2}-\dfrac{2a~{}\sin^{2}\theta[(r^{2}+a^{2})\Delta_{\theta}-\Delta]}{\Sigma^{2}\rho^{2}}dtd\varphi+\dfrac{\rho^{2}}{\Delta}dr^{2}$
(8)
$\displaystyle+\dfrac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}+\dfrac{\sin^{2}\theta\left[(r^{2}+a^{2})^{2}~{}\Delta_{\theta}-\Delta
a^{2}\sin^{2}\theta\right]}{\Sigma^{2}\rho^{2}}d\varphi^{2},$

where

$\displaystyle\Sigma=1+\dfrac{\Lambda
a^{2}}{3},~{}\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,~{}\Delta_{\theta}=1+\dfrac{\Lambda
a^{2}\cos^{2}\theta}{3},$ $\displaystyle\Delta=\left(1-\dfrac{\Lambda
a^{2}}{3}\right)\left(r^{2}+a^{2}\right)-2Mr+Q^{2}.$

The electromagnetic potential $A_{\mu}$ of KNdS black hole is given by

$\displaystyle
A_{\mu}=\dfrac{Qr\left(\delta_{\mu}^{t}-a\sin^{2}\theta\delta_{\mu}^{\varphi}\right)}{\rho^{2}\Sigma}.$
(9)

Eq. (8) describes a charged rotating black hole with mass $m$, spin parameter
$a$ and charge $Q$ with the cosmological constant $\Lambda$. The case
$\Lambda=0$ gives the solution of the Kerr-Newman black hole. For $\Lambda>0$
or $\Lambda<0$, eq. (8) represents KNdS black hole or anti KNdS black hole
respectively.

The horizons of the KNdS can be obtained from the event horizon equation as

$\Delta=\left(1-\dfrac{\Lambda
r^{2}}{3}\right)\left(r^{2}+a^{2}\right)-2Mr+Q^{2}=0.$ (10)

If $\Lambda>0$, Eq. (10) gives four real roots whenever $\frac{1}{\Lambda}\gg
M^{2}>Q^{2}+a^{2}$. The four roots are $r_{C}$, $r_{H}$, $r_{+}$ and $r_{-}$
$\left(r_{C}>r_{H}>r_{+}>r_{-}\right)$. Here $r_{C}$ denotes the cosmological
horizon, $r_{H}$ represents the event horizon and $r_{-}$ corresponds to the
Cauchy horizon. One reaches singularity $r=0$, $\theta=\frac{\pi}{2}$ and on
other side of $r=0$, $r=r_{-}$ is considered as another cosmological horizon
gibbons1 .

In this coordinate system, the event horizon and the infinite red-shift
surface do not coincide due to rotation. Because of this, it is inconvenient
to study the characteristic of tunneling radiation. So, we perform the
dragging coordinate transformation zhangC

$\dfrac{d\varphi}{dt}=\Omega=\dfrac{-g_{t\varphi}}{g_{\varphi\varphi}}.$ (11)

The line element (8) is reduced to

$\displaystyle
ds^{2}=-\dfrac{\Delta\Delta_{\theta}\rho^{2}}{\Sigma^{2}\left[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}dt^{2}+\dfrac{\rho^{2}}{\Delta}dr^{2}+\dfrac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}.$
(12)

The non-vanishing component of electromagnetic potential $A_{0}$ is given by

$\displaystyle
A_{0}=\dfrac{(r^{2}+a^{2})\Delta_{\theta}Qr}{\Sigma\left[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}.$ (13)

The dragging coordinate transformation makes the geometrical optical limit a
reliable approximation; hence, the WKB approximation can be applied.

Near the event horizon $r=r_{H}$, we use the approximation

$\displaystyle\Delta(r)=$
$\displaystyle\Delta(r_{H})+(r-r_{H})\Delta,_{r}(r_{H})+O\left((r-r_{H})^{2}\right)$
$\displaystyle\approx(r-r_{H})\Delta,_{r}(r_{H}),$ (14)

where $r_{H}$ is defined as

$\displaystyle r_{H}=\dfrac{1}{\alpha_{1}}\left(1+\dfrac{4\Lambda
M^{2}}{3\beta_{1}^{2}\alpha_{1}}+\dots\right)\left(M+\sqrt{M^{2}-(a^{2}+Q^{2})\alpha_{1}}\right),$
(15)

where $\beta=\sqrt{1-\frac{\Lambda a^{2}}{3}}$ and
$\alpha_{1}=\sqrt{\left(1+\frac{\Lambda a^{2}}{3}\right)^{2}+\frac{4\Lambda
Q^{2}}{3}}$. Then eq. (12) takes the form

$\displaystyle
ds^{2}=-\dfrac{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}dt^{2}+\dfrac{\rho^{2}(r_{H})}{(r-r_{H})\Delta,_{r}(r_{H})}dr^{2}+\dfrac{\rho^{2}(r_{H})}{\Delta_{\theta}}d\theta^{2}.$
(16)

The metric determinant and contravariant components of the line element (16)
are as follows

$\displaystyle
g=-\dfrac{\rho^{6}(r_{H})}{\Sigma^{2}\left(r_{H}^{2}+a^{2}\right)^{2}\Delta_{\theta}},$
$\displaystyle
g^{00}=-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})},$
$\displaystyle
g^{11}=\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})},~{}~{}g^{22}=\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})},$
$\displaystyle g^{01}=g^{02}=g^{10}=g^{12}=g^{20}=g^{21}=0.$ (17)

The angular velocity at the event horizon is given as

$\displaystyle\Omega=\dfrac{a}{r^{2}_{h}+a^{2}}.$ (18)

The surface gravity near the event horizon of the KNdS black hole is obtained
as

$\displaystyle\kappa=$ $\displaystyle\lim_{g_{00}\rightarrow
0}\Biggl{(}-\dfrac{1}{2}\sqrt{-\dfrac{g^{11}}{g_{00}}}\dfrac{dg_{00}}{dr}\Biggr{)}$
$\displaystyle=$
$\displaystyle\frac{\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}{\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}.$ (19)

The original Hawking temperature of the KNdS black hole is obtained from the
relation $T_{o}=\frac{\kappa}{2\pi}$ christina1 as

$\displaystyle
T_{o}=\frac{\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}{2\pi\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}.$ (20)

By using the expression of $M$ from eq. (10), we derive the heat capacity of
the black hole by using the relation $C_{o}=\frac{\partial M}{\partial T_{o}}$
priyo1 ; arun1 as

$\displaystyle C_{o}=$ $\displaystyle\left(\dfrac{\partial M}{\partial
r_{h}}\right)\left(\dfrac{\partial r_{h}}{\partial T_{o}}\right)$
$\displaystyle=$
$\displaystyle\dfrac{2\pi\left(a^{2}+r^{2}\right)^{2}\left(3+a^{2}\Lambda\right)\left[a^{2}\left(3+r^{2}\Lambda\right)+3\left(Q^{2}-r^{2}+r^{4}\Lambda\right)\right]}{3\left[a^{4}\left(r^{2}\Lambda-3\right)+3r^{2}\left(r^{2}+r^{4}\Lambda-3Q^{2}\right)+a^{2}\left(8r^{4}\Lambda-3Q^{2}-12r^{2}\right)\right]}.$
(21)

Applying WKB approximation, $\Psi_{\mu}$ given in eq. (2) is taken as

$\displaystyle\Psi_{\mu}=c_{\mu}(t,r,\theta)\exp\left[\frac{i}{\hbar}\mathcal{S}(t,r,\theta)\right],$
(22)

where $\mathcal{S}$ is defined as

$\displaystyle\mathcal{S}(t,r,\theta)=\mathcal{S}_{0}(t,r,\theta)+\hbar~{}\mathcal{S}_{1}(t,r,\theta)+\hbar^{2}~{}\mathcal{S}_{2}(t,r,\theta)+\cdots.$
(23)

By substituting eqs. (22), (23) and (16) in eq. (2) and keeping only the
lowest order in $\hbar$, we obtain the equations for the coefficients
$c_{\mu}$ as

$\displaystyle\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\left[c_{0}~{}\mathcal{B}_{1}^{2}\left(\partial_{r}\mathcal{S}_{0}\right)^{2}-c_{1}~{}\mathcal{B}_{0}\mathcal{B}_{1}\left(\partial_{r}\mathcal{S}_{0}\right)(\partial_{t}\mathcal{S}_{0}+eA_{0})\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\left[c_{0}~{}\mathcal{B}_{2}^{2}(\partial_{\theta}\mathcal{S}_{0})^{2}-c_{2}\mathcal{B}_{0}\mathcal{B}_{2}(\partial_{\theta}\mathcal{S}_{0})(\partial_{t}\mathcal{S}_{0}+eA_{0})\right]+m^{2}c_{0}~{}=~{}0,$
(24)
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}\left[c_{1}~{}\mathcal{B}_{0}^{2}(\partial_{t}\mathcal{S}_{0}+eA_{0})^{2}-c_{0}~{}\mathcal{B}_{0}\mathcal{B}_{1}\left(\partial_{r}\mathcal{S}_{0}\right)(\partial_{t}\mathcal{S}_{0}+eA_{0})\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\left[c_{1}~{}\mathcal{B}_{2}^{2}(\partial_{\theta}\mathcal{S}_{0})^{2}-c_{2}\mathcal{B}_{1}\mathcal{B}_{2}(\partial_{r}\mathcal{S}_{0})(\partial_{\theta}\mathcal{S}_{0})\right]+m^{2}c_{1}~{}=~{}0,$
(25)
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}\left[c_{2}~{}\mathcal{B}_{0}^{2}(\partial_{t}\mathcal{S}_{0}+eA_{0})^{2}-c_{0}~{}\mathcal{B}_{0}\mathcal{B}_{2}\left(\partial_{\theta}\mathcal{S}_{0}\right)(\partial_{t}\mathcal{S}_{0}+eA_{0})\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}+\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\left[c_{2}~{}\mathcal{B}_{1}^{2}(\partial_{r}\mathcal{S}_{0})^{2}-c_{1}\mathcal{B}_{1}\mathcal{B}_{2}(\partial_{r}\mathcal{S}_{0})(\partial_{\theta}\mathcal{S}_{0})\right]+m^{2}c_{2}~{}=~{}0,$
(26)

where the $\mathcal{B}_{\mu}$’s are defined as

$\displaystyle\mathcal{B}_{0}=1+\dfrac{\beta~{}\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}(\partial_{t}\mathcal{S}_{0}+eA_{0})^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})},$
$\displaystyle\mathcal{B}_{1}=1+\dfrac{\beta~{}(r-r_{H})\Delta,_{r}(r_{H})(\partial_{r}\mathcal{S}_{0})^{2}}{\rho^{2}(r_{H})},~{}~{}~{}\mathcal{B}_{2}=1+\dfrac{\beta\Delta_{\theta}(\partial_{\theta}\mathcal{S}_{0})^{2}}{\rho^{2}(r_{H})}.$
(27)

Considering the symmetries of the metric (16), we carry on the separation of
variables as follows

$\displaystyle\mathcal{S}_{0}=-(E-j\Omega)~{}t+R(r)+\Theta(\theta)+U,$ (28)

where $E$ is the energy of the emitted vector particles, $j$ is the angular
momentum and $U$ is a complex constant. On inserting eq. (28) in eqs. (3)-(3),
we get a matrix equation as

$\displaystyle F(c_{0},c_{1},c_{2})^{T}~{}=~{}0,$ (29)

where $F$ is a $3\times 3$ matrix and all the entries are as follows

$\displaystyle F_{11}=$
$\displaystyle\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\mathcal{B}_{1}^{2}\mathcal{R}^{\prime
2}+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\mathcal{B}_{2}^{2}J_{\theta}^{2}+m^{2},$
$\displaystyle F_{12}=$
$\displaystyle-\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})\mathcal{B}_{0}\mathcal{B}_{1}\mathcal{R}^{\prime},$
$\displaystyle F_{13}=$
$\displaystyle-\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})\mathcal{B}_{0}\mathcal{B}_{2}J_{\theta},$
$\displaystyle F_{21}=$
$\displaystyle\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})\mathcal{B}_{0}\mathcal{B}_{1}\mathcal{R}^{\prime},$
$\displaystyle F_{22}=$
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})^{2}\mathcal{B}_{0}^{2}+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\mathcal{B}_{2}^{2}J_{\theta}^{2}+m^{2},$
$\displaystyle F_{23}=$
$\displaystyle-\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\mathcal{B}_{1}\mathcal{B}_{2}J_{\theta}\mathcal{R}^{\prime},$
$\displaystyle F_{31}=$
$\displaystyle\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})\mathcal{B}_{0}\mathcal{B}_{2}J_{\theta},$
$\displaystyle F_{32}=$
$\displaystyle-\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\mathcal{B}_{1}\mathcal{B}_{2}J_{\theta}\mathcal{R}^{\prime},$
$\displaystyle F_{33}=$
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})^{2}\mathcal{B}_{2}+\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\mathcal{B}_{1}^{2}\mathcal{R}^{\prime
2}$ $\displaystyle+m^{2},$ (30)

where $\mathcal{R}^{\prime}=\partial_{r}\mathcal{R}$ and
$J_{\theta}=\partial_{\theta}\Theta$.

To find a non-trivial solution of eq. (29), we put the determinant of the
matrix $F$ equals zero, which in turn gives

$\displaystyle\mathcal{O}(\beta^{4})$
$\displaystyle(\partial_{r}\mathcal{R})^{12}+\mathcal{O}(\beta^{3})(\partial_{r}\mathcal{R})^{10}+\mathcal{O}(\beta^{2})(\partial_{r}\mathcal{R})^{8}+\left[H_{6}+\mathcal{O}(\beta^{2})\right](\partial_{r}\mathcal{R})^{6}$
$\displaystyle+\left[H_{4}+\mathcal{O}(\beta^{2})\right](\partial_{r}\mathcal{R})^{4}+\left[H_{2}+\mathcal{O}(\beta^{2})\right](\partial_{r}\mathcal{R})^{2}+H_{0}+\mathcal{O}(\beta^{2})~{}=~{}0,$
(31)

where

$\displaystyle H_{6}=$ $\displaystyle
4\beta\left[\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\right]^{3},$
$\displaystyle H_{4}=$
$\displaystyle\dfrac{(r-r_{H})^{2}\Delta^{2}_{,_{r}}(r_{H})}{\rho^{4}(r_{H})}\left[1+4\beta\left\\{\dfrac{-(-E+j\Omega+eA_{0})^{2}~{}\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}+\dfrac{\Delta_{\theta}J_{\theta}^{2}}{\rho^{2}(r_{H})}+m^{2}\right\\}\right],$
$\displaystyle H_{2}=$
$\displaystyle\dfrac{2(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\left[\dfrac{-(-E+j\Omega+eA_{0})^{2}~{}\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}+\dfrac{\Delta_{\theta}J_{\theta}^{2}}{\rho^{2}(r_{H})}+m^{2}\right.$
$\displaystyle\left.-2\beta\left\\{\dfrac{(-E+j\Omega+eA_{0})^{4}~{}\Sigma^{4}\left(r^{2}_{H}+a^{2}\right)^{4}}{(r-r_{H})^{2}\Delta^{2},_{r}(r_{H})~{}\rho^{4}(r_{H})}-\dfrac{\Delta_{\theta}^{2}J_{\theta}^{4}}{\rho^{2}(r_{H})}\right\\}\right],$
$\displaystyle H_{0}=$
$\displaystyle\left\\{\dfrac{-(-E+j\Omega+eA_{0})^{2}~{}\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}+\dfrac{\Delta_{\theta}J_{\theta}^{2}}{\rho^{2}(r_{H})}+m^{2}\right\\}\left[\dfrac{\Delta_{\theta}J_{\theta}^{2}}{\rho^{2}(r_{H})}+m^{2}\right.$
$\displaystyle\left.+\dfrac{(-E+j\Omega+eA_{0})^{2}~{}\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}-4\beta\left\\{\dfrac{(-E+j\Omega+eA_{0})^{4}~{}\Sigma^{4}\left(r^{2}_{H}+a^{2}\right)^{4}}{(r-r_{H})^{2}\Delta^{2},_{r}(r_{H})~{}\rho^{4}(r_{H})}\right.\right.$
$\displaystyle\left.\left.-\dfrac{\Delta^{2}_{\theta}J_{\theta}^{4}}{\rho^{4}(r_{H})}\right\\}\right].$

By neglecting the higher order term of $\beta$, eq. (3) becomes

$\displaystyle\dfrac{4\beta(r-r_{H})^{2}\Delta^{2},_{r}(r_{H})~{}(\partial_{r}\mathcal{R})^{4}}{\rho^{4}(r_{H})}+\dfrac{(r-r_{H})\Delta,_{r}(r_{H})~{}(\partial_{r}\mathcal{R})^{2}}{\rho^{2}(r_{H})}+\dfrac{\Delta_{\theta}J^{2}_{\theta}}{\rho^{2}(r_{H})}$
$\displaystyle+4\beta\left\\{\dfrac{\Delta^{2}_{\theta}J^{4}_{\theta}}{\rho^{4}(r_{H})}-\dfrac{(-E+j\Omega+eA_{0})^{4}{}{}{}{}{}{}\Sigma^{4}\left(r^{2}_{H}+a^{2}\right)^{4}}{(r-r_{H})^{2}\Delta^{2},_{r}(r_{H})~{}\rho^{4}(r_{H})}\right\\}+m^{2}$
$\displaystyle-\dfrac{(-E+j\Omega+eA_{0})^{2}\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}=0.$
(32)

We obtain the solution to the derivative of the radial action by neglecting
the higher power of $\beta$ as

$\displaystyle\partial_{r}\mathcal{R}=\pm\sqrt{-\dfrac{\left(J^{2}_{\theta}\Delta_{\theta}+m^{2}\rho^{2}\right)}{(r-r_{H})~{}\Delta,_{r}(r_{H})}+\dfrac{(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}~{}\Sigma^{2}}{(r-r_{H})^{2}~{}\left(\Delta,_{r}(r_{H})\right)^{2}}}\left(1+\beta~{}\dfrac{\chi_{1}}{\chi_{2}}\right),$
(33)

where

$\displaystyle\chi_{1}=$ $\displaystyle
2\Delta,_{r}(r_{H})(r-r_{H})\left(2J^{4}_{\theta}\Delta^{2}_{\theta}+2J^{2}_{\theta}m^{2}\Delta_{\theta}\rho^{2}+m^{4}\rho^{4}\right)$
$\displaystyle-4\Sigma^{2}(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\left(J^{2}_{\theta}\Delta_{\theta}+m^{2}\rho^{2}\right),$
$\displaystyle\chi_{2}=$
$\displaystyle\rho^{2}\left\\{(r-r_{H})\Delta,_{r}(r_{H})\left(J^{2}_{\theta}\Delta_{\theta}+m^{2}\rho^{2}\right)-\Sigma^{2}(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\right\\}.$
(34)

The solution of the radial action is obtained by integrating eq. (33) around
the pole $r=r_{H}$. The imaginary part of the radial action gives the
particle’s rate of tunneling as

$\displaystyle Im~{}\mathcal{R}_{\pm}=$ $\displaystyle\pm Im~{}\int
dr\sqrt{-\dfrac{\left(J^{2}_{\theta}\Delta_{\theta}+m^{2}\rho^{2}\right)}{(r-r_{H})~{}\Delta,_{r}(r_{H})}+\dfrac{(-E+j\Omega+eA_{0})^{2}~{}\left(r^{2}_{H}+a^{2}\right)^{2}~{}\Sigma^{2}}{(r-r_{H})^{2}~{}\left(\Delta,_{r}(r_{H})\right)^{2}}}~{}$
$\displaystyle\times\left(1+\beta~{}\dfrac{\chi_{1}}{\chi_{2}}\right)$
$\displaystyle=$ $\displaystyle\pm\dfrac{\pi(E-j\Omega-
eA_{0})\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}{2\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}\left(1+\beta~{}\Pi\right),$
(35)

where $\Pi=4m^{2}+\frac{4J_{\theta}^{2}\Delta_{\theta}}{\rho^{2}}$.
$\mathcal{R}_{+}$ and $\mathcal{R}_{-}$ stand for the radial action of the
outgoing and ingoing particles respectively. Based on WKB approximation, the
probabilities of the vector boson particles tunneling across the event horizon
$r_{H}$ are

$\displaystyle\mathcal{P}_{outgoing}=\exp\biggl{[}-2\biggl{\\{}Im(\mathcal{R}_{+})+Im(U)\biggr{\\}}\biggr{]}$
and
$\displaystyle\mathcal{P}_{ingoing}=\exp\biggl{[}-2\biggl{\\{}Im(\mathcal{R}_{-})+Im(U)\biggr{\\}}\biggr{]}.$
(36)

According to the semiclassical WKB approximation, the ingoing vector boson
particles have a $100\%$ chance of entering the black hole a6 . It shows that
$Im(\mathcal{R}_{+})=-Im(\mathcal{R}_{-})$. Thus the tunneling rate of the
vector boson particles is given by

$\displaystyle\Gamma=\dfrac{\mathcal{P}_{outgoing}}{\mathcal{P}_{ingoing}}=\exp$
$\displaystyle\left[-4~{}Im~{}\mathcal{R}_{+}\right]$ $\displaystyle=\exp$
$\displaystyle\left[\dfrac{-2\pi(E-j\Omega-
eA_{0})\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}{\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}(1+\beta~{}\Pi)\right].$
(37)

The Hawking temperature of the KNdS black hole under the quantum gravity
effect is obtained as

$\displaystyle T_{d}=$
$\displaystyle\dfrac{\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}{2\pi\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}\left(1-\beta~{}\Pi\right)$ $\displaystyle=$ $\displaystyle
T_{o}~{}\left(1-\beta~{}\Pi\right),$ (38)

where
$T_{o}=\frac{\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}{2\pi\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}$ is the original Hawking temperature. It is obvious from the
point $\Pi>0$ that the Hawking temperature is modified due to the quantum
gravity effects. The modifed Hawking temperature is lower than the original
Hawking temperature due to the presence of $\beta>0$. Further, the modified
Hawking temperature relies on the mass and angular momentum of the emitted
vector boson particles. In the absence of the quantum gravity effects i.e.
$\beta=0$, the original Hawking temperature of KNdS black hole is recovered.

The modified heat capacity is calculated as

$\displaystyle C_{H}=$
$\displaystyle\dfrac{2\pi\left(a^{2}+r^{2}\right)^{2}\left(3+a^{2}\Lambda\right)\left[a^{2}\left(3+r^{2}\Lambda\right)+3\left(Q^{2}-r^{2}+r^{4}\Lambda\right)\right]}{3\left[a^{4}\left(r^{2}\Lambda-3\right)+3r^{2}\left(r^{2}+r^{4}\Lambda-3Q^{2}\right)+a^{2}\left(8r^{4}\Lambda-3Q^{2}-12r^{2}\right)\right]}$
$\displaystyle\times\left(1+\beta~{}\Pi\right)$ $\displaystyle=$
$\displaystyle C_{o}\left(1+\beta~{}\Pi\right).$ (39)

From eq. (3), it is observed that the modified heat capacity reduced to the
original heat capacity ($C_{o}$) when $\beta=0$. The modified heat capacity of
KNdS black hole increases due to the quantum gravity effects.

Now using another coordinate transformation $\phi=\varphi-\Omega t$ chen3
where

$\displaystyle\Omega=\dfrac{a~{}[(r^{2}+a^{2})\Delta_{\theta}-\Delta]}{(r^{2}+a^{2})^{2}~{}\Delta_{\theta}-\Delta
a^{2}\sin^{2}\theta},$ (40)

then the line element (8) reduces to

$\displaystyle ds^{2}=$
$\displaystyle-\dfrac{\Delta\Delta_{\theta}\rho^{2}}{\Sigma^{2}\left[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}dt^{2}+\dfrac{\rho^{2}}{\Delta}dr^{2}+\dfrac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}$
$\displaystyle+\dfrac{\left[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]\sin^{2}\theta}{\rho^{2}\Sigma^{2}}d\phi^{2}.$ (41)

The corresponding non-zero electromagnetic potentials are given by

$\displaystyle
A_{0}=\dfrac{(r^{2}+a^{2})\Delta_{\theta}~{}Q~{}r}{\Sigma\left[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]},\qquad
A_{3}=-\dfrac{Qra\sin^{2}\theta}{(r^{2}+a^{2}\cos^{2}\theta)\Sigma}.$ (42)

Using eq. (3) in eq. (3), we get

$\displaystyle ds^{2}=$
$\displaystyle-\dfrac{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}dt^{2}+\dfrac{\rho^{2}(r_{H})}{(r-r_{H})\Delta,_{r}(r_{H})}dr^{2}+\dfrac{\rho^{2}(r_{H})}{\Delta_{\theta}}d\theta^{2}$
$\displaystyle+\dfrac{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}{\rho^{2}(r_{H})\Sigma^{2}}d\phi^{2}.$
(43)

The metric determinant and the nonzero contravariant components of the line
element given above are as follows

$\displaystyle g=\dfrac{\rho^{4}\sin^{2}\theta}{\Sigma^{4}},\quad
g^{00}=-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})},$
$\displaystyle
g^{11}=\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})},\quad
g^{22}=\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})},$ $\displaystyle
g^{33}=\dfrac{\rho^{2}(r_{H})\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}.$
(44)

According to WKB approximation, $\Psi_{\mu}$ from eq. (2) can be written as

$\displaystyle\Psi_{\mu}=c_{\mu}(t,r,\theta,\phi)\exp\left[\frac{i}{\hbar}\mathcal{S}(t,r,\theta,\phi)\right],$
(45)

where $\mathcal{S}$ is defined as

$\displaystyle\mathcal{S}(t,r,\theta)=\mathcal{S}_{0}(t,r,\theta,\phi)+\hbar~{}\mathcal{S}_{1}(t,r,\theta,\phi)+\hbar^{2}~{}\mathcal{S}_{2}(t,r,\theta,\phi)+\cdots.$
(46)

By substituting eqs. (45) and (46) in eq. (3), and keeping only the lowest
order in $\hbar$, we derive the equations for the coefficients $c_{\mu}$ as

$\displaystyle\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\left[c_{0}~{}\mathcal{H}_{1}^{2}\left(\partial_{r}\mathcal{S}_{0}\right)^{2}-c_{1}~{}\mathcal{H}_{0}\mathcal{H}_{1}\left(\partial_{r}\mathcal{S}_{0}\right)(\partial_{t}\mathcal{S}_{0}+eA_{0})\right]+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}$
$\displaystyle\left[c_{0}~{}\mathcal{H}_{2}^{2}~{}(\partial_{\theta}\mathcal{S}_{0})^{2}-c_{2}\mathcal{H}_{0}\mathcal{H}_{2}~{}(\partial_{\theta}\mathcal{H})(\partial_{t}\mathcal{S}_{0}+eA_{0})\right]+\dfrac{\rho^{2}(r_{H})~{}\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}$
$\displaystyle\left[c_{0}\mathcal{H}_{3}^{2}~{}(\partial_{\phi}\mathcal{S}_{0}+eA_{3})^{2}-c_{3}~{}\mathcal{H}_{0}\mathcal{H}_{3}\left(\partial_{t}\mathcal{S}_{0}+eA_{0}\right)(\partial_{\phi}\mathcal{S}_{0}+eA_{3})\right]+m^{2}c_{0}~{}=~{}0,$
(47)
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}\left[c_{1}~{}\mathcal{H}_{0}^{2}(\partial_{t}\mathcal{S}_{0}+eA_{0})^{2}-c_{0}~{}\mathcal{H}_{0}\mathcal{H}_{1}\left(\partial_{r}\mathcal{S}_{0}\right)(\partial_{t}\mathcal{S}_{0}+eA_{0})\right]$
$\displaystyle+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\left[c_{1}~{}\mathcal{H}_{2}^{2}(\partial_{\theta}\mathcal{S}_{0})^{2}-c_{2}\mathcal{H}_{1}\mathcal{H}_{2}(\partial_{r}\mathcal{S}_{0})(\partial_{\theta}\mathcal{S}_{0})\right]+\dfrac{\rho^{2}(r_{H})\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}$
$\displaystyle\left[c_{1}~{}\mathcal{H}_{3}^{2}(\partial_{\phi}\mathcal{S}_{0}+eA_{3})^{2}-c_{3}~{}\mathcal{H}_{1}\mathcal{H}_{3}\left(\partial_{r}\mathcal{S}_{0}\right)(\partial_{\phi}\mathcal{S}_{0}+eA_{3})\right]+m^{2}c_{1}~{}=~{}0,$
(48)
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}\left[c_{2}~{}\mathcal{H}_{0}^{2}(\partial_{t}\mathcal{S}_{0}+eA_{0})^{2}-c_{0}~{}\mathcal{H}_{0}\mathcal{H}_{2}\left(\partial_{\theta}\mathcal{S}_{0}\right)(\partial_{t}\mathcal{S}_{0}+eA_{0})\right]$
$\displaystyle+\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\left[c_{2}~{}\mathcal{H}_{1}^{2}(\partial_{r}\mathcal{S}_{0})^{2}-c_{1}\mathcal{H}_{1}\mathcal{H}_{2}(\partial_{r}\mathcal{S}_{0})(\partial_{\theta}\mathcal{S}_{0})\right]+\dfrac{\rho^{2}(r_{H})\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}$
$\displaystyle\left[c_{2}~{}\mathcal{H}_{3}^{2}(\partial_{\phi}\mathcal{S}_{0}+eA_{3})^{2}-c_{3}~{}\mathcal{H}_{2}\mathcal{H}_{3}\left(\partial_{\theta}\mathcal{S}_{0}\right)(\partial_{\phi}\mathcal{S}_{0}+eA_{3})\right]+m^{2}c_{2}~{}=~{}0,$
(49)

$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}\left[c_{3}~{}\mathcal{H}_{0}^{2}(\partial_{t}\mathcal{S}_{0}+eA_{0})^{2}-c_{0}~{}\mathcal{H}_{0}\mathcal{H}_{3}(\partial_{t}\mathcal{S}_{0}+eA_{0})\right.$
$\displaystyle\left.(\partial_{\phi}\mathcal{S}_{0}+eA_{3})\right]+\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\left[c_{3}~{}\mathcal{H}_{1}^{2}(\partial_{r}\mathcal{S}_{0})^{2}-c_{1}\mathcal{H}_{1}\mathcal{H}_{3}(\partial_{r}\mathcal{S}_{0})(\partial_{\phi}\mathcal{S}_{0}+eA_{3})\right]$
$\displaystyle+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\left[c_{3}~{}\mathcal{H}_{2}^{2}~{}(\partial_{\theta}\mathcal{S}_{0})^{2}-c_{2}\mathcal{H}_{2}\mathcal{H}_{3}~{}(\partial_{\theta}\mathcal{S}_{0})(\partial_{\phi}\mathcal{S}_{0}+eA_{3})\right]+m^{2}c_{3}~{}=~{}0,$
(50)

where the $\mathcal{H}_{\mu}$’s are defined as

$\displaystyle\mathcal{H}_{0}=1+\beta\dfrac{~{}\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}(\partial_{t}\mathcal{S}_{0}+eA_{0})^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})},~{}~{}\mathcal{H}_{1}=1+\beta\dfrac{(r-r_{H})\Delta,_{r}(r_{H})(\partial_{r}\mathcal{S}_{0})^{2}}{\rho^{2}(r_{H})},$
$\displaystyle\mathcal{H}_{2}=1+\beta~{}\dfrac{\Delta_{\theta}~{}(\partial_{\theta}\mathcal{S}_{0})^{2}}{\rho^{2}(r_{H})},~{}~{}\quad\mathcal{H}_{3}=1+\beta~{}\dfrac{\rho^{2}(r_{H})\Sigma^{2}~{}(\partial_{\phi}\mathcal{S}_{0}+eA_{3})^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}.$
(51)

Considering the symmetry of the black hole, the corresponding action of the
vector boson particles can be written as

$\displaystyle\mathcal{S}_{0}=-(E-j\Omega)~{}t+\mathcal{W}(r)+\Theta(\theta,\phi)+V,$
(52)

where $E$, $j$ and $V$ are the energy of the emitted vector particles, angular
momentum and complex constant respectively. On inserting eq. (52) in eqs.
(3)-(3), a matrix equation is obtained as

$\displaystyle\mathcal{F}\left(c_{0},c_{1},c_{2},c_{3}\right)^{T}=0,$ (53)

where $\mathcal{F}$ is a $4\times 4$ matrix, whose components of the matrix
are as follows

$\displaystyle\mathcal{F}_{11}=$
$\displaystyle\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\mathcal{H}_{1}^{2}~{}\mathcal{W}^{\prime
2}+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\mathcal{H}_{2}^{2}~{}J_{\theta}^{2}+\dfrac{\rho^{2}(r_{H})\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}$
$\displaystyle\mathcal{H}_{3}^{2}~{}\left(J_{\phi}+eA_{3}\right)+m^{2},$
$\displaystyle\mathcal{F}_{12}=$
$\displaystyle-\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})\mathcal{H}_{0}\mathcal{H}_{1}\mathcal{W}^{\prime},$
$\displaystyle\mathcal{F}_{13}=$
$\displaystyle-\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})\mathcal{H}_{0}\mathcal{H}_{2}~{}J_{\theta},$
$\displaystyle\mathcal{F}_{14}=$
$\displaystyle-\dfrac{\rho^{2}(r_{H})~{}\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}(-E+j\Omega+eA_{0})\left(J_{\phi}+eA_{3}\right)~{}\mathcal{H}_{0}\mathcal{H}_{3},$
$\displaystyle\mathcal{F}_{21}=$
$\displaystyle\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})\mathcal{H}_{0}\mathcal{H}_{1}\mathcal{W}^{\prime},$
$\displaystyle\mathcal{F}_{22}=$
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})^{2}\mathcal{H}_{0}^{2}+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\mathcal{H}_{2}^{2}J_{\theta}^{2}$
$\displaystyle+\dfrac{\rho^{2}(r_{H})\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}\left(J_{\phi}+eA_{3}\right)^{2}\mathcal{H}_{3}^{2}+m^{2},$
$\displaystyle\mathcal{F}_{23}=$
$\displaystyle-\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\mathcal{H}_{1}\mathcal{H}_{2}~{}J_{\theta}~{}\mathcal{W}^{\prime},$
$\displaystyle\mathcal{F}_{24}=$
$\displaystyle-\dfrac{\rho^{2}(r_{H})~{}\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}\left(J_{\phi}+eA_{3}\right)~{}\mathcal{W}^{\prime}~{}\mathcal{H}_{1}\mathcal{H}_{3},$
$\displaystyle\mathcal{F}_{31}=$
$\displaystyle\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})\mathcal{H}_{0}\mathcal{H}_{2}~{}J_{\theta},$
$\displaystyle\mathcal{F}_{32}=$
$\displaystyle-\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\mathcal{H}_{1}\mathcal{H}_{2}~{}J_{\theta}~{}\mathcal{W}^{\prime},$
$\displaystyle\mathcal{F}_{33}=$
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})^{2}~{}\mathcal{H}_{0}^{2}+\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\mathcal{H}_{1}^{2}~{}\mathcal{W}^{\prime
2}$
$\displaystyle+\dfrac{\rho^{2}(r_{H})\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}\left(J_{\phi}+eA_{3}\right)^{2}\mathcal{H}_{3}^{2}+m^{2},$
$\displaystyle\mathcal{F}_{34}=$
$\displaystyle-\dfrac{\rho^{2}(r_{H})~{}\Sigma^{2}}{\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\sin^{2}\theta}\left(J_{\phi}+eA_{3}\right)~{}J_{\theta}~{}\mathcal{H}_{2}\mathcal{H}_{3},$
$\displaystyle\mathcal{F}_{41}=$
$\displaystyle\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})~{}\left(J_{\phi}+eA_{3}\right)~{}\mathcal{H}_{0}\mathcal{H}_{3},$
$\displaystyle\mathcal{F}_{42}=$
$\displaystyle-\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\left(J_{\phi}+eA_{3}\right)\mathcal{H}_{1}\mathcal{H}_{2}~{}\mathcal{W}^{\prime},$
$\displaystyle\mathcal{F}_{43}=$
$\displaystyle-\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}~{}\left(J_{\phi}+eA_{3}\right)~{}J_{\theta}~{}\mathcal{H}_{2}\mathcal{H}_{3},$
$\displaystyle\mathcal{F}_{44}=$
$\displaystyle-\dfrac{\Sigma^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}(-E+j\Omega+eA_{0})^{2}~{}\mathcal{H}_{0}^{2}+\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\mathcal{H}_{1}^{2}~{}\mathcal{W}^{\prime
2}$
$\displaystyle+\dfrac{\Delta_{\theta}}{\rho^{2}(r_{H})}\mathcal{H}_{2}^{2}J_{\theta}^{2}+m^{2},$
(54)

where $\mathcal{W}^{\prime}=\partial_{r}\mathcal{W}$,
$J_{\theta}=\partial_{\theta}\Theta$ and $J_{\phi}=\partial_{\phi}\Theta$.

Eq. (53) has a non-trivial solution if the determinant of the matrix
$\mathcal{F}$ equals zero. If $det(\mathcal{F})=0$, then eq. (53) gives

$\displaystyle\mathcal{O}(\beta^{6})(\partial_{r}\mathcal{W})^{18}+\mathcal{O}(\beta^{5})(\partial_{r}\mathcal{W})^{16}+\mathcal{O}(\beta^{4})(\partial_{r}\mathcal{W})^{14}+\mathcal{O}(\beta^{3})(\partial_{r}\mathcal{W})^{12}+\mathcal{O}(\beta^{2})$
$\displaystyle(\partial_{r}\mathcal{W})^{10}+\mathcal{O}(\beta^{2})(\partial_{r}\mathcal{W})^{8}+\mathcal{O}(\beta^{2})(\partial_{r}\mathcal{W})^{6}+\mathcal{O}(\beta^{2})(\partial_{r}\mathcal{W})^{4}+\mathcal{O}(\beta^{2})(\partial_{r}\mathcal{W})^{2}+\mathcal{O}(\beta^{2})$
$\displaystyle+\left[A^{*}_{0}(\partial_{r}\mathcal{W})^{2}+A^{*}_{1}\right]\left\\{B_{0}+B_{2}(\partial_{r}\mathcal{W})^{2}+B_{4}(\partial_{r}\mathcal{W})^{4}+B_{6}(\partial_{r}\mathcal{W})^{6}\right\\}~{}=~{}0.$
(55)

(The expressions for $A_{i}$ and $B_{i}$ are given in Appendix A.)

Solving eq. (3) by neglecting the higher order terms of $\beta$, we obtain the
solution of the derivative of radial action as

$\displaystyle\partial_{r}\mathcal{W}=$
$\displaystyle\Biggl{(}-\dfrac{m^{2}\rho^{2}(r_{H})+J_{\theta}^{2}\Delta_{\theta}}{(r-r_{H})\Delta,_{r}(r_{H})}+\dfrac{(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Sigma^{2}}{(r-r_{H})^{2}\Delta^{2},_{r}(r_{H})}$
$\displaystyle-\dfrac{\left(J_{\phi}+eA_{3}\right)^{2}\rho^{4}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta}{(r-r_{H})\left(r^{2}_{H}+a^{2}\right)\Delta_{\theta}~{}\Delta,_{r}(r_{H})}\Biggr{)}^{\dfrac{1}{2}}~{}\times\left(1+\dfrac{\chi_{1}}{\chi_{2}}\beta\right).$
(56)

(The expressions of $\chi_{1}$ and $\chi_{2}$ are given in Appendix B.)

The imaginary part of the radial action is obtained by integrating eq. (3) at
the pole, $r=r_{H}$ as

$\displaystyle Im\mathcal{W}_{\pm}=$ $\displaystyle\pm Im\bigintss
dr~{}\Biggl{(}-\dfrac{m^{2}\rho^{2}(r_{H})+J_{\theta}^{2}\Delta_{\theta}}{(r-r_{H})\Delta,_{r}(r_{H})}+\dfrac{(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Sigma^{2}}{(r-r_{H})^{2}\Delta^{2},_{r}(r_{H})}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\dfrac{\left(J_{\phi}+eA_{3}\right)^{2}\rho^{4}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta}{(r-r_{H})\left(r^{2}_{H}+a^{2}\right)\Delta_{\theta}~{}\Delta,_{r}(r_{H})}\Biggr{)}^{\dfrac{1}{2}}~{}\times\left(1+\dfrac{\chi_{1}}{\chi_{2}}\beta\right)$
$\displaystyle=$ $\displaystyle\pm\dfrac{\pi(E-j\Omega-
eA_{0})\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}{2\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}~{}\left(1+\beta~{}\Xi\right),$
(57)

where

$\displaystyle\Xi=$
$\displaystyle-\dfrac{4\left(J_{\phi}+eA_{3}\right)^{2}\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}}+\dfrac{1}{2~{}\rho^{4}(r_{H})~{}\Sigma^{2}\left(J_{\phi}+eA_{3}\right)\left(J_{\phi}+eA_{3}-1\right)}$
$\displaystyle\Biggl{[}3m^{2}~{}\left(r^{2}_{H}+a^{2}\right)^{2}~{}\Delta_{\theta}~{}\sin^{2}\theta\bigl{\\{}5J^{2}_{\theta}\Delta_{\theta}+3m^{2}~{}\rho^{2}(r_{H})\bigr{\\}}+\left(J_{\phi}+eA_{3}\right)^{3}\rho^{2}(r_{H})~{}\Sigma^{2}$
$\displaystyle\biggl{[}-8J_{\theta}^{2}~{}\Delta_{\theta}\left(J_{\phi}+eA_{3}-1\right)+m^{2}~{}\rho^{2}(r_{H})\bigl{\\{}8+7\left(J_{\phi}+eA_{3}\right)\bigr{\\}}\biggr{]}\Biggr{]}.$
(58)

Here $\mathcal{W}_{+}$ and $\mathcal{W}_{-}$ indicate the radial action of the
outgoing and ingoing particles respectively. According to WKB approximation,
the tunneling probabilities are given by

$\displaystyle\mathcal{P}_{outgoing}=\exp\biggl{[}-2\biggl{\\{}Im(\mathcal{R}_{+})+Im(V)\biggr{\\}}\biggr{]}$
and
$\displaystyle\mathcal{P}_{ingoing}=\exp\biggl{[}-2\biggl{\\{}Im(\mathcal{R}_{-})+Im(V)\biggr{\\}}\biggr{]}.$
(59)

There is a $100\%$ probability of ingoing particle to enter the black hole in
accordance with the semiclassical WKB approximation. Thus, the tunneling rate
of $W^{+}$ boson particles is given by

$\displaystyle\Gamma_{rate}=\dfrac{\mathcal{P}_{outgoing}}{\mathcal{P}_{ingoing}}=$
$\displaystyle\exp\bigl{[}-4\\{Im(\mathcal{R}_{+})\bigr{]}$ $\displaystyle=$
$\displaystyle\exp\Biggl{[}\dfrac{-2~{}\pi(E-j\Omega-
eA_{0})\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}{\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}~{}\left(1+\beta~{}\Xi\right)\Biggr{]}.$
(60)

The Boltzman factor gives the Hawking temperature of the black hole kerner2 .
Thus, the GUP modified Hawking temperature is derived as

$\displaystyle T_{d4}=$
$\displaystyle\dfrac{\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}{2~{}\pi\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}~{}\left(1-\beta~{}\Xi\right)$ $\displaystyle=$
$\displaystyle~{}T_{o}~{}\left(1-\beta~{}\Xi\right),$ (61)

where
$T_{o}=\dfrac{\left[(r_{H}-M)-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}{2~{}\pi\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}$, is the original Hawking temperature of KNdS black hole
without any quantum gravity effects. The modified Hawking temperature $T_{d4}$
may be lower or greater than the original Hawking temperature $T_{o}$
according to $\Xi>0$ or $\Xi<0$ respectively. The modified Hawking temperature
depends on the quantum numbers (mass and angular momentum) of the emitted
vector boson particles.

The modified heat capacity is calculated as

$\displaystyle C_{H4}=$ $\displaystyle\left(\dfrac{\partial M}{\partial
r_{h}}\right)\left(\dfrac{\partial r_{h}}{\partial T_{d}}\right)$
$\displaystyle=$
$\displaystyle\dfrac{2\pi\left(a^{2}+r^{2}\right)^{2}\left(3+a^{2}\Lambda\right)\left[a^{2}\left(3+r^{2}\Lambda\right)+3\left(Q^{2}-r^{2}+r^{4}\Lambda\right)\right]}{3\left[a^{4}\left(r^{2}\Lambda-3\right)+3r^{2}\left(r^{2}+r^{4}\Lambda-3Q^{2}\right)+a^{2}\left(8r^{4}\Lambda-3Q^{2}-12r^{2}\right)\right]}$
$\displaystyle\times\left(1+\beta~{}\Xi\right).$ (62)

From eq. (3), it is observed that the modified heat capacity reduces to the
original heat capacity when $\beta=0$. Thus, the original heat capacity is
recovered in the absence of the quantum gravity effects. The modified heat
capacity $C_{H4}$ is higher or lower than the original heat capacity $C_{o}$,
according to $\Xi>0$ or $\Xi<0$ respectively.

##  4 Remnant of 3-dimensional KNdS black hole

Many studies have shown that the GUP effect could give a black hole remnant
chen1 ; chen2 ; myung1 ; gangopadhyay1 ; feng1 ; medved ; bargueno . For
3-dimensional KNdS black hole the quantum gravity effects slow down the
increase of Hawking temperature. This leads to the formation of remnants in
black hole evaporation. From this point, we will investigate the remnant of
3-dimensional KNdS black hole. The tunneling particle’s mass is no longer
considered in the following discussion since the tunneling particles at the
event horizon are effectively massless. According to the uncertainty
principle, the lower limit of the tunneling particle energy can be expressed
adler ; amelino as

$\displaystyle E\geq\dfrac{\hbar}{\Delta x}.$ (63)

Near the event horizon, one may take the uncertainty of the position as the
radius of the black hole adler ; amelino as

$\displaystyle\Delta x\approx r_{BH}=r_{H}.$ (64)

From eq. (3), we obtain

$\displaystyle T_{d}=$
$\displaystyle\frac{\left[r_{H}-\dfrac{1}{2}\left(1-\dfrac{\Lambda
r^{2}_{H}}{3}\right)\left(r_{H}+\dfrac{a^{2}}{r_{H}}\right)-\dfrac{Q^{2}}{2r_{H}}-\frac{\Lambda~{}r_{H}}{3}\left(2r_{H}^{2}+a^{2}\right)\right]}{2\pi\left(r^{2}_{H}+a^{2}\right)\left(1+\frac{\Lambda
a^{2}}{3}\right)}$ $\displaystyle\times\left(1-\frac{4\beta
J_{\theta}^{2}\Delta_{\theta}}{r^{2}_{H}+a^{2}\cos^{2}\theta}\right).$ (65)

From eq. (4), it is observed that when

$\displaystyle r_{H}\leq\sqrt{\dfrac{4\beta J^{2}_{\theta}\left(1+\Lambda
a^{2}\cos^{2}\theta\right)-3a^{2}\cos^{2}\theta}{3}},$ (66)

the modified Hawking temperature becomes negative. This violates the law of
black hole thermodynamics and thus has no physical meaning. It is clear that
the evaporation will stop under the effects of GUP. Thus the Hawking
temperature becomes zero when $r_{H}$ reaches the minimum radius, $r_{min}$ as

$\displaystyle r_{min}=$ $\displaystyle\sqrt{\dfrac{4\beta
J^{2}_{\theta}\left(1+\Lambda
a^{2}\cos^{2}\theta\right)-3a^{2}\cos^{2}\theta}{3}}.$ (67)

Using eq. (15) in eq. (3), we obtain the expression of $T_{d}$ in terms of the
mass of the black hole as

$\displaystyle T_{d}\approx\dfrac{\zeta_{1}}{\zeta_{2}}\left(1-\frac{4\beta
J_{\theta}^{2}\Delta_{\theta}}{\dfrac{1}{\alpha^{2}_{1}}\left(1+\dfrac{4\Lambda
M^{2}}{3\beta_{1}^{2}\alpha_{1}}\right)^{2}\left(M+\sqrt{M^{2}-(a^{2}+Q^{2})\alpha_{1}}\right)^{2}+a^{2}\cos^{2}\theta}\right),$
(68)

where

$\displaystyle\zeta_{1}=$
$\displaystyle\dfrac{1}{\alpha_{1}}\left(1+\dfrac{4\Lambda
M^{2}}{3\beta_{1}^{2}\alpha_{1}}\right)\left(M+\sqrt{M^{2}-(a^{2}+Q^{2})\alpha_{1}}\right)$
$\displaystyle\left[1-\dfrac{\Lambda
a^{2}}{3}-\dfrac{2\Lambda}{3~{}\alpha^{2}_{1}}\left(1+\dfrac{4\Lambda
M^{2}}{3\beta_{1}^{2}\alpha_{1}}\right)^{2}\left(M+\sqrt{M^{2}-(a^{2}+Q^{2})\alpha_{1}}\right)^{2}\right]-M,$
$\displaystyle\zeta_{2}=$ $\displaystyle 2\pi\left(1+\dfrac{\Lambda
a^{2}}{3}\right)\left[a^{2}+\dfrac{1}{\alpha^{2}_{1}}\left(1+\dfrac{4\Lambda
M^{2}}{3\beta_{1}^{2}\alpha_{1}}\right)^{2}\left(M+\sqrt{M^{2}-(a^{2}+Q^{2})\alpha_{1}}\right)^{2}\right].$
(69)

To make the Hawking temperature $T\geq 0$ i.e. to ensure the GUP corrected
temperature has a physical meaning, the mass of the black hole must hold the
inequality

$\displaystyle M\geq$
$\displaystyle\dfrac{\beta^{2}_{1}}{8(a^{2}+Q^{2})\Lambda-6\beta^{2}_{1}}\left[3\sqrt{-\alpha_{1}(a^{2}+Q^{2})}+\dfrac{3}{2}\biggl{\\{}-4\alpha_{1}(a^{2}+Q^{2})\right.$
$\displaystyle\left.+\dfrac{8\alpha_{1}\zeta_{3}\left\\{3\beta^{2}_{1}-4(a^{2}+Q^{2})\Lambda\right\\}\left(-a^{2}+Q^{2}+4J^{2}_{\theta}\alpha_{1}\beta\Delta_{\theta}-a^{2}\alpha_{1}\cos^{2}\theta\right)}{3\beta^{2}_{1}}\biggr{\\}}^{\frac{1}{2}}\right].$
(70)

It is noted that the mass of the black hole has a minimum value which is given
by

$\displaystyle M_{min}=$
$\displaystyle\dfrac{\beta^{2}_{1}}{8(a^{2}+Q^{2})\Lambda-6\beta^{2}_{1}}\left[3\sqrt{-\alpha_{1}(a^{2}+Q^{2})}+\dfrac{3}{2}\biggl{\\{}-4\alpha_{1}(a^{2}+Q^{2})\right.$
$\displaystyle\left.+\dfrac{8\alpha_{1}\zeta_{3}\left\\{3\beta^{2}_{1}-4(a^{2}+Q^{2})\Lambda\right\\}\left(-a^{2}+Q^{2}+4J^{2}_{\theta}\alpha_{1}\beta\Delta_{\theta}-a^{2}\alpha_{1}\cos^{2}\theta\right)}{3\beta^{2}_{1}}\biggr{\\}}^{\frac{1}{2}}\right].$
(71)

##  5 Graphical Analysis

In this section, we will examine graphically the effects of parameters
$\beta$, $\Lambda$ and $m$ on the modified Hawking temperatures and modified
heat capacities with respect to event horizon $r_{H}$.

###  5.1 Temperature $T_{d}$ with radius of event horizon $r_{H}$ for
3-dimensional KNdS black hole

This subsection is devoted to analysing the behaviour of modified Hawking
temperature. The parameters are taken as follows $a=0.3$, $Q=1$,
$\theta=\frac{\pi}{2}$ and $J_{\theta}=0.1$.

Figure 1:  Hawking temperature $T_{d}$ with respect to radius of event horizon
$r_{H}$ for different values of $\beta$.

Figure 2:  Hawking temperature $T_{d}$ with respect to the radius of event
horizon $r_{H}$ for different values of cosmological constant $\Lambda$.

  1. 1.

Figure 1 indicates the variation of the modified Hawking temperature with
$r_{H}>0$, for different values of $\beta$ with fixed value of the
cosmological constant $\Lambda=1$ and $m=0.1$. We observe that for $\beta=25$,
the temperature decreases and tends to zero. As the radius of horizon
increases for $\beta<25$, the modified Hawking temperatures become negative.
This negative temperature and divergent behaviour reveals the nonphysically
unstable state of the black hole wajiha1 . Moreover, for $\beta>25$, as the
horizon increases, the temperature decreases and once the minimum value is
reached, the temperature increases. It is worth mentioning that, for
$\beta<25$, we observe the nonphysical behaviour with negative temperature and
for $\beta=25$, the temperature vanishes. Furthermore, the temperatures are
positive when $\beta>25$. The $\beta$ effects decelerate the increase in
Hawking temperature which is also shown numerically in table 1.

  2. 2.

Figure 2 shows the behaviour of modified Hawking temperature with $r_{H}>0$,
for different values of positive cosmological constant $\Lambda>0$. The
parameters are taken as follows: $m=0.1$ and $\beta=40$. It is observed that
the modified Hawking temperatures decrease and attain its minimum values,
which is also calculated numerically in table 2. Later, it keeps on increasing
as $r_{H}$ increases and its behaviour is linear. The change in $\Lambda$
gives the diverging temperature $T_{d}$ as $r_{H}$ increases.

Figure 3: Hawking temperature $T_{d}$ with respect to radius of event horizon
$r_{H}$ for different values of $m$.

  3. 3.

Figure 3 indicates the behaviour of Hawking temperature for different values
of $m$. At first, the Hawking temperature drops suddenly and after attaining
its minimum point, it keeps on increasing with increasing the horizon radius.
Increasing the values of parameter $m$ tend to increase the modified Hawking
temperatures of the black hole which is also shown numerically in table 3.

Figure 4: Hawking temperature $T_{d}$ versus radius of event horizon $r_{H}$
for different values of $a$.

  4. 4.

Figure 4 provides the graphical analysis of $T_{d}$ via horizon radius $r_{H}$
for different values of spin parameter $a$. The pink line, which corresponds
to $a=0$ represents the Hawking temperature graph for Reissner-Nordstrom-de
Sitter (RNdS) black hole. The graph shows that the Hawking temperature of the
RNdS black hole is greater than that of the KNdS black hole. The effect of
spin parameter $a$ decelerates the increase in Hawking temperature. This
graphical presentation is compatible with the numerical calculation of table
4.

From figures 1 to 4 show that the temperature cools down to the minimum
$T_{d}^{C}$, at $r_{H}=r_{H}^{C}$ and keeps on increasing as
$r_{H}>r_{H}^{C}$.

a=0.3; m=0.1; $\Lambda=1$
---
$\beta$ |  25 |  30 |  50 |  70 |  90
$r_{H}^{c}$ |  7.97272 |  1.81347 |  1.43917 |  1.3935 |  1.37473
$T_{d}^{c}$ |  0.00953 |  0.06208 |  0.16458 |  0.26337 |  0.36169
Table 1:  The tabulated values of the critical radius $r_{H}^{C}$ and the
critical temperature $T_{H}^{C}$ for different values of $\beta$. a=0.3;
m=0.1; $\beta=40$
---
$\Lambda$ |  0.3 |  0.4 |  0.6 |  0.8 |  1
$r_{H}^{c}$ |  1.4842 |  1.4876 |  1.49087 |  1.49664 |  1.5016
$T_{d}^{c}$ |  0.01115 |  0.02619 |  0.05600 |  0.08596 |  0.11457
Table 2:  The tabulated values of the critical radius $r_{H}^{C}$ and the
critical temperature $T_{H}^{C}$ for different values of $\Lambda$. a=0.3;
$\Lambda=1$; $\beta=40$
---
$m$ |  0.1 |  0.3 |  0.5 |  0.7 |  0.9
$r_{H}^{c}$ |  1.5016 |  1.18474 |  1.16561 |  1.16027 |  1.15807
$T_{d}^{c}$ |  0.11457 |  1.07833 |  2.97447 |  5.81695 |  9.60655
Table 3:  The tabulated values of the critical radius $r_{H}^{C}$ and the
critical temperature $T_{H}^{C}$ for different values of emitted particle mass
$m$.  m=0.1; $\Lambda=1$; $\beta=40$
---
$a$ |  0.3 |  0.4 |  0.6 |  0.8 |  1
$r_{H}^{c}$ |  1.44254 |  1.86004 |  1.96988 |  2.07671 |  2.17653
$T_{d}^{c}$ |  0.11791 |  0.05845 |  0.04575 |  0.03314 |  0.02359
Table 4:  The tabulated values of the critical radius $r_{H}^{C}$ and the
critical temperature $T_{H}^{C}$ for different values of $a$.

###  5.2 Heat capacity $C_{H}$ with radius of event horizon $r_{H}$ for
3-dimensional KNdS black hole

This subsection focuses on analysing modified heat capacity in different
domains of event horizon radius $r_{H}$ with fixed parameters: $a=0.3$, $Q=1$,
$\theta=\frac{\pi}{2}$ and $J=0.1$. The heat capacity is connected to the
local thermal stability. If the black hole has a negative heat capacity, it is
unstable to thermal radiation and if it has a positive heat capacity, it is
stable to thermal radiation.

  1. Figure 5: Heat capacity $C_{H}$ versus radius of event horizon $r_{H}$ for different values of $\beta$.
  2. (i)

For fixed parameters $a=0.3$, $\Lambda=1$ and $m=0.5$, the variation of heat
capacity $C_{H}$ with the horizon radius $r_{H}$ for different values of
$\beta$ is shown in figure 5. It is observed that the phase transition occurs
at $r_{H}=r_{H}^{*}=1.15467$, $r_{H}^{*}$ denotes the position of the horizon
radius at which the phase transition takes place. The position of phase
transition remains unchanged in 3-dimensional KNdS black hole for different
values of $\beta$.

Figure 6: Heat capacity $C_{H}$ versus radius of event horizon $r_{H}$ for
different values of $\Lambda$.

  3. (ii)

Figure 6 illustrates the behaviour of $C_{H}$ w.r.t $r_{H}$ for fixed
parameters $a=0.3$, $\beta=40$, $m=0.5$ and for different values of $\Lambda$.
For different values of cosmological constant $\Lambda$, there are different
positions of phase transition for 3-dimensional KNdS black hole. Increasing
the values of $\Lambda$, the positions of phase transition $r_{H}^{*}$ are
shifted towards the origin, which ensures the black hole faster stability for
larger value of $\Lambda$. Different positions of phase transition for
different black hole parameters are presented in table 5.

Figure 7: Heat capacity $C_{H}$ versus radius of event horizon $r_{H}$ for
different values of $a$.

  4. (iii)

Figure 7 shows the variation of $C_{H}$ for different values of spin parameter
$a$ and for fixed $\Lambda=1$, $\beta=40$, $m=0.5$. Increasing the value of
rotation parameter $a$, the phase transition occurs at larger value of horizon
radius $r_{H}^{*}$ which is also shown in table 6. Moreover, larger the value
of $a$ delays the stability of the black hole.

Figures 5 to 7 show that there is a phase transition when $r_{H}=r_{H}^{*}$.
The positions of phase transition are shown in tables 5 and 6. The black holes
are unstable in the region $0<r_{H}<r_{H}^{*}$ and stable in the region
$r_{H}^{*}<r_{H}<\infty$ i.e. the smaller black holes are less stable than
larger black holes and vice versa.

a=0.3; $m=0.5$; $\beta=40$
---
$\Lambda$ |  0.3 |  0.5 |  0.7 |  0.9
$r_{H}^{*}$ |  1.42403 |  1.31282 |  1.23652 |  1.17887
Table 5:  The tabulated values of the position of phase transition $r_{H}^{*}$
for different values of $\Lambda$.  $\Lambda=0.1$; $m=0.5$; $\beta=40$
---
$a$ |  0.5 |  1 |  1.5 |  2
$r_{H}^{*}$ |  1.17485 |  1.23815 |  1.29861 |  1.35098
Table 6:  The tabulated values of the position of phase transition $r_{H}^{*}$
for different values of $a$.

###  5.3  Hawking temperature $T_{d4}$ with radius of event horizon for
4-dimensional KNdS black hole

This subsection is devoted to analysing the behaviour of modified Hawking
temperatures of KNdS black hole. The parameters are taken as follows: $Q=0.1$,
$\theta=\frac{\pi}{2}$, $e=1$, $J_{\theta}=0.1$ and $J_{\phi}=0.2$.

  1. Figure 8: Hawking temperature $T_{d4}$ with respect to radius of event horizon $r_{H}$ for different values of $m$.
  2. (i)

Figure 8 shows the behaviour of $T_{d4}$ for different values of $\beta$ and
for fixed values of $a=0.2$, $\Lambda=1$ and $m=0.1$. $T_{d4}$ increases
exponentially and reaches its maximum height $T_{d4}^{max}$ for different
values of $\beta$. Further, the temperature keeps on decreasing with
increasing the horizon radius. It is noteworthy to mention that the
temperature increases with increasing the values of $\beta$. Hence, the
$\beta$ effects accelerate the increase in $T_{d4}$. The proof of the above
statement is also calculated in table 7.

Figure 9: Hawking temperature $T_{d4}$ with respect to radius of event horizon
$r_{H}$ for different values of $\beta$.

  3. (ii)

The behaviour of $T_{d4}$ for varying cosmological constant $\Lambda$ is
depicted in figure 9. The parameters are as follows: $a=0.2$, $\beta=50$ and
$m=0.1$. The temperature increases to a certain height and attains its peak
point $T_{d4}^{~{}max}$ at $r_{H}=r_{H}^{c}$, then $T_{d4}$ decreases as
$r_{H}$ increases. It is noted that the $\Lambda$ effects decelerate the
increase in $T_{d4}$ which is also shown numerically in table 8.

  4. (iii)

The behaviour of $T_{d4}$ w.r.t $r_{H}$ for varying mass of the vector boson
particle $m$ is depicted in Figure 10. The temperature increases to a certain
height $T_{d4}^{~{}max}$ and then decreases with increasing $r_{H}$. The rate
of increase of temperature $T_{d4}$ is dependent on the increase of $m$. The
validity of the above statements is calculated numerically in table 9.

  5. (iv)

Figure 11 shows the behaviour of $T_{d4}$ w.r.t $r_{H}$ for varying $a$ and
fixed $\beta=50$, $\Lambda=1$ and $m=0.1$. The temperatures increase
exponentially upto $T_{d4}^{~{}max}$ and decrease with increasing the horizon
radius. The temperature gradually increases with decreasing the values of
rotation parameter, $a$. The numerical calculation shown in table 10 supports
the above statement.

Figure 10: Hawking temperature $T_{d4}$ with respect to radius of event
horizon $r_{H}$ for different values of $m$.

Figure 11: Hawking temperature $T_{d4}$ with respect to radius of event
horizon $r_{H}$ for different values of $a$.

We present the following tables to see the effects of $\beta$, $\Lambda$, $m$
and $a$ on the Hawking temperature. Table 7 confirms that $T_{d4}^{~{}max}$
gradually increases on increasing $\beta$ (for fixed values of $\lambda$, $m$
and $a$). Similarly table 9 also confirms that $T_{d4}^{~{}max}$ gradually
increases on increasing $m$ (for fixed values of $\beta$, $\lambda$, and $a$).
On the contrary, tables 8 and 10 confirm that $T_{d4}^{~{}max}$ gradually
decreases with increasing $\lambda$ (for fixed values of $\beta$, $m$ and $a$)
and $a$ (for fixed values of $\beta$, $\lambda$, and $m$) respectively. From
table 9, it is observed that $T_{d4}^{~{}max}$ is highly dependent on $m$
compared to that of $\beta$, $\Lambda$ and $a$.

$a$=0.2; $\Lambda=1$; $m=0.1$
---
$\beta$ |  10 |  30 |  50 |  70 |  90
$r_{H}^{c}$ |  0.34861 |  0.34849 |  0.34847 |  0.34846 |  0.34845
$T_{d4}^{~{}max}$ |  14.2974 |  42.7385 |  71.1796 |  99.6206 |  128.062
Table 7:  The tabulated values of $r_{H}^{C}$ and $T_{d4}^{~{}max}$ for
different values of $\beta$.  $a$=0.2; $\Lambda=1$; $\beta=50$
---
$\Lambda$ |  0.2 |  0.4 |  0.6 |  0.8 |  1
$r_{H}^{c}$ |  0.35376 |  0.35244 |  0.35117 |  0.34979 |  0.34847
$T_{d4}^{~{}max}$ |  87.2216 |  82.793 |  78.4615 |  75.4107 |  71.1796
Table 8:  The tabulated values of $r_{H}^{C}$ and $T_{d4}^{~{}max}$ for
different values of $\Lambda$.  $a=0.2$; $\Lambda=1$; $\beta=50$
---
$m$ |  0.1 |  0.3 |  0.5 |  0.7 |  0.9
$r_{H}^{c}$ |  0.34847 |  0.39069 |  0.45936 |  0.50492 |  0.53291
$T_{d4}^{~{}max}$ |  71.1796 |  111.502 |  211.269 |  386.449 |  652.623
Table 9:  The tabulated values of $r_{H}^{C}$ and $T_{d4}^{~{}max}$ for
different values of $m$.  $\beta=20$; $\Lambda=1$; $m=0.1$
---
$a$ |  0.1 |  0.15 |  0.2 |  0.25 |  0.3
$r_{H}^{c}$ |  0.2075 |  0.27555 |  0.34852 |  0.42236 |  0.49421
$T_{d4}^{~{}max}$ |  176.342 |  65.3181 |  28.518 |  13.7598 |  6.89087
Table 10:  The tabulated values of $r_{H}^{C}$ and $T_{d4}^{~{}max}$ for
different values of $a$.

###  5.4 Heat Capacity $C_{H4}$ versus horizon radius $r_{H}$ for
4-dimensional KNdS black hole

This subsection studies the modified heat capacity for different values of
$\beta$, $\Lambda$ and $a$ for fixed values of $Q=1$, $\theta=\frac{\pi}{2}$,
$e=1$, $J_{\theta}=0.1$ and $J_{\phi}=0.2$.

  1. (i)

The variation of modified heat capacity for different values of $\beta$ is
shown in figure 12. The parameters are taken as $a=0.2$, $\Lambda=0.5$ and
$m=0.1$. It is observed that there is one position of phase transition in the
absence of GUP, but there are two positions of phase transition under the
influence of GUP. The first phase transition in figure 12 is due to the
quantum gravity effects and it occurs at $r_{H}=r_{H_{1}}=0.99338$. The
position of second phase transition is at $r_{H}=r_{H_{2}}=1.29667$. The
variation of $\beta$ doesn’t affect the position of phase transition.

  2. (ii)

For $a=0.2$, $\beta=15$ and $m=0.1$, the variation of $C_{H4}$ w.r.t $r_{H}$
changing the values of $\Lambda$ is illustrated in figure 13. Varying the
values of cosmological constant $\Lambda$, the positions of phase transition
are also varied. With increasing the values of $\Lambda$, the position of
phase transition is shifted toward the origin, which implies a slower rate of
becoming a stable black hole. Table 11 is constructed numerically to show the
different positions of phase transitions.

Figure 12: Heat Capacity $C_{H4}$ with respect to radius of event horizon
$r_{H}$ for different values of $\beta$.

Figure 13: Heat Capacity $C_{H4}$ with respect to radius of event horizon
$r_{H}$ for different values of $\Lambda$.

  3. (iii)

Figure 14 represents the behaviour of $C_{H4}$ versus $r_{H}$ for fixed
$\beta=15$, $\Lambda=0.5$ and $m=0.1$. The position of phase transition is
shifted to far away from the origin toward the positive direction of $r_{H}$
with increasing the value of spin parameter $a$. It shows that the black hole
becomes stable faster with increasing the value of rotational parameter $a$
which is also indicated numerically by table 11.

Figure 14: Heat Capacity $C_{H4}$ with respect to radius of event horizon
$r_{H}$ for different values of $a$.

$a=0.2$; $\beta=15$; $m=0.1$ |  $\Lambda=0.5$; $\beta=15$; $m=0.1$
---|---
$\Lambda$ |  $r_{H_{1}}$ |  $r_{H_{2}}$ |  $a$ |  $r_{H_{1}}$ |  $r_{H_{2}}$
0.3 |  0.99602 |  1.40094 |  0.1 |  0.49917 |  1.2864
0.5 |  0.99338 |  1.29667 |  0.4 |  0.7472 |  1.29073
0.7 |  0.99075 |  1.22479 |  0.6 |  0.99338 |  1.29667
Table 11:  The tabulated values of $r_{H_{1}}$ and $r_{H_{2}}$ for different
values of $\Lambda$ and $a$.

##  6 Conclusion

This work studies the GUP effects on tunneling of massive vector boson
particles from KNdS black hole. Firstly, we utilize the GUP-corrected
Lagrangian of massive vector field and derive the modified wave equation for
massive vector boson particles. Using the modified wave equation, the quantum
tunneling of KNdS black hole in 3-dimensional and 4-dimensional frame dragging
coordinates are investigated. Further, the modified Hawking temperatures and
heat capacities due to GUP are derived. They depend not only on the quantum
gravity parameter $\beta$, spin parameter $a$, mass of the emitted particle
$m$, cosmological constant $\Lambda$, charge of the black hole $Q$ but also on
angular coordinates $\theta$, $J_{\theta}$ and $J_{\phi}$. For 3-dimensional
KNdS black hole, the corrected Hawking temperature is lower than the original
Hawking temperature which shows that the quantum gravity effects slow down the
increase of the Hawking radiation temperature. Moreover, the modified heat
capacity of 3-dimensional KNdS black hole is greater than the original heat
capacity. In the case of 4-dimensional KNdS black hole, the modified Hawking
temperature is either lower or higher than the original Hawking temperature
according to $\Xi>0$ or $\Xi<0$ respectively. Further, the modified heat
capacity is either lower or higher than the original Hawking temperature
according to $\Xi<0$ or $\Xi>0$ respectively. The stable and unstable
formation of black hole are studied in quantum gravity effects. The remnant of
3-dimensional KNdS black hole is also discussed in the presence of quantum
gravity effects. We also illustrate the graphs of modified Hawking
temperatures and heat capacities and explore the effects of $\beta$,
$\Lambda$, $a$ and $m$.  If the radius of event horizon $r_{H}$ increases, the
modified Hawking temperature of a 3-dimensional KNdS black hole tends to
decrease for $\beta<25$, but for $\beta>25$, the temperature cools down till
it reaches its minimum point and then increases, which leads to the formation
of stable black hole. For a 4-dimensional KNdS black hole with the above fixed
parameters, the modified Hawking temperature increases w.r.t $r_{H}$ and after
attaining maximum height, the temperature eventually goes down. It is worth
noting that there are one phase transition and two phase transitions for a
non-zero horizon of 3-dimensional KNdS black hole and 4-dimensional KNdS black
hole respectively. Different positions of phase transitions are due to the
quantum gravity effects. It is noted that for different values of
dimensionless parameter $\beta$, the position of phase transitions remain the
same in 3-dimensional and 4-dimensional KNdS black hole under the influence of
quantum gravity effects. The modified Hawking temperatures and heat capacities
tend to the original Hawking temperature and heat capacity of KNdS black hole
in the absence of quantum gravity effects. Hence quantum gravity effects
modified the Hawking temperature and heat capacity of the black hole.

##  Appendix A Coefficients of eq. (3)

$\displaystyle A^{*}_{0}=$
$\displaystyle\dfrac{(r-r_{H})\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})},$ (72)
$\displaystyle A^{*}_{1}=$ $\displaystyle
m^{2}+\dfrac{J_{\theta}^{2}\Delta_{\theta}}{\rho^{2}(r_{H})}-\dfrac{(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Sigma^{2}}{(r-r_{H})\Delta,_{r}(r_{H})\rho^{2}(r_{H})}+\dfrac{\left(J_{\phi}+eA_{3}\right)^{2}\rho^{2}(r_{H})\Sigma^{2}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}},$
(73) $\displaystyle B_{6}=$
$\displaystyle\dfrac{6m^{2}\beta~{}(r-r_{H})^{3}~{}\Delta^{2},_{r}(r_{H})}{\rho^{6}(r_{H})},$
(74) $\displaystyle B_{4}=$
$\displaystyle\dfrac{m^{2}~{}(r-r_{H})^{2}~{}\Delta^{2},_{r}(r_{H})}{\rho^{4}(r_{H})}+\dfrac{2\beta~{}(r-r_{H})^{2}~{}\Delta^{2},_{r}(r_{H})}{\rho^{4}(r_{H})}\Biggr{[}-\Biggl{\\{}\dfrac{1}{(r-r_{H})~{}\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}$
$\displaystyle\Bigg{(}(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Sigma^{2}\bigg{[}-\dfrac{2\left(J_{\phi}+eA_{3}-1\right)\left(J_{\phi}+eA_{3}\right)\rho^{2}(r_{H})\Sigma^{2}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}}$
$\displaystyle+3m^{2}\biggr{]}\Bigg{)}\Biggr{\\}}+m^{2}\biggl{[}3m^{2}+\dfrac{3J^{2}_{\theta}\Delta_{\theta}}{\rho^{2}(r_{H})}+\dfrac{\left(J_{\phi}+eA_{3}\right)\left(2+J_{\phi}+eA_{3}\right)\rho^{2}(r_{H})\Sigma^{2}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}}\biggr{]}\Biggr{]},$
(75) $\displaystyle B_{2}=$
$\displaystyle\dfrac{(r-r_{H})~{}\Delta,_{r}(r_{H})}{\rho^{2}(r_{H})}\Biggl{[}\dfrac{(-E+j\Omega+eA_{0})^{2}\left(J_{\phi}+eA_{3}\right)\left(J_{\phi}+eA_{3}-1\right)\Sigma^{4}\csc^{2}\theta}{(r-r_{H})~{}\Delta^{2},_{r}(r_{H})}$
$\displaystyle+2m^{4}+\dfrac{2~{}J_{\theta}^{2}m^{2}\Delta_{\theta}}{\rho^{2}(r_{H})}+\dfrac{\left(J_{\phi}+eA_{3}\right)\left(1+J_{\phi}+eA_{3}\right)m^{2}\rho^{2}(r_{H})\Sigma^{2}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}}$
$\displaystyle-\dfrac{2m^{2}(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Sigma^{2}}{(r-r_{H})~{}\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}+2\beta\Biggl{\\{}\dfrac{1}{(r-r_{H})^{2}~{}\Delta^{2},_{r}(r_{H})~{}\rho^{4}(r_{H})}$
$\displaystyle\biggl{[}(-E+j\Omega+eA_{0})^{4}\left(r^{2}_{H}+a^{2}\right)^{4}\Sigma^{4}\biggl{\\{}\dfrac{\left(J_{\phi}+eA_{3}-1\right)\left(J_{\phi}+eA_{3}\right)\rho^{2}(r_{H})\Sigma^{2}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}}$
$\displaystyle-3m^{2}\biggr{\\}}\biggr{]}+\dfrac{(-E+j\Omega+eA_{0})^{2}\left(J_{\phi}+eA_{3}\right)^{3}\left(J_{\phi}+eA_{3}-1\right)\rho^{2}(r_{H})\Sigma^{6}\csc^{4}\theta}{(r-r_{H})~{}\left(r^{2}_{H}+a^{2}\right)^{2}\Delta^{2}_{\theta}~{}\Delta,_{r}(r_{H})}$
$\displaystyle+m^{2}\bigg{[}\dfrac{3J^{4}_{\theta}\Delta_{\theta}^{2}}{\rho^{4}(r_{H})}+\dfrac{\left(J_{\phi}+eA_{3}\right)^{3}\left\\{1+2\left(J_{\phi}+eA_{3}\right)\right\\}\rho^{4}(r_{H})\Sigma^{4}\csc^{4}\theta}{\left(r^{2}_{H}+a^{2}\right)^{4}\Delta^{2}_{\theta}}\biggr{]}\Biggr{\\}}\Biggr{]},$
(76) $\displaystyle B_{0}=$
$\displaystyle-\Biggl{[}\dfrac{J_{\theta}^{2}\Delta_{\theta}}{\rho^{2}(r_{H})}-\dfrac{(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Sigma^{2}}{(r-r_{H})\Delta,_{r}(r_{H})\rho^{2}(r_{H})}+\dfrac{\left(J_{\phi}+eA_{3}\right)^{2}\rho^{2}(r_{H})\Sigma^{2}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}}$
$\displaystyle+m^{2}\Biggl{]}+\dfrac{2~{}\beta}{(r-r_{H})^{3}\left(r^{2}_{H}+a^{2}\right)^{6}\Delta^{3}_{\theta}~{}\Delta^{3},_{r}(r_{H})~{}\rho^{6}(r_{H})}\Biggr{[}3(-E+j\Omega+eA_{0})^{6}\Delta_{\theta}^{6}\Sigma^{6}$
$\displaystyle\left(r^{2}_{H}+a^{2}\right)^{10}\biggl{\\{}m^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}-\left(J_{\phi}+eA_{3}\right)\left(J_{\phi}+eA_{3}-1\right)\rho^{2}(r_{H})\Sigma^{2}\csc^{2}\theta\biggr{\\}}$
$\displaystyle-(-E+j\Omega+eA_{0})^{4}(r-r_{H})\left(r^{2}_{H}+a^{2}\right)^{8}\Delta^{2}_{\theta}~{}\Delta,_{r}(r_{H})\Sigma^{4}\Biggr{(}3m^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}$
$\displaystyle\left(J_{\theta}^{2}\Delta_{\theta}+m^{2}\rho^{2}(r_{H})\right)-\left(J_{\phi}+eA_{3}\right)^{2}\rho^{2}(r_{H})\biggl{\\{}J_{\theta}^{2}\left(J_{\phi}+eA_{3}-1\right)\Delta_{\theta}-3m^{2}\rho^{2}(r_{H})\biggr{\\}}$
$\displaystyle\Sigma^{2}\csc^{2}\theta\Biggr{)}+m^{2}(r-r_{H})^{3}\Delta^{3},_{r}(r_{H})\Biggl{(}3J_{\theta}^{6}\left(r^{2}_{H}+a^{2}\right)^{6}\Delta^{6}_{\theta}+J_{\theta}^{2}~{}\Delta^{2}_{\theta}~{}\rho^{8}(r_{H})~{}\Sigma^{4}\csc^{4}\theta$
$\displaystyle\left(J_{\phi}+eA_{3}\right)^{3}\biggl{\\{}1+2\left(J_{\phi}+eA_{3}\right)\biggr{\\}}+\left(J_{\phi}+eA_{3}\right)^{3}\rho^{10}(r_{H})~{}\Sigma^{4}\csc^{4}\theta\biggl{[}m^{2}\Delta_{\theta}$
$\displaystyle\biggl{\\{}1+2\left(J_{\phi}+eA_{3}\right)\biggr{\\}}\left(r^{2}_{H}+a^{2}\right)^{2}+3\rho^{2}(r_{H})\Sigma^{2}\csc^{2}\theta\left(J_{\phi}+eA_{3}\right)^{2}\biggr{]}+J_{\theta}^{4}~{}\Delta_{\theta}^{4}~{}\rho^{2}(r_{H})$
$\displaystyle\left(r^{2}_{H}+a^{2}\right)^{4}\biggl{\\{}3m^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}+\left(J_{\phi}+eA_{3}\right)\left(2+J_{\phi}+eA_{3}\right)~{}\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta\biggr{\\}}\Biggr{)}$
$\displaystyle-(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}(r-r_{H})^{2}\Delta^{2},_{r}(r_{H})\Sigma^{2}\Biggl{(}\left(1-J_{\phi}-eA_{3}\right)\left(J_{\phi}+eA_{3}\right)$
$\displaystyle\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta\biggl{\\{}2J^{2}_{\theta}\left(r^{2}_{H}+a^{2}\right)^{4}\Delta_{\theta}^{4}+J_{\theta}^{2}\left(J_{\phi}+eA_{3}\right)^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}^{2}\rho^{4}(r_{H})~{}\Sigma^{2}$
$\displaystyle\csc^{2}\theta+3\left(J_{\phi}+eA_{3}\right)^{4}\rho^{8}(r_{H})~{}\Sigma^{4}~{}\csc^{4}\theta\biggr{\\}}+m^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}\biggl{\\{}3J_{\theta}^{4}\Delta_{\theta}^{4}\left(r^{2}_{H}+a^{2}\right)^{4}$
$\displaystyle+\left(J_{\phi}+eA_{3}\right)^{3}\left(2+J_{\phi}+eA_{3}\right)\rho^{8}(r_{H})~{}\Sigma^{4}~{}\csc^{4}\theta\biggr{\\}}\Biggr{)}\Biggr{]}.$
(77)

##  Appendix B The expressions of $\chi_{1}$ and $\chi_{2}$ given in eq. (3)

$\displaystyle\chi_{1}=$
$\displaystyle-\dfrac{3(-E+j\Omega+eA_{0})^{6}m^{2}\left(r^{2}_{H}+a^{2}\right)^{6}\Sigma^{6}}{(r-r_{H})^{3}\Delta^{3},_{r}(r_{H})~{}\rho^{6}(r_{H})}+\dfrac{(-E+j\Omega+eA_{0})^{4}~{}\Sigma^{4}}{(r-r_{H})^{2}~{}\Delta_{\theta}^{2}~{}\Delta^{2},_{r}(r_{H})~{}\rho^{6}(r_{H})}$
$\displaystyle\Biggl{[}9m^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}-4\left(J_{\phi}+eA_{3}-1\right)\left(J_{\phi}+eA_{3}\right)\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta\Biggr{]}$
$\displaystyle\Biggl{[}\Delta_{\theta}\left(r^{2}_{H}+a^{2}\right)^{2}\left(J_{\theta}^{2}\Delta_{\theta}+m^{2}~{}\rho^{2}(r_{H})\right)+\left(J_{\phi}+eA_{3}\right)^{2}\rho^{4}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta\Biggr{]}$
$\displaystyle+m^{2}\Biggl{[}\dfrac{3\left(J_{\theta}^{2}\Delta_{\theta}+m^{2}~{}\rho^{2}(r_{H})\right)^{3}}{\rho^{6}(r_{H})}+\dfrac{\left(J_{\phi}+eA_{3}\right)\Sigma^{2}~{}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}~{}\rho^{2}(r_{H})}\Biggl{\\{}2J_{\theta}^{2}~{}m^{2}~{}\Delta_{\theta}~{}\rho^{2}(r_{H})$
$\displaystyle\bigl{[}2+7\left(J_{\phi}+eA_{3}\right)\bigr{]}+J_{\theta}^{2}~{}\Delta_{\theta}^{2}\bigl{[}4+5\left(J_{\phi}+eA_{3}\right)\bigr{]}+m^{4}~{}\rho^{4}(r_{H})\bigl{[}4+5\left(J_{\phi}+eA_{3}\right)\bigr{]}\Biggr{\\}}$
$\displaystyle+\dfrac{\rho^{2}(r_{H})~{}\Sigma^{4}~{}\csc^{4}\theta\left(J_{\phi}+eA_{3}\right)^{3}\bigl{[}4+5\left(J_{\phi}+eA_{3}\right)\bigr{]}\left(J_{\theta}^{2}~{}\Delta_{\theta}+m^{2}~{}\rho^{2}(r_{H})\right)}{\left(r^{2}_{H}+a^{2}\right)^{4}\Delta_{\theta}^{2}}$
$\displaystyle-\dfrac{\left(J_{\phi}+eA_{3}-4\right)\left(J_{\phi}+eA_{3}\right)^{5}\rho^{6}(r_{H})\Sigma^{6}\csc^{6}\theta}{\left(r^{2}_{H}+a^{2}\right)^{6}~{}\Delta_{\theta}^{3}}\Biggr{]}-\dfrac{(j\Omega+eA_{0}-E)^{2}\left(r^{2}_{H}+a^{2}\right)^{2}\Sigma^{2}}{(r-r_{H})~{}\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}$
$\displaystyle\Biggl{[}9m^{6}+m^{2}\biggl{\\{}\dfrac{J_{\theta}^{2}~{}\Delta_{\theta}}{\rho^{2}(r_{H})}+\dfrac{\left(J_{\phi}+eA_{3}\right)^{2}~{}\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta}{\left(r^{2}_{H}+a^{2}\right)^{2}~{}\Delta_{\theta}}\biggr{\\}}$
$\displaystyle\biggl{\\{}\dfrac{9J_{\theta}\Delta_{1}}{\rho^{2}(r_{H})}+\dfrac{\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta\left(J_{\phi}+eA_{3}\right)\left(8+J_{\phi}+eA_{3}\right)}{\left(r^{2}_{H}+a^{2}\right)^{2}~{}\Delta_{\theta}}\biggr{\\}}$
$\displaystyle+6m^{4}\biggl{\\{}\dfrac{3J_{\theta}^{2}\Delta_{\phi}}{\rho^{2}(r_{H})}+\dfrac{\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta\left(J_{\phi}+eA_{3}\right)\left[1+2(J_{\phi}+eA_{3})\right]}{\left(r^{2}_{H}+a^{2}\right)^{2}~{}\Delta_{\theta}}\biggr{\\}}$
$\displaystyle-\dfrac{4~{}\Sigma^{2}~{}\csc^{2}\theta\left(J_{\phi}+eA_{3}-1\right)\left(J_{\phi}+eA_{3}\right)}{\left(r^{2}_{H}+a^{2}\right)^{6}~{}\Delta_{\theta}^{3}~{}\rho^{2}(r_{H})}\biggl{\\{}J_{\theta}^{4}~{}\Delta_{\theta}^{4}\left(r^{2}_{H}+a^{2}\right)^{4}+J_{\theta}^{2}~{}\Delta_{\theta}^{2}~{}\rho^{4}(r_{H})$
$\displaystyle\Sigma^{2}~{}\csc^{2}\theta\left(J_{\phi}+eA_{3}\right)^{2}\left(r^{2}_{H}+a^{2}\right)^{2}+~{}\rho^{8}(r_{H})\Sigma^{4}~{}\csc^{4}\theta\left(J_{\phi}+eA_{3}\right)^{4}\biggr{\\}}\Biggr{]},$
(78) $\displaystyle\chi_{2}=$
$\displaystyle\dfrac{\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta\left(1-J_{\phi}-eA_{3}\right)\left(J_{\phi}+eA_{3}\right)}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}}\Biggl{[}m^{2}-\dfrac{\Sigma^{2}(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})~{}\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}\Biggr{]}$
$\displaystyle\Biggl{[}m^{2}+\dfrac{J_{\theta}^{2}\Delta_{\theta}}{\rho^{2}(r_{H})}-\dfrac{\Sigma^{2}(-E+j\Omega+eA_{0})^{2}\left(r^{2}_{H}+a^{2}\right)^{2}}{(r-r_{H})~{}\Delta,_{r}(r_{H})~{}\rho^{2}(r_{H})}+\dfrac{\rho^{2}(r_{H})~{}\Sigma^{2}~{}\csc^{2}\theta\left(J_{\phi}+eA_{3}\right)^{2}}{\left(r^{2}_{H}+a^{2}\right)^{2}\Delta_{\theta}}\Biggr{]}.$
(79)

###### Acknowledgements.

The first author would like to thank the Manipur University, Canchipur for
providing Non-NET fellowship. The authors also acknowledge the anonymous
reviewers for valuable suggestions and comments to improve the paper.

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