∎

11institutetext:  Y. Onika Laxmi 22institutetext: Department of Mathematics,
Manipur University, Canchipur, 795003, India
22email<EMAIL_ADDRESS>33institutetext: T. Ibungochouba Sing${\rm
h}^{*}$ 44institutetext: Department of Mathematics, Manipur University,
Canchipur, 795003, India
44email<EMAIL_ADDRESS>55institutetext: I. Ablu Meitei
66institutetext: Physics Department, D.M. College of Science, Imphal, Manipur,
795001, India
66email<EMAIL_ADDRESS>
# Modified entropy of Kerr-de Sitter black hole in Lorentz symmetry violation
theory

Y. Onika Laxmi  T. Ibungochouba Sing${\rm h}^{*}$ I. Ablu Meitei

(Received: date / Accepted: date)

###### Abstract

The quantum tunneling radiation of scalar particles near the event horizon of
Kerr-de Sitter black hole is investigated in three systems of coordinates
namely naive coordinate system, Painleve coordinate system and Eddington
coordinate system using Lorentz violation theory in curved space time. The
Klein-Gordon equation of scalar particles is transformed into Hamilton-Jacobi
equation by using Lorentz violation theory in curved space time. We observe
that due to Lorentz violation theory, the expressions of Hawking temperatures,
the Bekenstein-Hawking entropies and heat capacities near the event horizon of
Kerr-de Sitter black hole are modified. The Hawking temperatures, entropies
and heat capacities increase or decrease depending upon the choices of ether
like vectors $u^{\alpha}$.

###### Keywords:

Kerr-de Sitter solution Bekenstein-Hawking entropy Klein-Gordon equation
Lorentz violation theory in curved space time

###### pacs:

4.20.Gz 04.20.-q 03.65.-w

##  1 Introduction

Using quantum field theory in curved space time, Hawking [1, 2] proposed that
a black hole emits radiation like a blackbody radiation. Refs. [3-5] lead to
connect black hole with thermodynamics. They showed that the entropy of black
hole is proportional to its horizon area. Since then, many researchers studied
the Hawking radiation from the static, stationary and nonstationary black
hole. Refs. [6, 7] developed a new technique to investigate the thermal
radiation of stationary and non stationary black holes. In this technique, the
radial part derived from Dirac equation and Maxwell’s equation can be recast
into a single form of wave equation at the event horizon of the black hole.
Then the ingoing and outgoing wave equations can be derived and the
corresponding Hawking radiation of black hole can be studied. More interesting
results of Hawking radiations of stationary and nonstationary black holes have
been obtained in [8-11]. Taking self gravitational interaction and the change
of curved space time, Refs. [12-14] proposed the modification of Hawking pure
thermal radiation spectrum and they found the conservation of information
during the thermal radiation from stationary black hole. Zhang and Zhao
[15-17] have generalized the Parikh-Wilczek’s technique to investigate the
tunneling of charged massive particle near the event horizon of black hole
where the geodesic equation has been derived by studying the relation between
the phase and group velocities of the tunneling particle. The tunneling of
massless particle through quantum horizon of Schwarzscild black hole is
investigated by taking quantum gravity into account. The minimal length,
minimal momentum and maximal momentum are also discussed using quantum gravity
effects in the framework of the generalized uncertainty principle. There seems
to be some connection between Lorentz symmetry violation and quantum gravity
effects [18]. Nozari and Mehdipour [19] also investigated the Hawking
radiation as a tunneling near the horizon of Schwarzscild black hole in
noncommutative space time. They showed that there is no correlation between
different modes of radiation in the noncommutative space time. Ref. [20] as an
extension of complex path analysis [21], discussed the Hawking radiation of
black hole using semi classical Hamilton-Jacobi equation, Feynman prescription
and WKB approximation ignoring the particle back reaction effect. The Hawking
radiation of Schwarzschild black hole has been discussed in naive coordinate
system and isotropic coordinate or invariant coordinate system. The Hawking
radiations in different black holes have been derived using Hamilton-Jacobi
equation [22-25 ]. Kerner and Mann [26] proposed the tunneling of spin-1/2
fermions near the event horizon of black hole using Dirac equation, Pauli
Sigma matrices and WKB approximation. To choose appropriate Gamma matrix from
the line element and substituting to the Dirac equation, the imaginary part of
the action related to the Boltzmann factor for emission at the Hawking
temperature is obtained. The tunneling of vector boson particle near the event
horizon of black hole is investigated using Hamilton-Jacobi ansatz to Proca
equation, Feynman presciption and WKB aproximation [27, 28]. They show that
the emission temperature is consistent with the Hawking temperature
corresponding to scalar particle for the Schwarzschild black hole geometry.
Applying Proca equation, the Hawking temperature of different black holes have
been obtained in [29-32]. The change in Bekenstein-Hawking entropy of black
hole has been investigated using Hamilton-Jacobi equation beyond the
semiclassical approximation and first law of black hole thermodynamics [33].
Refs. [34-37] extended the change in Bekenstein-Hawking entropy by tunneling
of fermions across the event horizon of black hole using Dirac equation in
curved space time. The general theory of relativity is based on gravity which
is impossible to renormalise, therefore many researchers proposed different
modified gravity theories. One of the theories which can break at high energy
levels is Lorentz symmetry. Refs. [38-42] proposed the Lorentz symmetry
violation based on various gravity models. Refs. [43, 44] investigated the
Lorentz violation theory of Dirac equation in flat Euclidean space using ether
like vectors $u^{\alpha}$. Many interesting results of Hawking radiation of
black hole with Lorentz violation theory have also been discussed in [45-48].
Liu, et al. [49] discussed the entropy correction of black hole using the
modified Hamilton-Jacobi equation in Lorentz violation theory and beyond the
semiclassical approximation. They show that the modified entropy depends on
the choices of ether like vector $u^{\alpha}$.

A quantum theory of gravity would be required to unify the general relativity
which is a classical theory and the standard model of particle Physics which
is a quantum theory. The relevant scale of quantum theory of gravity is the
Planck scale $(\sim 10^{19}\rm{GeV})$. It is not possible to conduct
experiments at such a scale. Violation of Lorentz symmetry could provide a key
to experimental probes at the Planck scale [50]. If Lorentz invariance is
violated at the Planck scale, it is likely that there must be an interpolation
at much lower energy so that a small amount of Lorentz violation should be
present at all energies [51]. Some experiments designed to test low energy
residual effects of Planck scale Lorentz violation are- (i) Penning traps
[52-54] (ii) Clock comparison experiments [55, 56] (iii) Spin polarized
torsion pendulum [57, 58] (iv) Meson tests [59-61] (v) Muon tests [62, 63]
(vi) Photon tests [64, 65] (vii) Neutrino tests [66, 67] (viii) Gravity tests
[68, 69] (ix) Cosmic rays [70, 71].

To study the modified Hawking temperature, entropy and heat capacity near the
event horizon of KdS black hole in Lorentz violation theory, we use three
coordinate systems namely naive, Painleve and Eddington coordinate systems.
The Painleve coordinate system has many attractive features. Firstly, the
metric components are regular near the event horizon of black hole. Secondly,
the constant time slice is flat Euclidean space. The importance of the second
one is that WKB approximation can be utilized to calculate the emission rate.
Using the quantum tunneling of flat space time, the WKB approximation can be
obtained. Thirdly, the time like Killing vector field keeps the spacetime
stationary. The advantage of a new coordinate system is to eliminate the
singularity near the event horizon of KdS black hole. The components of metric
in new coordinate system satisfied the Landau’s condition of the coordinate
clock synchronization [72-75].

The paper is organised as follows: In section 2, we discuss the derivation of
modified Hamilton-Jacobi equation from Klein-Gordon equation using Lorentz
violation theory in curved space time. In section 3, the actual Hawking
temperature and the modified Hawking temperature are obtained from direct
calculation and naive coordinate system using modified Hamilton-Jacobi
equation with Lorentz violation theory in curved space time respectively.
Using Painleve coordinates and Hamilton-Jacobi equation with Lorentz violation
theory in curved space time, the Hawking temperature, change in Bekenstein-
Hawking entropy and heat capacity of KdS black hole near the event horizon is
investigated in section 4. In section 5, we discuss the modified Hawking
temperature, change in Bekenstein-Hawking entropy and heat capacity of KdS
black hole using Eddington coordinate and Hamilton-Jacobi equation in Lorentz
violation theory. Some discussions and conclusions are given in section 6.

##  2 Lorentz violation theory and the modified Hamilton-Jacobi equation

The recent researches on high energy quantum gravity theory show that the
Lorentz-violation is needed to be modified in the Planck scale. Gomes, et al.
[76] developed a new Lorentz violation scalar equation in which the Lagrangian
${\mathcal{L}}$ is given as

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\partial_{\mu}\psi\partial^{\mu}\psi+\lambda(u^{\alpha}\partial_{\alpha}\psi)^{2}+m^{2}\psi^{2}],$
(1)

where $m$ and $u^{\alpha}$ are the mass of the scalar particle and etheric
like vector respectively and $\lambda$ is a proportionality constant and
$u^{\alpha}$ also satisfies

$\displaystyle u^{\alpha}u_{\alpha}$ $\displaystyle=$ $\displaystyle{\rm
const.}$ (2)

Coupling of the electromagnetic field with the ether like vector gives rise to
a preferred direction in space time leading to Lorentz symmetry violation [77,
78]. The scalar field equation in flat space time in accordance with the
principle of least action is

$\displaystyle\partial_{\mu}\partial^{\mu}\psi+\lambda
u^{\alpha}u^{\beta}\partial_{\alpha}\partial_{\beta}\psi+m^{2}\psi$
$\displaystyle=$ $\displaystyle 0.$ (3)

The etheric-like vector $u^{\alpha}$ may not be taken as a constant quantity
in curved space time but it can hold Eq. (2). The action of scalar particle in
which the scalar field theory under Lorentz-violation is taken into
consideration can be written as

$\displaystyle{\mathcal{S}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\frac{1}{2}[\partial_{\mu}\psi\partial^{\mu}\psi+\lambda(u^{\alpha}\partial_{\alpha}\psi)^{2}+m^{2}\psi^{2}].$
(4)

From Eq. (4), the Lorentz-violation of scalar field equation in the curved
space time can be written as

$\displaystyle-\frac{1}{\sqrt{-g}}\partial_{\mu}[\sqrt{-g}(g^{\mu\nu}+\lambda
u^{\mu}u^{\nu})\partial_{\nu}\psi]+m^{2}\psi$ $\displaystyle=$ $\displaystyle
0.$ (5)

Eq. (5) is the modified form of Klein-Gordon equation of scalar particle with
mass $m$ in the Lorentz-violation theory. In Eq. (5), if the constant term
$\lambda$ tends to zero, the original Klein-Gordon equation is recovered. To
derive modified Hamilton-Jacobi equation in the curved space time, the wave
function $\psi$ in Eq. (5) can be written as

$\psi=\psi_{c}e^{\frac{i}{\hbar}S},$ (6)

where $\mathcal{S}$=$\mathcal{S}$$(t,r,\theta,\phi)$ and $\hbar$ are the
Hamiltonian principal function of scalar field and Planck constant
respectively. Using Eq. (6) in Eq. (5) and neglecting higher order terms of
$\hbar$ in accordance with semiclassical theory, we obtain

$\displaystyle(g^{\mu\nu}+\lambda
u^{\mu}u^{\nu})\partial_{\mu}S\partial_{\nu}S+m^{2}$ $\displaystyle=$
$\displaystyle 0.$ (7)

Eq. (7) represents not only the dynamical equation of a scalar particle of
mass $m$ but also modified form of Hamilton-Jacobi equation of scalar particle
in the Lorentz-violation theory of curved space time. This modified Hamilton-
Jacobi equation will be used to investigate the entropy, modified Hawking
temperature and specific heat capacity of Kerr-de Sitter black hole in the
next sections.

##  3 Kerr-de Sitter solution and naive coordinate

The Kerr-de Sitter (KdS) black hole is a stationary rotating black hole. The
vacuum solution of of KdS black hole in $(t,r,\theta,\phi)$ coordinate system
with cosmological constant $\Lambda$ is given by [79]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{r}-\Delta_{\theta}a^{2}\sin^{2}\theta}{\rho^{2}\Xi^{2}}dt^{2}+\frac{\rho^{2}}{\Delta_{r}}dr^{2}-\frac{2a[\Delta_{\theta}(r^{2}+a^{2})-\Delta_{r}]\sin^{2}\theta}{\rho^{2}\Xi^{2}}dtd\phi$
(9)
$\displaystyle+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta_{r}a^{2}\sin^{2}\theta}{\rho^{2}\Xi^{2}}\sin^{2}\theta
d\phi^{2}\,$

where

$\displaystyle\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,\,\,\,\,\,\Xi=1+\frac{1}{3}\Lambda
a^{2},$ (10) $\displaystyle\Delta_{\theta}=1+\frac{1}{3}\Lambda
a^{2}\cos^{2}\theta,$ (11)
$\displaystyle\Delta_{r}=(r^{2}+a^{2})\Big{(}1-\frac{1}{3}\Lambda
r^{2}\Big{)}-2Mr.$ (12)

Here $\Lambda$, $M$ and $a$ represent the cosmological constant, mass of the
black hole and rotating parameter. The event horizon equation of KdS black
hole can be obtained from the null hypersurface equation [9, 10 ]

$\displaystyle g^{\mu\nu}\frac{\partial H}{\partial x^{\mu}}\frac{\partial
H}{\partial x^{\nu}}$ $\displaystyle=$ $\displaystyle 0,$ (13)

where $H=H(r)$. Then the event horizon equation is

$\Delta_{r}=(r^{2}+a^{2})\Big{(}1-\frac{1}{3}\Lambda r^{2}\Big{)}-2Mr=0.$ (14)

Equation (11) has four real roots for $\Lambda>0$ namely $r_{+}$, $r_{h}$,
$r_{-}$ and $r_{--}$ $(r_{+}>r_{h}>r_{-}>0>r_{--})$. The locations of $r_{+}$,
$r_{h}$ and $r_{-}$ represent the cosmological horizon, event horizon and
Cauchy horizon respectively. To investigate the Hawking radiation near the
event horizon, $\Delta_{r}$ can be factorised as follows [25]

$\displaystyle\Delta_{r}=(r-r_{h})\Xi^{2}L(r),$ (15)

with

$\displaystyle
L(r)=-\frac{\Lambda}{3}\Big{[}r^{3}+r_{h}r^{2}+\Big{(}a^{2}+r^{2}_{h}-\frac{3}{\Lambda}\Big{)}r+\frac{3a^{2}}{\Lambda
r_{h}}\Big{]}.$ (16)

The entropy of KdS black hole near the event horizon is calculated as

$\displaystyle S_{BH}=\frac{A}{4}=\frac{\pi(r^{2}_{h}+a^{2})}{\Xi}.$ (17)

The KdS black hole has a coordinate singularity near the event horizon. Now
the dragging co-ordinate system will be applied in KdS black hole. Let
$\frac{d\phi}{dt}=-\frac{g_{14}}{g_{44}}$ [17, 29], the space time metric of
KdS black hole (8) reduces to

$\displaystyle
ds^{2}=\hat{g}_{11}dt^{2}+\frac{\rho^{2}}{\Delta_{r}}dr^{2}+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2},$
(18)

where

$\displaystyle\hat{g}_{11}=-\frac{\Delta_{r}\Delta_{\theta}\rho^{2}}{\Xi^{2}[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta_{r}a^{2}\sin^{2}\theta]}.$
(19)

The angular velocity of KdS black hole near the event horizon is defined by

$\displaystyle\Omega_{h}=\frac{a}{r^{2}_{h}+a^{2}}.$ (20)

Using Eq. (15), the contravariant components of $g^{ab}$ are derived as

$\displaystyle\hat{g}^{11}=-\frac{\Xi^{2}[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta_{r}a^{2}\sin^{2}\theta]}{\Delta_{r}\Delta_{\theta}\rho^{2}},\,\,\,\,\,\,\hat{g}^{22}=\frac{\Delta_{r}}{\rho^{2}},\,\,\,\,\,\hat{g}^{33}=\frac{\Delta_{\theta}}{\rho^{2}}.$
(21)

The surface gravity near the event horizon of KdS black hole can be obtained
as [80, 81]

$\displaystyle\kappa=$ $\displaystyle\lim_{\hat{g}_{11}\rightarrow
0}\Bigg{(}-\frac{1}{2}\sqrt{-\frac{g^{22}}{\hat{g}_{11}}}\frac{d\hat{g}_{11}}{dr}\Bigg{)}=\frac{\Xi(3r_{h}^{2}-3\Lambda
r_{h}^{4}-\Lambda a^{2}r_{h}^{2}-3a^{2})}{6r_{h}(r_{h}^{2}+a^{2})}.$ (22)

The Hawking temperature near the event horizon of KdS black hole is given by
$T_{H}=\frac{\kappa}{2\pi}$

$\displaystyle T_{H}$ $\displaystyle=$
$\displaystyle\frac{\Xi(3r_{h}^{2}-3\Lambda r_{h}^{4}-\Lambda
a^{2}r_{h}^{2}-3a^{2})}{12\pi r_{h}(r_{h}^{2}+a^{2})}.$ (23)

To obtain the heat capacity of KdS black hole, the black hole mass can be
derived from $\Delta_{r}(r_{h})=0$ as

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{r_{h}}{2}+\frac{a^{2}}{2r_{h}}-\frac{\Lambda
r_{h}^{3}}{6}-\frac{\Lambda a^{2}r_{h}}{3}.$ (24)

The heat capacity $(C_{h})$ of blak hole [82, 83] is defined by

$\displaystyle C_{h}$ $\displaystyle=$ $\displaystyle\frac{\partial
M}{\partial T_{h}}=\Bigg{(}\frac{\partial M}{\partial
r_{h}}\Bigg{)}\Bigg{(}\frac{\partial r_{h}}{\partial T_{h}}\Bigg{)}.$ (25)

The heat capacity of KdS black hole near the event horizon $r=r_{h}$ is
obtained as

$\displaystyle C_{h}$ $\displaystyle=$
$\displaystyle\frac{2\pi(r_{h}^{2}+a^{2})^{2}F_{1}}{\Xi[F_{2}(3r_{h}^{2}+a^{2})-r_{h}^{2}(r_{h}^{2}+a^{2})F_{3}]},$
(26)

where the values of $F_{1}$, $F_{2}$ and $F_{3}$ are defined by

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle 3r_{h}^{2}-3\Lambda
r_{h}^{4}-3a^{2}-2\Lambda a^{2}r_{h}^{2},$ (27) $\displaystyle F_{2}$
$\displaystyle=$ $\displaystyle 3\Lambda r_{h}^{4}+\Lambda
a^{2}r_{h}^{2}-3r_{h}^{2}+3a^{2},$ (28) $\displaystyle F_{3}$ $\displaystyle=$
$\displaystyle 12\Lambda r_{h}^{2}+2\Lambda a^{2}-6.$ (29)

The Landau’s condition of the coordinate clock synchronization [84] is
satisfied by the space-time (15). It is obvious that event horizon and
infinite redshift surface of KdS black hole are concordant with each other.
This indicates that the geometrical optics limit can be used. The relationship
between the imaginary part of the radial action and tunneling probability can
be derived using WKB approximation [85]. To study Hawking radiation near the
event horizon of KdS black hole, the ether like vector $u^{\alpha}$ from (15)
can be taken as

$\displaystyle u^{t}$ $\displaystyle=$
$\displaystyle\frac{c_{t}}{\sqrt{-g_{11}}}=c_{t}\sqrt{\frac{\Xi^{2}K}{\Delta_{r}\Delta_{\theta}\rho^{2}}},$
(30) $\displaystyle u^{r}$ $\displaystyle=$
$\displaystyle\frac{c_{r}}{\sqrt{g_{rr}}}=\frac{c_{r}\sqrt{\Delta_{r}}}{\sqrt{\rho^{2}}},$
(31) $\displaystyle u^{\theta}$ $\displaystyle=$
$\displaystyle\frac{c_{\theta}}{\sqrt{g_{\theta\theta}}}=\frac{c_{\theta}\sqrt{\Delta_{\theta}}}{\sqrt{\rho^{2}}},$
(32)

where $c_{t}$, $c_{r}$, $c_{\theta}$ are the arbitrary constants and
$K=\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta_{r}a^{2}\sin^{2}\theta$. Near the
event horizon of black hole, $u^{\alpha}$ satisfies the following condition as

$\displaystyle u^{\alpha}u_{\alpha}=-c_{t}^{2}+c_{r}^{2}+c_{\theta}^{2}={\rm
constant}.$ (33)

Again using Eqs. (25) and (26) in Eq. (7), we obtain

$\displaystyle\hat{g}^{11}(\partial_{t}S)^{2}+\hat{g}^{22}(\partial_{r}S)^{2}+\hat{g}^{33}(\partial_{\theta}S)^{2}+\lambda
u^{t}u^{t}(\partial_{t}S)^{2}+\lambda u^{r}u^{r}(\partial_{r}S)^{2}$ (34)
$\displaystyle+\lambda u^{\theta}u^{\theta}(\partial_{\theta}S)^{2}+2\lambda
u^{t}u^{r}(\partial_{t}S)(\partial_{r}S)+2\lambda
u^{t}u^{\theta}(\partial_{t}S)(\partial_{\theta}S)$ (35)
$\displaystyle+2\lambda
u^{r}u^{\theta}(\partial_{r}S)(\partial_{\theta}S)+m^{2}=0.$ (36)

Eq. (27) contains the three independent variables namely $t$, $r$ and
$\theta$. To study the entropy of black hole, the action $S$ can be written as

$\displaystyle S=-\omega t+R(r,\theta)+j\phi,$ (37)

where $R(r,\theta)$, $\omega$ and $j$ denote the generalized momentum, energy
of the emitted particle and angular momentum with respect to $\phi$-axis
respectively. Using Eq. (27) in (28), the action $S$ can be written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle-\omega
t+\int\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}dr+j\phi,$ (38)

where

$\displaystyle A=$ $\displaystyle\frac{\Delta_{r}(1+\lambda
c_{r}^{2})}{\rho^{2}},$ $\displaystyle B=$ $\displaystyle
2\lambda\Big{[}\frac{c_{r}c_{\theta}\sqrt{\Delta_{r}\Delta_{\theta}}}{\rho^{2}}\Big{(}\frac{\partial
R}{\partial\theta}\Big{)}-\frac{c_{t}c_{r}\Xi\sqrt{K}(w-j\Omega_{h})}{\sqrt{\Delta_{\theta}}\rho^{2}}\Big{]},$
$\displaystyle C=$ $\displaystyle\frac{\Delta_{\theta}(1+\lambda
c_{\theta}^{2})}{\rho^{2}}\Big{(}\frac{\partial
R}{\partial\theta}\Big{)}^{2}-\frac{\Xi^{2}K}{\Delta_{r}\Delta_{\theta}\rho^{2}}(\omega-j\Omega_{h})^{2}(1-\lambda
c_{t}^{2})$ $\displaystyle-2\lambda
c_{t}c_{\theta}\frac{\Xi\sqrt{K}}{\sqrt{\Delta_{r}}\rho^{2}}(\omega-j\Omega_{h})\Big{(}\frac{\partial
R}{\partial\theta}\Big{)}+m^{2}.$

The KdS black hole has a pole at the event horizon $r=r_{h}$. Applying residue
theorem of complex analysis and Feynman prescription, the imaginary part of
the radial action will be evaluated. The imaginary part of the radial action
can be found as

$\displaystyle{\rm ImS}=\pi\frac{(r^{2}_{h}+a^{2})(\omega-j\Omega_{h})(\lambda
c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda c_{t}^{2}})}{\Xi
L(r_{h})(1+\lambda c_{r}^{2})}$ (40)

Using WKB approximation, the tunneling rate is obtained as

$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle e^{-2{\rm ImS}}.$
(41)

Then the Hawking temperature of KdS black hole near the event horizon
$r=r_{h}$ due to naive coordinate in Lorentz violation is given by

$\displaystyle T_{n}$ $\displaystyle=$ $\displaystyle\frac{\Xi(1+\lambda
c_{r}^{2})}{(\lambda c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda
c_{t}^{2}})}\frac{(3r_{h}^{2}-3\Lambda r_{h}^{4}-\Lambda
a^{2}r_{h}^{2}-3a^{2})}{6\pi r_{h}(r_{h}^{2}+a^{2})}.$ (42)

The tunneling probability and its radiation spectrum can be derived using WKB
approximation. Taking the conservation of energy and angular momentum, the
mass and angular momentum of black hole is allowed to fluctuate. If the
particle of energy $\omega$ and angular momentum $j$ radiate from the black
hole, the mass and angular momentum of KdS black hole will be $M-\omega$ and
$J-j$ respectively. The imaginary part of the radial action is calculated as
follows

$\displaystyle{\rm ImS}$ $\displaystyle=$
$\displaystyle\pi\int^{(\omega,J)}_{(0,0)}\frac{(r^{2}_{h}+a^{2})}{\Xi
L(r_{h})}\frac{[\lambda c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda
c_{t}^{2}}]}{(1+\lambda c_{r}^{2})}(d\omega-\Omega_{h}dj),$ (43)
$\displaystyle=$ $\displaystyle-\pi\frac{[\lambda c_{t}c_{r}+\sqrt{1+\lambda
c_{r}^{2}-\lambda c_{t}^{2}}]}{(1+\lambda
c_{r}^{2})}\int^{M-\omega}_{M}\frac{r^{2}_{h}}{\Xi L(r_{h})}dM,$ (44)

where $\Omega_{h}=\frac{a}{r^{2}_{h}+a^{2}}$ and $j=Ma$. From $\Delta_{r}=0$,
the change of mass is calculated as

$\displaystyle
dM=\Big{[}-\frac{\Lambda}{2}r^{2}_{h}+\Big{(}\frac{1}{2}-\frac{\Lambda
a^{2}}{6}\Big{)}-\frac{a^{2}}{2r^{2}_{h}}\Big{]}dr_{h}=\frac{L(r_{h})}{2r_{h}}dr_{h}.$
(45)

Then, the Eq. (33) can be written as

$\displaystyle{\rm ImS}$ $\displaystyle=$ $\displaystyle-\frac{\pi(\lambda
c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda c_{t}^{2}})}{2\Xi(1+\lambda
c_{r}^{2})}\int^{r_{e}}_{r_{f}}r_{h}dr_{h}$ (47)
$\displaystyle=-\frac{\pi(r^{2}_{e}-r^{2}_{f})}{4\Xi}\frac{(\lambda
c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda c_{t}^{2}})}{(1+\lambda
c_{r}^{2})},$

where $r_{e}$ and $r_{f}$ are the locations of horizons of KdS black hole
before and after the particle emission. The tunneling probability can be
calculated using WKB approximation as

$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle
exp\Bigg{[}\frac{[\lambda c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda
c_{t}^{2}}]}{(1+\lambda
c_{r}^{2})}\Bigg{(}\frac{\pi(r^{2}_{e}-r^{2}_{f})}{2\Xi}\Bigg{)}\Bigg{]},$
(48) $\displaystyle=$ $\displaystyle exp\Bigg{[}\frac{[\lambda
c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda c_{t}^{2}}]}{(1+\lambda
c_{r}^{2})}\Bigg{(}\frac{\pi(r^{2}_{e}-a^{2})}{2\Xi}-\frac{\pi(r^{2}_{f}-a^{2})}{2\Xi}\Bigg{)}\Bigg{]},$
(49) $\displaystyle=$ $\displaystyle exp\Bigg{[}\frac{[\lambda
c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda c_{t}^{2}}]}{(1+\lambda
c_{r}^{2})}\Big{(}\frac{1}{2}\Delta S_{BH}\Big{)}\Bigg{]}.$ (50)

The change in Bekenstein-Hawking entropy of KdS black hole due to Lorentz
violation theory in naive coordinate is given by

$\displaystyle\frac{1}{2}\frac{[\lambda c_{t}c_{r}+\sqrt{1+\lambda
c_{r}^{2}-\lambda c_{t}^{2}}]}{(1+\lambda c_{r}^{2})}\Delta S_{BH}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\frac{[\lambda
c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda c_{t}^{2}}]}{(1+\lambda
c_{r}^{2})}$ (51) $\displaystyle\times$
$\displaystyle[S_{BH}(M-\omega)-S_{BH}(M)].$ (52)

It shows that the change in Bekenstein-Hawking entropy is modified due to
presence of Lorentz violation parameter $\lambda$ and ether like vectors
$u^{\alpha}$. When $\lambda=0$, Eq. (37) becomes $\frac{1}{2}\Delta
S_{BH}=\frac{1}{2}[S_{BH}(M-\omega)-S_{BH}(M)]$ which is half of correct value
of Bekenstein-Hawking entropy near the event horizon of KdS black hole.

We calculate the heat capacity of KdS black hole at the event horizon
$r=r_{h}$ due to Lorentz violation in naive coordinte as

$\displaystyle C_{n}$ $\displaystyle=$ $\displaystyle\frac{\pi(\lambda
c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda
c_{t}^{2}})(r_{h}^{2}+a^{2})^{2}F_{1}}{\Xi(1+\lambda
c_{r}^{2})[F_{2}(3r_{h}^{2}+a^{2})-r_{h}^{2}(r_{h}^{2}+a^{2})F_{3}]},$ (53)

where the values of $F_{1}$, $F_{2}$ and $F_{3}$ are given in (24). From Eqs.
(32), (37) and (38), we observe that the Hawking temperature, entropy and heat
capacity depend on Lorentz violation parameter $\lambda$ and ether like
vectors $u^{\alpha}$.

##  4 Painleve coordinate

To discuss the tunneling rate near the event horizon $r=r_{h}$ of KdS black
hole, the well-defined coordinate system known as Painleve coordinate system
is considered. Let

$\displaystyle dt=dT+F(r,\theta)dr+G(r,\theta)d\theta,$ (54)

where $F(r,\theta)$ and $G(r,\theta)$ represent the two arbitrary functions
satisfying the following condition

$\displaystyle\frac{\partial F(r,\theta)}{\partial\theta}=\frac{\partial
G(r,\theta)}{\partial r}.$ (55)

It is obvious that the constant-time slice is a flat Euclidean space and
taking

$\displaystyle g_{22}+F^{2}(r,\theta)\hat{g}_{11}=1,$ (56)

the KdS black hole in Painleve coordinate is found as

$\displaystyle ds^{2}$
$\displaystyle=\hat{g}_{11}dT^{2}+2\sqrt{\hat{g}_{11}\Big{(}1-\frac{\rho^{2}}{\Delta_{r}}\Big{)}}dTdr+dr^{2}+\Big{(}\frac{\rho^{2}}{\Delta_{\theta}}+G^{2}\hat{g}_{11}\Big{)}d\theta^{2}$
(58)
$\displaystyle+2\hat{g}_{11}GdTd\theta+2G\sqrt{\hat{g}_{11}\Big{(}1-\frac{\rho^{2}}{\Delta_{r}}\Big{)}}drd\theta.$

The contravariant components of $g^{ab}$ from Eq. (42) are found as

$\displaystyle g^{11}$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{g}_{11}\rho^{2}}[\Delta_{r}+\Delta_{\theta}G^{2}\hat{g}_{11}],\,\,\,\,\,\,g^{23}=g^{32}=0,\,\,\,\,\,\,g^{22}=\frac{\Delta_{r}}{\rho^{2}},$
(59) $\displaystyle g^{12}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{r}}{\hat{g}_{11}\rho^{2}}\sqrt{\hat{g}_{11}\Big{(}1-\frac{\rho^{2}}{\Delta_{r}}\Big{)}},\,\,\,\,\,\,g^{31}=-\frac{G\Delta_{\theta}}{\rho^{2}},\,\,\,\,\,\,g^{33}=\frac{\Delta_{\theta}}{\rho^{2}}.$
(60)

To derive the modified Hawking temperature, entropy and heat capacity near the
event horizon of KdS black hole, the ether like vectors $u^{\alpha}$ can be
constructed from Eq. (42) as follows

$\displaystyle u^{T}$ $\displaystyle=$
$\displaystyle\frac{c_{T}}{\sqrt{\hat{g}_{11}}},$ (61) $\displaystyle u^{r}$
$\displaystyle=$
$\displaystyle\frac{c_{r}\sqrt{\hat{g}_{11}}}{g_{12}}=\frac{c_{r}\sqrt{\Delta_{r}}}{\sqrt{(\Delta_{r}-\rho^{2})}},$
(62) $\displaystyle u^{\theta}$ $\displaystyle=$
$\displaystyle\frac{c_{\theta}}{\sqrt{g_{33}}}=\frac{c_{\theta}}{\sqrt{\frac{\rho^{2}}{\Delta_{\theta}}+G^{2}\hat{g}_{11}}},$
(63)

where $c_{T}$, $c_{r}$ and $c_{\theta}$ are arbitrary constants and
$u^{\alpha}u_{\alpha}=c^{2}_{T}+2c_{T}c_{r}+c^{2}_{\theta}={\rm constant}$
(near the event horizon of black hole). Using Eqs. (43) and (44) in Eq. (7),
we get

$\displaystyle\frac{1}{\hat{g}_{11}\rho^{2}}\Big{(}\Delta_{r}+\Delta_{\theta}G^{2}\hat{g}_{11}\Big{)}\Big{(}\frac{\partial
S}{\partial
T}\Big{)}^{2}-\frac{2\Delta_{r}}{\hat{g}_{11}\rho^{2}}\sqrt{\hat{g}_{11}\Big{(}1-\frac{\rho^{2}}{\Delta_{r}}\Big{)}}\Big{(}\frac{\partial
S}{\partial T}\Big{)}\Big{(}\frac{\partial S}{\partial r}\Big{)}$ (64)
$\displaystyle+\frac{\Delta_{r}}{\rho^{2}}\Big{(}\frac{\partial S}{\partial
r}\Big{)}^{2}+\frac{\Delta_{\theta}}{\rho^{2}}\Big{(}\frac{\partial
S}{\partial\theta}\Big{)}^{2}-\frac{2G\Delta_{\theta}}{\rho^{2}}\Big{(}\frac{\partial
S}{\partial\theta}\Big{)}\Big{(}\frac{\partial S}{\partial T}\Big{)}+\lambda
u^{\mu}u^{\nu}\partial_{\mu}S\partial_{\nu}S+m^{2}=0.$ (65)

The above equation contains three independent variables namely $T$, $r$ and
$\theta$. The Hawking radiation occurs along the radial direction only. To
separate the variables from Eq. (45), we write the action $S$ as follows

$\displaystyle S=-\omega T+W(r,\theta)+j\phi.$ (67)

Using Eqs. (44) and (46) in Eq. (45), a quadratic equation in $(\frac{\partial
W}{\partial r})$ is obtained as follows

$\displaystyle M\Big{(}\frac{\partial W}{\partial
r}\Big{)}^{2}+N\Big{(}\frac{\partial W}{\partial r}\Big{)}+P=0,$ (68)

where

$\displaystyle M=$ $\displaystyle\frac{\Delta_{r}}{\rho^{2}}+\frac{\lambda
c_{r}^{2}\Delta_{r}}{\Delta_{r}-\rho^{2}},$ $\displaystyle N=$
$\displaystyle(w-j\Omega_{h})\Bigg{(}\frac{(2\Delta_{r}-2\rho^{2}-2\lambda
c_{T}c_{r}\rho^{2})\Xi\sqrt{K}}{\rho^{2}\sqrt{-\Delta_{\theta}\rho^{2}(\Delta_{r}-\rho^{2})}}\Bigg{)}+$
$\displaystyle\frac{2\lambda
c_{r}c_{\theta}\sqrt{\Delta_{r}\Delta_{\theta}}}{\sqrt{(\Delta_{r}-\rho^{2})(\rho^{2}+G^{2}\hat{g_{11}}\Delta_{\theta})}}\Big{(}\frac{\partial
W}{\partial\theta}\Big{)},$ $\displaystyle P=$
$\displaystyle\frac{\Delta_{\theta}}{\rho^{2}}\Bigg{(}G(w-j\Omega_{h})+\frac{\partial
W}{\partial\theta}\Bigg{)}^{2}+\frac{\Delta_{r}}{\rho^{2}}\frac{1}{\hat{g}_{11}}(w-j\Omega_{h})^{2}+\frac{\lambda
c_{\theta}^{2}\Delta_{\theta}}{\rho^{2}+\Delta_{\theta}G^{2}\hat{g}_{11}}\Big{(}\frac{\partial
W}{\partial\theta}\Big{)}^{2}$ $\displaystyle+\frac{\lambda
c_{T}^{2}(w-j\Omega_{h})^{2}}{\hat{g}_{11}}-\frac{2\lambda
c_{T}c_{\theta}(w-j\Omega_{h})}{\sqrt{\hat{g}_{11}\Big{(}\frac{\rho^{2}}{\Delta_{\theta}}+G^{2}\hat{g}_{11}}\Big{)}}\Big{(}\frac{\partial
W}{\partial\theta}\Big{)}+m^{2}.$

Substituting Eq. (47) in Eq. (46), the outgoing action S can be expressed as

$\displaystyle S=-\omega T+\int\frac{-N\pm\sqrt{N^{2}-4MP}}{2M}dr+j\phi.$ (69)

On integration of Eq. (48) near the event horizon $r=r_{h}$ of KdS black hole
using residue theorem of complex analysis, the imaginary part of outgoing
action S is found as

$\displaystyle{\rm
ImS}=\frac{2\pi(r^{2}_{h}+a^{2})(\omega-j\Omega_{h})(1+\lambda
c_{T}c_{r})}{\Xi L(r_{h})(1-\lambda c_{r}^{2})}.$ (70)

Using WKB approximation, the tunneling rate is

$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle e^{-2{\rm ImS}}.$
(71)

The Hawking temperature is given by

$\displaystyle T_{p}=$ $\displaystyle\frac{\Xi(1-\lambda
c_{r}^{2})(3r_{h}^{2}-3\Lambda r_{h}^{4}-\Lambda
a^{2}r_{h}^{2}-3a^{2})}{12\pi[(1+\lambda c_{T}c_{r})]r_{h}(r_{h}^{2}+a^{2})}.$
(72)

If $\lambda=0$ and $c_{T}+c_{r}=0$, the Lorentz violation has been cancelled
and the original Hawking temperature which is similar to the actual one given
in Eq. (20) is obtained.

The imaginary part of the outgoing action is calculated as

$\displaystyle{\rm ImS}$ $\displaystyle=$
$\displaystyle-2\pi\int^{M-\omega}_{M}\frac{(r^{2}_{h}+a^{2})(dw-\Omega_{h}dj)}{\Xi
L(r_{h})}\frac{(1+\lambda c_{T}c_{r})}{(1-\lambda c_{r}^{2})}.$

From Eqs. (34) and (52), we have

$\displaystyle{\rm ImS}=-\frac{\pi(r^{2}_{e}-r^{2}_{f})}{2\Xi}\frac{(1+\lambda
c_{T}c_{r})}{(1-\lambda c_{r}^{2})}.$ (74)

By using Eq. (53) in (50), we get

$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle
exp\Bigg{[}\frac{(1+\lambda c_{T}c_{r})}{(1-\lambda
c_{r}^{2})}\Bigg{(}\frac{\pi(r^{2}_{e}-r^{2}_{f})}{\Xi}\Bigg{)}\Bigg{]},$ (75)
$\displaystyle=$ $\displaystyle exp\Bigg{[}\frac{(1+\lambda
c_{T}c_{r})}{(1-\lambda
c_{r}^{2})}\Bigg{(}\frac{\pi(r^{2}_{e}-a^{2})}{\Xi}-\frac{\pi(r^{2}_{f}-a^{2})}{\Xi}\Bigg{)}\Bigg{]},$
(76) $\displaystyle=$ $\displaystyle exp\Bigg{[}\frac{(1+\lambda
c_{T}c_{r})}{(1-\lambda c_{r}^{2})}(\Delta S_{BH})\Bigg{]}.$ (77)

The change in Bekenstein-Hawking entropy of KdS black hole due to Lorentz
violation theory in Painleve coordinate is given by

$\displaystyle\frac{(1+\lambda c_{T}c_{r})}{(1-\lambda c_{r}^{2})}\Delta
S_{BH}$ $\displaystyle=$ $\displaystyle\frac{(1+\lambda
c_{T}c_{r})}{(1-\lambda c_{r}^{2})}\Big{[}S_{BH}(M-\omega)-S_{BH}(M)\Big{]}$
(78)

We calculate the heat capacity of KdS black hole near the event horizon
$r=r_{h}$ due to Lorentz violation in Painleve coordinte as

$\displaystyle C_{p}$ $\displaystyle=$ $\displaystyle\frac{2\pi(1+\lambda
c_{T}c_{r})(r_{h}^{2}+a^{2})^{2}F_{1}}{\Xi(1-\lambda
c_{r}^{2})[F_{2}(3r_{h}^{2}+a^{2})-r_{h}^{2}(r_{h}^{2}+a^{2})F_{3}]}.$ (79)

The values of $F_{1}$, $F_{2}$ and $F_{3}$ near the event horizon $r=r_{h}$ of
KdS black hole are given in Eq. (24). We observe that the modified Hawking
temperature (51), change in Bekenstein-Hawking entropy (55) and heat capacity
(56) near the event horizon of KdS black hole depend on the Lorentz violation
parameter $\lambda$ and ether like vectors $u^{\alpha}$. If $\lambda$ tends to
zero, then the original Hawking temperature, change in Bekenstein-Hawking
entropy and heat capacity near the event horizon of KdS black hole are
recovered.

##  5  Eddington coordinate

To find the entropy of KdS black hole near the event horizon $r=r_{h}$, the
well-behaved coordinate system named as Eddington coordinate will be used. We
consider the tansformation

$\displaystyle
dt=du-\frac{\Xi}{\Delta_{r}}\sqrt{\frac{[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta_{r}a^{2}\sin^{2}\theta]}{\Delta_{\theta}}}dr,$

then the space time (15) is transformed as

$\displaystyle
ds^{2}=-\frac{\Delta_{r}\Delta_{\theta}\rho^{2}}{\Xi^{2}K}du^{2}+\frac{2\rho^{2}}{\Xi}\sqrt{\frac{\Delta_{\theta}}{K}}dudr+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}.$
(80)

The contravariant components of $g^{ab}$ of the above equation can be written
as

$\displaystyle
g^{11}=0,\,\,g^{12}=g^{21}=\frac{\Xi}{\rho^{2}}\sqrt{\frac{K}{\Delta_{\theta}}},\,\,g^{22}=\frac{\Delta_{r}}{\rho^{2}},\,\,\,g^{33}=\frac{\Delta_{\theta}}{\rho^{2}}.$
(81)

We construct ether like vectors $u^{\alpha}$ from Eq. (57) as

$\displaystyle u^{u}$ $\displaystyle=$
$\displaystyle\frac{c_{u}}{\sqrt{g_{11}}}=c_{u}\sqrt{-\frac{\Xi^{2}K}{\Delta_{r}\Delta_{\theta}\rho^{2}}},$
(82) $\displaystyle u^{r}$ $\displaystyle=$
$\displaystyle\frac{c_{r}\sqrt{g_{11}}}{g_{12}}=\frac{c_{r}}{\rho^{2}}\sqrt{-\Delta_{r}\rho^{2}K},$
(83) $\displaystyle u^{\theta}$ $\displaystyle=$
$\displaystyle\frac{c_{\theta}}{\sqrt{g_{33}}}=c_{\theta}\Bigg{(}\sqrt{\frac{\Delta_{\theta}}{\rho^{2}}}\Bigg{)},$
(84)

where $c_{u}$, $c_{r}$ and $c_{\theta}$ are arbitrary constant and
$u^{\alpha}u_{\alpha}=c_{u}^{2}+2c_{r}c_{u}+c_{\theta}^{2}={\rm constant}$.
Using Eqs. (58) and (7), we obtain

$\displaystyle\frac{\Delta_{r}}{\rho^{2}}\Big{(}\frac{\partial S}{\partial
r}\Big{)}^{2}+\frac{2\Xi}{\rho^{2}}\sqrt{\frac{K}{\Delta_{\theta}}}\Big{(}\frac{\partial
S}{\partial u}\Big{)}\Big{(}\frac{\partial S}{\partial
r}\Big{)}+\frac{\Delta_{\theta}}{\rho^{2}}\Big{(}\frac{\partial
S}{\partial{\theta}}\Big{)}^{2}$ (85) $\displaystyle+\lambda
u^{\mu}u^{\nu}\partial_{\mu}S\partial_{\nu}S+m^{2}=0.$ (86)

The above equation involves the variables $u$, $r$ and $\theta$. To find
radial action of the space time, the action $S$ can be written as

$\displaystyle S=-\omega u+R(r)+\Theta(\theta)+j\phi.$ (87)

Using Eqs. (59) and (61) in Eq. (60), a quadratic equation in
$\Big{(}\frac{\partial R}{\partial r}\Big{)}$ is obtained as

$\displaystyle 2\lambda
c_{r}c_{\theta}\frac{\sqrt{-\Delta_{r}\Delta_{\theta}}}{\rho^{2}}\Big{(}\frac{\partial\Theta}{\partial\theta}\Big{)}\Big{(}\frac{\partial
R}{\partial
r}\Big{)}-\frac{2\Xi}{\rho^{2}}\sqrt{\frac{K}{\Delta_{\theta}}}(\omega-j\Omega_{h})(1+\lambda
c_{u}c_{r})\Big{(}\frac{\partial R}{\partial r}\Big{)}$ (88)
$\displaystyle+\frac{\Delta_{r}}{\rho^{2}}(1-\lambda
c_{r}^{2})\Big{(}\frac{\partial R}{\partial
r}\Big{)}^{2}+\frac{\Delta_{\theta}}{\rho^{2}}(1+\lambda
c_{\theta}^{2})\Big{(}\frac{\partial\Theta}{\partial\theta}\Big{)}^{2}-\lambda
c_{u}^{2}\frac{\Xi^{2}K}{\Delta_{r}\Delta_{\theta}\rho^{2}}(\omega-j\Omega_{h})^{2}$
(89) $\displaystyle-2\lambda
c_{u}c_{\theta}\frac{\Xi}{\rho^{2}}\sqrt{\frac{-K}{\Delta_{r}}}(\omega-j\Omega_{h})\Big{(}\frac{\partial\Theta}{\partial\theta}\Big{)}+m^{2}=0.$
(90)

Solving $\Big{(}\frac{\partial R}{\partial r}\Big{)}$ from Eq. (62), the
action $S$ can be expressed as

$\displaystyle S=-\omega
u+\int\frac{-B^{{}^{\prime}}\pm\sqrt{B^{{}^{\prime}2}-4A^{{}^{\prime}}D^{{}^{\prime}}}}{2A^{{}^{\prime}}}dr+j\phi,$
(91)

where

$\displaystyle A^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{r}}{\rho^{2}}(1-\lambda c_{r}^{2}),$
$\displaystyle B^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{2\Xi}{\rho^{2}}\sqrt{\frac{K}{\Delta_{\theta}}}(\omega-j\Omega_{h})(1+\lambda
c_{u}c_{r})+2\lambda
c_{r}c_{\theta}\frac{\sqrt{-\Delta_{r}\Delta_{\theta}}}{\rho^{2}}\Big{(}\frac{\partial\Theta}{\partial\theta}\Big{)},$
$\displaystyle D^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\Big{[}\frac{\Delta_{\theta}}{\rho^{2}}(1+\lambda
c_{\theta}^{2})\Big{(}\frac{\partial\Theta}{\partial\theta}\Big{)}-2\lambda
c_{u}c_{\theta}\frac{\Xi}{\rho^{2}}\sqrt{\frac{-K}{\Delta_{r}}}(\omega-j\Omega_{h})\Big{]}\Big{(}\frac{\partial\Theta}{\partial\theta}\Big{)}$
$\displaystyle-\lambda
c_{u}^{2}\frac{\Xi^{2}K}{\Delta_{r}\Delta_{\theta}\rho^{2}}(\omega-j\Omega_{h})^{2}+m^{2}.$

Completing the integration of Eq. (63) near the event horizon $r=r_{h}$ of KdS
black hole using residue theorem of complex analysis, the outgoing action $S$
is found as

$\displaystyle{\rm ImS}=2\pi\frac{(r^{2}_{h}+a^{2})(1+\lambda
c_{u}c_{r})(\omega-j\Omega_{h})}{\Xi L(r_{h})(1-\lambda c_{r}^{2})}.$ (93)

Using WKB approximation, the tunneling rate is obtained as

$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle e^{-2{\rm ImS}}.$
(94)

The Hawking temperature near the event horizon of KdS black hole due to
Lorentz violation theory in Eddington coordinate is obtained as

$\displaystyle T_{u}$ $\displaystyle=$ $\displaystyle\frac{\Xi(1-\lambda
c_{r}^{2})(3r_{h}^{2}-3\Lambda r_{h}^{4}-\Lambda a^{2}r_{h}^{2}-3a^{2})}{12\pi
r_{h}(r_{h}^{2}+a^{2})(1+\lambda c_{u}c_{r})}.$ (95)

If $\lambda=0$ and $c_{u}+c_{r}=0$, the Lorentz violation has been cancelled
and the original Hawking temperature which is similar to the temperature given
in Eq. (20) is obtained. Using WKB approximation the tunneling rate is
obtained as

$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle
exp\Bigg{[}\frac{(1+\lambda c_{u}c_{r})}{(1-\lambda
c_{r}^{2})}\Bigg{(}\frac{\pi(r^{2}_{e}-r^{2}_{f})}{\Xi}\Bigg{)}\Bigg{]}$ (96)
$\displaystyle=$ $\displaystyle exp\Bigg{[}\frac{(1+\lambda
c_{u}c_{r})}{(1-\lambda c_{r}^{2})}\Delta S_{BH}\Bigg{]}.$ (97)

The change in Bekenstein-Hawking entropy near the event horizon of KdS black
hole due to Lorentz violation theory in Eddington coordinate is given by

$\displaystyle\frac{(1+\lambda c_{u}c_{r})}{(1-\lambda c_{r}^{2})}\Delta
S_{BH}=\frac{(1+\lambda c_{u}c_{r})}{(1-\lambda
c_{r}^{2})}\Big{[}S_{BH}(M-\omega)-S_{BH}(M)\Big{]}.$ (98)

We observe that the modified entropy depends on the Lorentz violation
parameter $\lambda$ and ether like vector $u^{\alpha}$. If $\lambda=0$ and
$c_{u}+c_{r}=0$ in Eq. (68), the Lorentz violation theory has been cancelled.
In such case, the actual Bekenstein-Hawking entropy is obtained near the event
horizon of KdS black hole (20).

We calculate the heat capacity near the event horizon of KdS black hole at the
event horizon $r=r_{h}$ due to Lorentz violation theory in Eddington
coordinate as

$\displaystyle C_{u}$ $\displaystyle=$
$\displaystyle\frac{2\pi(r_{h}^{2}+a^{2})^{2}(1+\lambda
c_{u}c_{r})F_{1}}{\Xi(1-\lambda
c_{r}^{2})[F_{2}(3r_{h}^{2}+a^{2})-r_{h}^{2}(r_{h}^{2}+a^{2})F_{3}]}.$ (99)

Near the event horizon $r=r_{h}$ of KdS black hole, the values of $F_{1}$,
$F_{2}$ and $F_{3}$ are given in Eq. (24). The modified heat capacity depends
on Lorentz violation parameter $\lambda$ and the ether like vector
$u^{\alpha}$. In the absence of Lorentz violation theory the original heat
capacity is obtained. It is observed that the well-behaved co-ordinate is
necesssary to obtain the original Hawking temperature, change in Bekenstein-
Hawking entropy and heat capacity of KdS black hole via Hamilton-Jacobi
equation.

##  6 Discussion and Conclusion

The Hawking temperature near the event horizon of KdS black hole derived from
the naive coordinate (32), Painleve coordinate (51) and Eddington coordinate
(66) can be recast having different conditions of constant terms as follows

$\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle\delta\frac{\Xi(3r_{h}^{2}-3\Lambda r_{h}^{4}-\Lambda
a^{2}r_{h}^{2}-3a^{2})}{6\pi r_{h}(r_{h}^{2}+a^{2})}.$ (100)

Case (i): If

$\displaystyle\delta=\frac{(1+\lambda c_{r}^{2})}{(\lambda
c_{t}c_{r}+\sqrt{1+\lambda c_{r}^{2}-\lambda c_{t}^{2}})},$ (101)

$T_{c}$ becomes the modified Hawking temperature of KdS black hole given in
Eq. (32). If $\lambda=0$ and $c_{r}=c_{t}$, $\lambda\neq 0$, the Lorentz-
violation is cancelled. In such case, $T_{c}$ will be two times of the correct
Hawking temperature given in Eq. (20). Our result is consistent with earlier
literatures [20, 23, 74, 75].

Case (ii): If

$\displaystyle\delta=\frac{1}{2}\frac{(1-\lambda c_{r}^{2})}{(1+\lambda
c_{T}c_{r})},$ (102)

Eq. (70) is equal to the modified Hawking temperature of KdS black hole given
in Eq. (51). If $\lambda=0$ and $c_{T}+c_{r}=0$, the Lorentz violation has
been cancelled and $T_{c}$ becomes the original Hawking temperature given in
Eq. (20). If $\delta>\frac{1}{2}$, $c_{T}+c_{r}<0$, the Hawking temperature
near the event horizon of KdS black hole increases $(T_{c}=T_{p}>T_{H})$ and
if $\delta<\frac{1}{2}$, $c_{T}+c_{r}>0$, the Hawking temperature decreases
$(T_{c}=T_{p}<T_{H})$, where $T_{p}$ and $T_{H}$ denote the modified Hawking
temperature (51) and original Hawking temperature (20) respectively near the
event horizon of KdS black hole.

Case (iii): If

$\displaystyle\delta=\frac{1}{2}\frac{(1-\lambda c_{r}^{2})}{(1+\lambda
c_{u}c_{r})},$ (103)

$T_{c}$ is the modified Hawking temperature of KdS black hole given in Eq.
(66). If $\lambda=0$ and $c_{u}+c_{r}=0$, the Lorentz violation has been
cancelled and $T_{c}$ tends to the original Hawking temperature of KdS black
hole given in Eq. (20). If $\delta>\frac{1}{2}$, $c_{u}+c_{r}<0$, the Hawking
temperature near the event horizon of KdS black hole increases
$(T_{c}=T_{u}>T_{H})$ and if $c_{u}+c_{r}>0$, the Hawking temperature
decreases $(T_{c}=T_{u}<T_{H})$, where $T_{u}$ denotes the modified Hawking
temperature near the event horizon of KdS black hole given by Eq. (66). The
modified Hawking temperatures in the three different coordinate systems and
the original Hawking temperature of KdS black hole are plotted with the radius
of event horizon in Fig. 1.

Figure 1: Plot of modified Hawking temperature, $T$ with radius of event
horizon, ${r_{h}}$ of KdS black hole for different coordinate systems- naive,
Painleve and Eddington. The original Hawking temperature is also shown. Here,
$a=0.2$, $\Lambda=0.6$, $\lambda=1$, $c_{t}=0.4$, $c_{r}=0.5$, $c_{T}=0.2$,
$c_{u}=0.9.$

To investigate the Hawking temperature of Schwarzschild black hole in static
coordinate using the Hamilton-Jacobi equation, the temperature is obtained as
$T=1/(4\pi M)$ which is different from actually calculated by other methods
[21, 86, 87]. It is called factor of 2 problem [88]. If Hamilton-Jacobi
equation is used to investigate the Hawking temperature of black hole in
Painleve coordinate, the correct Hawking temperature is obtained [89]. It
shows that the factor of 2 problem is a problem depending upon the choice of
coordinate. Akhmedov et al. [90] showed that the factor of 2 problem can not
be solved because it may be seen from the fact that the value of the integral
should not be changed by simple changing spatial variable. The same
observations can be seen in the study of Hawking radiation and change in
Bekenstein-Hawking entropy of KdS black hole using Hamilton-Jacobi equation in
Lorentz violation theory.

The change in Bekenstein-Hawking entropies near the event horizon of KdS black
hole derived from naive coordinate (37), Painleve coordinate (55) and
Eddington coordinate (68) can be combined into a single expression as follows

$\displaystyle\beta\Delta
S_{BH}=\beta\Big{[}S_{BH}(M-\omega)-S_{BH}(M)\Big{]}$ (104)

Case (i): If

$\displaystyle\beta=\frac{1}{2}\frac{[\lambda c_{t}c_{r}+\sqrt{1+\lambda
c_{r}^{2}-\lambda c_{t}^{2}}]}{(1+\lambda c_{r}^{2})},$ (105)

Eq. (74) is the change in Bekenstein-Hawking entropy given in Eq. (37). If
$\lambda=0$ and $c_{r}=c_{t}$, $\lambda\neq 0$, then Eq. (74) becomes
$\frac{1}{2}\Delta S_{BH}=\frac{1}{2}S_{BH}(M-w)-\frac{1}{2}S_{BH}(M),$ which
gives half of the Bekenstein-Hawking entropy near the event horizon of KdS
black hole [25].

Case (ii): If

$\displaystyle\beta=\frac{(1+\lambda c_{T}c_{r})}{(1-\lambda c_{r}^{2})},$
(106)

Eq. (74) becomes the change in Bekenstein-Hawking entropy given in Eq. (55).
When $\lambda=0$ and $c_{T}+c_{r}=0$, the Lorentz violation has been cancelled
and Eq. (74) becomes $\Delta S_{BH}=S_{BH}(M-w)-S_{BH}(M),$ which is the
actual value of change in Bekenstein-Hawking entropy of KdS black hole. The
Bekenstein-Hawking entropy (BHE) of KdS black hole increases if
$c_{T}+c_{r}>0$ $(\beta>1)$ and the BHE decreases if $c_{T}+c_{r}<0$
$(\beta<1)$.

Case (iii): If

$\displaystyle\beta=\frac{(1+\lambda c_{u}c_{r})}{(1-\lambda c_{r}^{2})},$
(107)

Eq. (74) is reduced to the change in Bekenstein-Hawking entropy given in Eq.
(68). When $\lambda=0$ and $c_{u}+c_{r}=0$, the Lorentz violation has been
cancelled. Then Eq. (74) becomes $\Delta S_{BH}=S_{BH}(M-w)-S_{BH}(M),$ which
gives the actual change in Bekenstein-Hawking entropy of KdS black hole. The
Bekenstein-Hawking entropy (BHE) of KdS black hole increases if
$c_{u}+c_{r}>0$ $(\beta>1)$ and the BHE decreases if $c_{u}+c_{r}<0$
$(\beta<1)$.

The heat capacity near the event horizon of KdS black hole derived from naive
coordinate (38), Painleve coordinate (56) and Eddington coordinate (69) can be
written into a single expression as follows

$\displaystyle C_{c}$ $\displaystyle=$
$\displaystyle\gamma\frac{\pi(r_{h}^{2}+a^{2})^{2}F_{1}}{\Xi[F_{2}(3r_{h}^{2}+a^{2})-r_{h}^{2}(r_{h}^{2}+a^{2})F_{3}]}.$
(108)

Case (i): If

$\displaystyle\gamma=\frac{(\lambda c_{t}c_{r}+\sqrt{1+\lambda
c_{r}^{2}-\lambda c_{t}^{2}})}{(1+\lambda c_{r}^{2})},$ (109)

Eq. (78) is the modified heat capacity of KdS black hole given in Eq. (38). If
$\lambda=0$ and $c_{r}=c_{t}$, $\lambda\neq 0$, the Lorentz-violation theory
is cancelled. In such case, the value of heat capacity, $C_{c}$ is the half of
the correct one given in Eq. (23).

Case (ii): If

$\displaystyle\gamma=\frac{2(1+\lambda c_{T}c_{r})}{(1-\lambda c_{r}^{2})},$
(110)

Eq. (78) represents the heat capacity of KdS black hole given in Eq. (56). If
$\lambda=0$ and $c_{T}+c_{r}=0$, the Lorentz violation has been cancelled and
Eq. (78) is equal to the heat capacity of KdS black hole given in Eq. (23).
The heat capacity near the event horizon of KdS black hole increases,
$C_{c}=C_{p}>C_{h}$ if $c_{T}+c_{r}>0$ $(\gamma>2)$ and the heat capacity
decreases, $C_{c}=C_{p}<C_{h}$ if $c_{T}+c_{r}<0$ $(\gamma<2)$, where $C_{p}$
and $C_{h}$ are the modified heat capacity (56) and original specific heat
(23) respectively near the event horizon of KdS black hole.

Case (iii): If

$\displaystyle\gamma=\frac{2(1+\lambda c_{u}c_{r})}{(1-\lambda c_{r}^{2})},$
(111)

Eq. (78) becomes the heat capacity of KdS black hole given in Eq. (69). If
$\lambda=0$ and $c_{u}+c_{r}=0$, the Lorentz violation has been cancelled and
$C_{c}$ tends to the original heat capacity given in Eq. (23). The heat
capacity near the event horizon of KdS black hole increases,
$C_{c}=C_{u}>C_{h}$ if $c_{u}+c_{r}>0$ $(\gamma>2)$ and the heat capacity
decreases, $C_{c}=C_{u}<C_{h}$ if $c_{u}+c_{r}<0$ $(\gamma<2)$, where $C_{u}$
represents the modified heat capacity (69) near the event horizon of KdS black
hole.

In this paper, we discuss the tunneling of scalar particles across the event
horizon of KdS black hole using Klein-Gordon equation with the Lorentz
violation theory of curved space time in three co-ordinate systems namely
naive coordinate system, Painleve coordinate system and Eddington coordinate
system.

The Hawking temperatures, entropies and heat capacities are modified due to
the presence of Lorentz violation parameter $\lambda$ and ether like vectors
$u^{\alpha}$. It is shown that the Hawking temperatures, entropies and heat
capacities increase or decrease depending upon the choices of ether like
vectors $u^{\alpha}$. When Lorentz violation theory is cancelled, the naive
coordinate gives the incorrect result of Hawking temperature, Bekenstein-
Hawking entropy and specific heat capacity but Painleve coordinate and
Eddington coordinate give the exact Hawking temperatures, entropies and heat
capacities of KdS black hole.

The variation between naive and Painleve or Eddington coordinate may be seen
in a curved manifold, a covariant distribution of the form $1/(r+i0)$ cannot
be derived from the non-locally integrable function $1/r$.

In the Painleve coordinate and Eddington coordinate systems, the metric
components are analytic at the event horizon $r=r_{h}$ of the black hole and
there is a time-like Killing vector, which keeps the space time stationary.

Hence the well-behaved coordinates are suitable to study the Hawking
radiation, Bekenstein-Hawking entropy and heat capacity of KdS black hole near
the event horizon via Klein-Gordon equation.

## Acknowledgements

This work is supported by the DST INSPIRE fellowship, New Delhi, India (Grant
No. IF190759). The authors would like to thank the anonymous reviewers for
valuable comments and suggestions to improve the paper.

## Competing Interests

The authors declare that there is no conflict of interests regarding the
publication of this paper

## Data Availability

No data were used to support the findings of this study.

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