#  Incorporating intrinsic compressibility effects in velocity transformations
for wall-bounded turbulent flows

Asif Manzoor Hasan<EMAIL_ADDRESS>Process and Energy Department, Delft
University of Technology, Leeghwaterstraat 39, 2628 CB, Delft, The Netherlands
Johan Larsson  Department of Mechanical Engineering, University of Maryland,
College Park, MD 20742, USA  Sergio Pirozzoli  Dipartimento di Ingegneria
Meccanica e Aerospaziale, Sapienza Università di Roma, Via Eudossiana 18,
00184 Roma, Italy  Rene Pecnik<EMAIL_ADDRESS>Process and Energy
Department, Delft University of Technology, Leeghwaterstraat 39, 2628 CB,
Delft, The Netherlands

###### Abstract

A transformation that relates a compressible wall-bounded turbulent flow with
non-uniform fluid properties to an equivalent incompressible flow with uniform
fluid properties is derived and validated. The transformation accounts for
both variable-property and intrinsic compressibility effects, the latter being
the key improvement over the current state-of-the-art. The importance of
intrinsic compressibility effects contradicts the renowned Morkovin’s
hypothesis.

††preprint: APS/123-QED

The law of the wall for incompressible turbulent flows is one of the
cornerstones of fluid dynamics [1]. Such a universal law is still missing for
compressible flows, because the interplay of thermodynamics and hydrodynamics
leads to significantly richer flow physics and even more intricate phenomena
in turbulence. Efforts have long been devoted to find a transformation that
reduces the mean velocity profile of compressible wall-bounded flows to that
of incompressible, constant-property flows [2]. Such a transformation can
assist in extending the incompressible modeling techniques to compressible
flows, eventually enabling better flow and heat transfer predictions for a
range of applications.

The history of velocity transformations dates back to the 1950s, when Van
Driest [3] (hereafter VD) proposed a correction to the incompressible law of
the wall, accounting for mean density variations in the friction velocity
scale. Zhang  _et al._ [2] proposed a transformation that improves the
collapse in the wake region of compressible boundary layers. However, both
transformations were developed for adiabatic boundary layers, and as such,
they fail for diabatic flows. In 2016, Trettel and Larsson [4] (hereafter TL)
formally derived an alternative to the VD transformation, suggesting that the
semi-local wall coordinate, previously defined on intuitive grounds by Huang
_et al._ [5], is the correct scaling to account for changes in the viscous
length scale. Patel  _et al._ [6] developed a mathematically equivalent
velocity transformation by studying the effect of variable density and
viscosity on turbulence in channel flows at the zero Mach number limit. Their
findings revealed that the primary influence of variable properties on the
velocity transformation can be effectively characterized by the semi-local
Reynolds number. Despite being accurate for channel flows, these
transformations are inaccurate for high-speed boundary layers [4, 7, 8, 9],
typically yielding higher log-law intercept as compared to incompressible
flows. Recently, Griffin  _et al._ [8] (hereafter GFM) derived a new
transformation based on the universality of the ratio of production and
dissipation, and a stress-based blending function. The GFM transformation
improves the collapse of the velocity profile for compressible boundary
layers, however, it is inaccurate for ideal gas flows with non-air-like
viscosity laws, and for flows with fluids at supercritical pressures [10].
Volpiani  _et al._ [11] proposed a data-driven transformation which also
improves the results for compressible boundary layers, although not rooted in
physical principles. The lack of a universal, physics-driven compressibility
transformation for turbulent wall-bounded flows sets up the motivation for
this Letter.

Figure 1: (a) Mean velocity profiles in constant-property compressible channel
flows, after TL transformation [4, 6] defined as
$d\bar{U}_{TL}^{+}=\left(1-y/\delta_{v}^{*}\,d\delta_{v}^{*}/dy\right)\sqrt{\bar{\rho}/\rho_{w}}d\bar{u}^{+}$
(since property variations are absent in these cases, the transformed velocity
is equivalent to $\bar{u}^{+}$). (Inset): Log-law constant $C_{TL}$ as a
function of $M_{\tau}$. Ideal gas: (Large closed stars) constant-property
compressible channels; (open $\triangle$) cooled channels [4]; (closed
$\triangle$) adiabatic channels with pseudo-heat sources (present authors,
unpublished); (open $\square$) cooled and (closed $\square$) adiabatic
boundary layers ([12, 13, 14]; [7] Mach 2 and 14 cases only; A. Ceci, private
communication); (closed $\circ$) channels with non-air-like viscosity power-
law exponent of -0.5 ([9]; present authors, unpublished). The dashed line
shows a fit for the constant-property cases, whereas the gray shaded area
indicates an error bar of +/-5%. Note that low-Reynolds number cases (less
than 300) are excluded. (b) turbulent shear stress for the constant-property
compressible cases. (Inset): Kolmogorov length scale, scaled by the semi-local
viscous length scale $\delta_{v}^{*}$. The black symbols are the
incompressible case of Moser  _et al._ [15], in both (a) and (b).

All transformations outlined above rely on the implicit assumption that
intrinsic compressibility effects are insignificant, and that only mean fluid
property variations matter for this problem. This is Morkovin’s hypothesis
[16], often advocated in the theoretical analysis of compressible turbulent
wall-bounded flows. However, depending on the flow conditions, intrinsic
compressibility effects associated with density changes of fluid elements in
response to changes in pressure can also be important [17]. These density
changes generate dilatational velocity fluctuations that can modulate the
near-wall dynamics, thereby affecting the turbulence statistics [18, 19, 20,
21, 22]. Yet, their influence on mean velocity scaling is unclear.

Objectives.—The first objective of this Letter is to argue that Morkovin’s
hypothesis is not quite accurate and that intrinsic compressibility effects
can modify the mean velocity profile, and the second objective is to derive a
transformation that accounts for these effects. To address the first goal, we
perform Direct Numerical Simulations (DNS) of compressible channel flow at
high Mach number, in which we isolate intrinsic compressibility effects by
eliminating mean property variations. To attain approximately constant mean
properties, we follow the method proposed by Coleman  _et al._ [23], in which
viscous heating is removed from the energy equation. These “constant property”
(CP) simulations are performed at bulk Mach numbers ($M_{b}$, namely the ratio
of the bulk velocity to the speed of sound based on the wall temperature) of
0.3, 2.28, 3, and 4, and a friction Reynolds number ($Re_{\tau}$, defined
below) of 550, using STREAmS [24] with an ideal gas equation of state and a
power-law for the dynamic viscosity.

Definitions.—Before analyzing the flow cases, it is necessary to introduce a
few important quantities. The friction velocity and the viscous length scales
at the wall are defined as $u_{\tau}=\sqrt{\tau_{w}/\rho_{w}}$ and
$\delta_{v}=\mu_{w}/(\rho_{w}u_{\tau})$, respectively, where $\tau_{w}$ is the
wall shear stress, and $\rho_{w}$, $\mu_{w}$ are the wall density and
viscosity. The friction Reynolds number is defined as
$Re_{\tau}=\delta/\delta_{v}$, where $\delta$ is either the channel half-
height or the thickness of the boundary layer. To account for variations in
the fluid properties, the semi-local friction velocity and viscous length
scales are defined based on the local density and viscosity as
$u_{\tau}^{*}=\sqrt{\tau_{w}/\bar{\rho}}$ and
$\delta_{v}^{*}=\bar{\mu}/(\bar{\rho}u_{\tau}^{*})$. Hence, both these scales
vary in the wall-normal direction. Superscripts $+$ and $*$ are used to denote
scaling with wall or semi-local quantities, respectively. The overbar symbol
is used to denote Reynolds averaging, and single and double primes are used to
denote fluctuations from Reynolds averages and from Favre (density-weighted)
averages, respectively.

Intrinsic compressibility effects on mean velocity.—Figure 1(a) shows the mean
velocity profiles for the four CP cases. The velocity profile for the low-Mach
number case ($M_{b}=0.3$) collapses with the incompressible case of Moser  _et
al._ [15] at a similar Reynolds number. However, as the Mach number increases,
a clear increase in the log-law intercept is observed. Due to roughly constant
mean properties and negligible fluctuations generated by heat transfer, the
log-law shift can be solely attributed to intrinsic compressibility effects,
which contradicts Morkovin’s hypothesis.

After identifying the impact of intrinsic compressibility effects, the next
crucial step is to determine the most suitable parameter for quantifying them.
From dimensional analysis in the near-wall region of compressible boundary
layers, Bradshaw [16] deduced $u_{\tau}=\sqrt{\tau_{w}/\rho_{w}}$ to be the
relevant velocity scale, and $a_{w}=\sqrt{\gamma RT_{w}}$ as the relevant
sound speed, respectively. Thus, $M_{\tau}=u_{\tau}/a_{w}$ was identified as
the most suitable Mach number, as also supported by Smits and Dussauge [25].
In the semi-local scaling framework, these scales can be redefined using local
properties, such that the semi-local friction Mach number
$M_{\tau}^{*}=u_{\tau}^{*}/\bar{a}$ becomes the relevant parameter, with
$\bar{a}=\sqrt{\gamma R\bar{T}}$ being the speed of sound based on the local
temperature. However, for ideal gas flows, $u_{\tau}^{*}\sim\sqrt{\bar{T}}$
(due to constant mean pressure), implying that the semi-local friction Mach
number is nearly constant across the boundary layer, and hence approximately
equal to $M_{\tau}$.

Another argument in support of $M_{\tau}$ being the most suitable parameter
can be provided as follows. Coleman  _et al._ [23] suggested that the pressure
fluctuations in relation to the mean thermodynamic pressure,
$p^{\prime}/\bar{p}$, is an appropriate indicator of intrinsic compressibility
effects because the isentropic changes in fluid volume (dilatational
fluctuations) are related to pressure fluctuations through $\partial
u_{i}^{\prime}/\partial x_{i}\approx-1/\gamma D(p^{\prime}/\bar{p})/Dt$ [17],
where $D()/Dt$ denotes material derivative, and Einstein summation is implied.
On the other hand, Bernardini and Pirozzoli [12] and Duan  _et al._ [26]
noticed that $p^{\prime}$ scaled by the hydrodynamic scale $\tau_{w}$ shows a
weak Mach number dependence. Combining these statements, we can write
$p^{\prime}/\bar{p}=\tau_{w}/\bar{p}\,(p^{\prime}/\tau_{w})\approx\gamma
M_{\tau}^{2}(p^{\prime}/\tau_{w})$. This shows that the Mach number dependence
of the quantity $p^{\prime}/\bar{p}$, and consequently dilatation, can be
attributed to the factor $\gamma M_{\tau}^{2}$. It further underscores the
importance of $M_{\tau}$ as the correct parameter for gauging intrinsic
compressibility effects. These arguments are substantiated by the observations
made by Yu  _et al._ [21], who found that intrinsic compressibility effects on
the wall-shear-stress and wall-pressure fluctuations accurately scale with
$M_{\tau}$.

The inset in Figure 1(a) shows the variation of the log-law intercept of the
transformed mean velocity profile ($C_{TL}$, evaluated as in Trettel and
Larsson [4] using integration bounds from $y^{*}=y/\delta_{v}^{*}\approx 50$
to $y/\delta\approx 0.1$) as a function of the friction Mach number $M_{\tau}$
for the four CP cases, and for several compressible ideal gas channel flows
and boundary layers available in the literature. To account for mean property
variations, the semi-local velocity transformation [4, 6] (also known as the
TL transformation) is utilized. This transformation does not incorporate
intrinsic compressibility effects and is thus strictly valid for low-speed
heated/cooled wall-bounded flows only, such as the zero-Mach number cases of
Patel  _et al._ [6]. Hence, its application to high-speed flows can help
isolate intrinsic compressibility effects. The trend line in the inset of
Figure 1(a) is obtained by considering the CP cases only, hence it is a
measure of the log-law shift due to intrinsic compressibility alone.
Interestingly, the majority of the other cases follow the trend line,
suggesting that increase in the log-law intercept, as also observed in the
literature [see, e.g. 4, 7, 8, 9], is mainly due to intrinsic compressibility
effects. Deviations from the common trend can be attributed to effects other
than those directly related to mean property variations (within the
assumptions of the semi-local scaling theory) and intrinsic compressibility,
as we briefly discuss at the end of this Letter. Note that non-negligible
scatter is also observed in incompressible flows [27], which suggests that the
precise determination of the log-law constant is sensitive to
numerical/experimental uncertainties.

The subsequent discussion outlines the physical mechanism for the occurrence
of a log-law shift. The TL transformation assumes universality of the
turbulent shear stress (and hence, viscous shear stress) in the inner layer.
However, Figure 1(b) shows an outward shift of the turbulent shear stress for
the CP cases, with increasing Mach number. This shift can be explained in
terms of delayed development of active turbulence (wall-normal fluctuations)
away from the wall, as observed in literature [7, 13, 28], because it directly
controls the production of the turbulent shear stress [29]. The delay in
active fluctuations is caused by reduced inter-component energy transfer from
the streamwise to the lateral components, as observed previously in
compressible channels [30] and boundary layers [28]. The outward shift in the
turbulent shear stress implies an analogous outward shift in the viscous shear
stress, such that the total shear stress remains unchanged. Since the TL-
transformed mean velocity profile results from the integration of the viscous
shear stress [4, 6], its outward shift is responsible for increase of the log-
law intercept.

To account for the outward shift outlined above, we drop the universality
assumption of the turbulent shear stress, made in Trettel and Larsson [4], and
derive a mean velocity transformation accounting for intrinsic compressibility
effects.

Derivation.—In the inner layer of parallel (or quasi-parallel) shear flows,
integration of the mean momentum equation implies that the sum of viscous and
turbulent shear stresses is equal to the total shear stress, given as

$\bar{\mu}\frac{d\bar{u}}{dy}-\overline{\rho
u^{\prime\prime}v^{\prime\prime}}\approx{\tau_{t}},$ (1)

where $\tau_{t}\approx\tau_{w}$ in boundary layers and it varies linearly in
channel flows. Note that terms due to viscosity fluctuations are neglected.
Normalizing Eq. (1) by $\tau_{w}$ and using the definitions of $u^{*}_{\tau}$
and $\delta^{*}_{v}$, we get the non-dimensional form as

$\underbrace{\frac{\delta_{v}^{*}}{u_{\tau}^{*}}\frac{d\bar{u}}{dy}}_{{\color[rgb]{0,0,0}d\bar{U}_{TL}^{+}/dy^{*}}}+r_{uv}^{+}\approx{\tau_{t}^{+}},$
(2)

where $r_{uv}^{+}=-{\overline{\rho
u^{\prime\prime}v^{\prime\prime}}}/\tau_{w}$ and
$\tau_{t}^{+}=\tau_{t}/\tau_{w}$. Next, following Trettel and Larsson [4], we
assume universality of the total shear stress and equate Eq. (2) with its
incompressible counterpart to get

$\frac{d\bar{U}^{+}}{dY^{+}}+R_{uv}^{+}=\frac{\delta_{v}^{*}}{u_{\tau}^{*}}\frac{d\bar{u}}{dy}+r_{uv}^{+},$
(3)

where $\bar{U}^{+}=\bar{U}/u_{\tau}$ and $Y^{+}=Y/\delta_{v}$ denote the non-
dimensional velocity and wall-normal coordinate of an incompressible flow,
that constitute the universal law of the wall.

Introducing the definition of the eddy viscosity for incompressible flows
(superscript ‘$i$’) as $\mu_{t}^{i}/\mu_{w}=R_{uv}^{+}/(d\bar{U}^{+}/dY^{+})$,
and similarly for compressible flows (superscript ‘$c$’) as
$\mu_{t}^{c}/\bar{\mu}=r_{uv}^{+}/([\delta_{v}^{*}/u_{\tau}^{*}]d\bar{u}/dy)$,
Eq. (3) can be written as

$\left(1+\mu_{t}^{i}/\mu_{w}\right)\left(\frac{d\bar{U}^{+}}{dY^{+}}\right)=\left(1+\mu_{t}^{c}/\bar{\mu}\right)\left(\frac{\delta_{v}^{*}}{u_{\tau}^{*}}\frac{d\bar{u}}{dy}\right),$
(4)

which upon rearrangement yields

$\frac{d\bar{U}^{+}}{d\bar{u}^{+}}=\left(\frac{1+\mu_{t}^{c}/\bar{\mu}}{1+\mu_{t}^{i}/\mu_{w}}\right)\frac{\delta_{v}^{*}}{\delta_{v}}\frac{dY^{+}}{dy^{+}}\frac{u_{\tau}}{u_{\tau}^{*}}.$
(5)

Eq. (5) offers a very general eddy-viscosity-based framework for deriving
compressibility transformations for wall-bounded flows that satisfy Eq. (1).
This equation in dimensional form is similar to that proposed in Iyer  _et
al._ [31], where it is employed to deduce an eddy viscosity model, provided a
velocity transformation kernel is known (see also Ref. [32]).

In order to fully define the velocity transformation, a relationship between
$Y^{+}$ and $y^{+}$ shall be established. Assuming that the turbulent shear
stress is universal in the inner layer, Trettel and Larsson [4] deduced that
$Y^{+}=(\delta_{v}/\delta_{v}^{*})y^{+}=y^{*}$. However, as seen in Figure
1(b), the turbulent shear stress is not universal in the presence of intrinsic
compressibility effects, hence the question of whether or not $Y^{+}=y^{*}$
still holds has to be reassessed. Indeed, $Y/\delta_{v}=y/\delta_{v}^{*}$
implies that $\delta_{v}^{*}$ is the proper length scale for small-scale
turbulence and viscous effects in compressible flows, just like $\delta_{v}$
in incompressible flows. This was first proposed by Huang  _et al._ [5] and
later verified for a range of turbulence statistics by Patel  _et al._ [6].
The inset of Figure 1(b) shows the distribution of the Kolmogorov length scale
in semi-local units [6] for the four CP cases and the incompressible case of
Moser  _et al._ [15] at a similar Reynolds number. The nearly universal
distribution throughout the inner layer, despite non-universality of the
turbulent shear stress, supports the validity of $Y^{+}=y^{*}$ also in the
presence of intrinsic compressibility effects.

Exploiting the coordinate transformation $Y^{+}=y^{*}$, and using
$dy^{*}/dy^{+}=\left(1-y^{*}\,d\delta_{v}^{*}/dy\right)\delta_{v}/\delta_{v}^{*}$,
and $u_{\tau}/u_{\tau}^{*}=\sqrt{\bar{\rho}/\rho_{w}}$, we obtain the final
proposed velocity transformation kernel from Eq. (5) as

$\frac{d\bar{U}^{+}}{d\bar{u}^{+}}=\underbrace{\left(\frac{1+\mu_{t}^{c}/\bar{\mu}}{1+\mu_{t}^{i}/\mu_{w}}\right)}_{3}\underbrace{\left({1-\frac{y}{\delta_{v}^{*}}\frac{d\delta_{v}^{*}}{dy}}\right)}_{2}\underbrace{{\sqrt{\frac{\bar{\rho}}{\rho_{w}}}}}_{1}.$
(6)

Eq. (6) embodies a sequence of velocity transformations, as outlined below:

  * •

Factor 1 is the correction proposed by Van Driest [3] to account for the
change in the friction velocity scale from $u_{\tau}=\sqrt{\tau_{w}/\rho_{w}}$
to $u_{\tau}^{*}=\sqrt{\tau_{w}/\bar{\rho}}$.

  * •

Factor 2 is the correction proposed in Trettel and Larsson [4] and Patel  _et
al._ [6] to account for the change in the viscous length scale from
$\delta_{v}$ to $\delta_{v}^{*}$. Factors 1 and 2 combined form the TL
transformation kernel, but written in terms of the semi-local viscous length
scale, equivalent to that proposed in Patel  _et al._ [6]. These factors
account for the effects of mean property variations on the velocity
transformation.

  * •

Factor 3 is the proposed correction which accounts for additional physics
beyond those captured by the TL transformation.

Figure 2: Assessment of the (a) TL, (b) GFM, and (c) proposed transformations
for 55 ideal gas and 2 supercritical fluid cases. Ideal gas: (red solid lines)
constant-property compressible channels; (gray solid lines) cooled channels
[4, 20], adiabatic channels with pseudo-heat sources ([18, 20]; present
authors, unpublished), cooled and adiabatic boundary layers ([12, 13, 33, 34,
14]; [7] Mach 2 and 14 cases only; A. Ceci, private communication); (gray
dashed lines) channels and boundary layers with non-air-like viscosity power-
law exponents of -0.5 and -1.75 ([9]; present authors, unpublished).
Supercritical fluid: (gray dash-dotted lines) channel flows [35]. (Insets):
Percent error ($\varepsilon$) in the velocity transformation computed with
respect to the incompressible reference [36], as described in Griffin  _et
al._ [8]. Note that the inset for GFM has larger axis limits, and that the
non-air-like case with the largest error of 44% is not shown. Symbols are as
in Figure 1(a). Additionally, supercritical cases are denoted using closed
$\lozenge$. Shaded region indicates an error bar of +/-3%. As in Figure 1,
low-Reynolds number cases (less than 300) are excluded.

In order to obtain a closed form of the transformation, the eddy viscosities
$\mu_{t}^{i}$ and $\mu_{t}^{c}$ must be prescribed. Out of the many possible
eddy viscosity models, we consider the Johnson-King (JK) model [37] to achieve
an explicit expression of the transformation. The JK model is built on Van
Driest’s mixing-length arguments in the logarithmic region [3], with a damping
function [38] to reproduce the correct near-wall behaviour. The eddy viscosity
is defined as $\mu_{t}^{i}=\sqrt{\tau_{w}\rho_{w}}\kappa YD^{i}$, with
$D^{i}=[1-\mathrm{exp}({-Y^{+}}/{A^{+}})]^{2}$. The set of constants
$\kappa=0.41$, $A^{+}=17$ is commonly used [39], and yields an incompressible
log-law intercept 5.2. Similarly, for compressible flows,
$\mu_{t}^{c}=\sqrt{\tau_{w}\bar{\rho}}\kappa yD^{c}$, with the damping
function $D^{c}$ defined based on the the semi-local wall distance ($y^{*}$)
to account for mean property variations [40, 32]. As outlined previously,
intrinsic compressibility effects modulate the near-wall damping of
turbulence, causing the turbulent shear stress to shift outwards (Figure
1(b)). Thus, we modify the damping constant to depend on $M_{\tau}$, such that
the compressible eddy viscosity model reads

$\mu_{t}=\sqrt{\tau_{w}\bar{\rho}}\kappa
y\underbrace{\left[1-\mathrm{exp}\left({\frac{-y^{*}}{A^{+}+f(M_{\tau})}}\right)\right]^{2}}_{D^{c}}.$
(7)

The increase in the effective damping constant ($A^{+}+f(M_{\tau})$) in Eq.
(7), implies that the eddy viscosity (and hence, turbulent shear stress)
shifts outwards, with subsequent increase of the log-law intercept ($C_{TL}$).
In fact, it can be readily checked that $C_{TL}$ grows linearly with the
damping constant. Since the log-law intercept also depends linearly on
$M_{\tau}$ (see inset of Figure 1(a)), we argue that the corrective term
$f(M_{\tau})$ should be linear. Here, we use $f(M_{\tau})=19.3\,M_{\tau}$ to
reproduce the linear curve-fit presented in Figure 1(a).

Writing eddy viscosities in the non-dimensional form as
$\mu_{t}^{i}/\mu_{w}=\kappa Y^{+}D^{i}$ and $\mu_{t}^{c}/\bar{\mu}=\kappa
y^{*}D^{c}$, and replacing $Y^{+}$ by $y^{*}$ in $\mu_{t}^{i}/\mu_{w}$ yields
the final velocity transformation

$\bar{U}^{+}=\int_{0}^{\bar{u}^{+}}\\!\\!\left({\frac{1+\kappa
y^{*}{D^{c}}}{1+\kappa{y^{*}}{D^{i}}}}\right){\left({1-\frac{y}{\delta_{v}^{*}}\frac{d\delta_{v}^{*}}{dy}}\right)}\sqrt{\frac{\bar{\rho}}{\rho_{w}}}\,{d\bar{u}^{+}}.$
(8)

Results and Discussion.—This transformation is tested and compared to the TL
and GFM transformations in Figure 2 for fifty-seven flow cases, including
adiabatic and cooled boundary layers, cooled channels, and non-ideal flows,
covering a wide range of Mach numbers. The three transformations are
equivalent in the viscous sublayer, because the GFM log-layer transformation
is blended with TL, whereas the current transformation naturally reduces to TL
in the viscous sublayer, where $\mu_{t}\approx 0$ and factor 3 in Eq. (6)
reduces to unity. The log-law shift in the TL transformation is apparent. Such
a selective upward shift is not seen in the GFM transformation for the
conventional ideal gas cases. However, it fails for the constant-property
cases, ideal gas cases with non-air-like viscosity laws, and supercritical
fluid cases. The present transformation shows the least spread for the flow
cases considered herein, and it effectively removes the log-law shift observed
in the TL-transformation. Note that for all the transformations, the spread is
larger in the outer part of boundary layers and channels, which is arguably
beyond their scope, all being focused on the inner, constant-stress layer.

Despite the improved accuracy, the proposed transformation is only as accurate
as the assumptions made in its derivation. For instance, the transformation
might be inaccurate for cases where Eq. (1) does not hold, such as in
supercritical boundary layers where large density fluctuations induce a near-
wall convective flux in the stress balance equation [41]. Also, we have
assumed that variable-property effects are limited to factors 1 and 2 in Eq.
(6), and that these effects do not contribute to non-universality of the
turbulent shear stress (factor 3 in Eq. (6)). However, this is not always the
case, as suggested by the scatter in the log-law intercept with respect to the
fitted curve in Figure 1(a), which is eventually reflected in the new
transformation (see Figure 2(c)). We suspect cancellation between these
unincorporated effects and intrinsic compressibility effects to be the reason
why the TL transformation was found to be very accurate for ideal-gas channel
flows, but not for boundary layers. Incorporating these additional physics is
the next step for future studies aimed at developing an even more general
transformation. Lastly, it is important to note that for ideal gas cases, as
outlined above, the semi-local friction Mach number ($M_{\tau}^{*}$) is
roughly constant in the wall-normal direction and is equal to $M_{\tau}$,
however, for cases in which $M_{\tau}^{*}$ varies significantly in the domain,
$M_{\tau}$ may not be the most suitable parameter.

Implications on turbulence modeling.—The eddy viscosity with modified damping
function (Eq. (7)) can be interpreted as a compressibility-corrected wall
model for Large Eddy Simulations (LES), which can be implemented in current
codes by simply changing the damping function. The proposed transformation can
also help developing Reynolds Averaged Navier–Stokes (RANS) turbulence models
sensitized to compressibility effects. For instance, the modified damping
function in Eq. (7) can inspire modifications in the mixing lengths of
algebraic models [42, 43, 44]. Last, the inverse of the current transformation
can be leveraged to improve the drag and heat transfer predictive theories
[45, 46, 47].

To summarize, the log-law shift observed in the TL transformation can be
primarily attributed to the non-universality of the turbulent shear stress
caused by intrinsic compressibility effects. We ascertain this based on our
tailored constant-property compressible cases, in which the only dominant
effect is due to intrinsic compressibility. Taking $M_{\tau}$ as the most
suitable parameter to quantify these effects, we propose a new transformation
that effectively removes the log-law shift. The proposed transformation
accounts for the changes in friction velocity and viscous length scales due to
variations in mean properties, and for intrinsic compressibility effects.
Thus, it applies to a wide variety of cases. We anticipate that it may serve
as a building block for improved turbulence models; for example, it could be
used directly as an equilibrium wall-model.

We thank Dr. P. Costa for the insightful discussions and for reviewing the
manuscript. We thank A. Ceci for performing 2 boundary layer simulations for
this work. Dr. A. Trettel is thanked for discussions on computing the log-law
constant. Finally, we thank all the authors [4, 12, 7, 13, 14, 33, 34, 18, 20,
35, 9] for sharing their data with us. This work was supported by the European
Research Council grant no. ERC-2019-CoG-864660, Critical; and the Air Force
Office of Scientific Research under grants FA9550-19-1-0210 and
FA9550-19-1-7029.

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# Modified Hawking temperature and entropy of Kerr-de Sitter black hole in
Lorentz violation theory

Y. Onika Laxmi  Mathematics Department, Manipur University
Canchipur, Manipur, India
T. Ibungochouba Singh  Mathematics Department, Manipur University
Canchipur, Manipur, India
<EMAIL_ADDRESS>
###### Abstract

In this paper, we discuss the tunneling of scalar particles near the event
horizon of stationary and nonstationary Kerr-de Sitter black hole using
Lorentz violation theory in curved space time. The modified form of Hamilton-
Jacobi equation is derived from the Klein-Gordon equation by applying Lorentz
violation theory. The Hawking temperatures derived from stationary and
nonstationary Kerr-de Sitter black holes are modified due to Lorentz violation
theory. It is noted that the change in Bekenstein-Hawking entropy and modified
Hawking temperatures of stationary and nonstationary Kerr-de Sitter black hole
not only depend on the black hole parameters but also on ether like vectors
${\rm u^{\alpha}}$.

###### keywords:

Hawking radiation; Bekenstein-Hawking entropy; Specific Heat; Lorentz
violation theory

Received (Day Month Year)Revised (Day Month Year)

PACS Nos.: 04.70.Dy, 04.20.Gz, 03.65.-w.

##  1 Introduction

Hawkin${\rm g}^{1,2}$ showed theoretically that a black hole radiates like a
black body in which the temperature of radiation is dependent on the surface
gravity of black hole. The discovery of Hawking radiation leads to black hole
thermodynamic${\rm s.}^{3-5}$ Since then, many scientists have proposed
different techniques to study the quantum tunneling of black holes. Refs.
[6,7] proposed a new method to investigate the quantum thermal and nonthermal
radiations of stationary and nonstationary black holes by applying the
tortoise coordinate transformation, Maxwell’s electromagnetic field equation,
Klein-Gordon equation and Dirac equation. Applying this method, many
interesting results in different black holes have been derived i${\rm
n.}^{8-11}$ Refs. [12-14] introduced the semiclassical tunneling technique to
investigate the Hawking radiation of black hole. In this method, the Hawking
radiation is taken as a tunneling process near the event horizon of the black
hole and the outgoing particle produces the tunneling barrier of black hole.
They obtain well-behaved coordinate system which has no singularity near the
event horizon to derive the emission rate. Refs. [15-17], as an extension of
Parikh and Wilczek approach, studied the Hawking radiation as a tunneling of
charged massive particle at the event horizon of black hole by developing the
relation between phase and group velocity of the tunneling particle. Applying
the Hamilton-Jacobi equation, Feynman prescription and WKB approximation,
Refs. [18] investigated the Hawking radiation at the event horizon of rotating
and nonrotating black holes. They showed that the isotropic coordinate or
invariant coordinate gives the correct Hawking temperature whereas naive
coordinate leads to half of correct Hawking temperature. Ref. [19] studied the
Hawking radiation at the event horizon of different black holes by using Dirac
equation, Feynman prescription and Pauli Sigma matrices. In this method, the
appropriate gamma matrices obtained from black hole and suitable wave function
are substituted in Dirac equations, then the action related to Boltzman factor
for emission in accordance with semiclassical approximation is obtained.

Refs. [20-23] discussed the corrected Hawking temperature and entropy of black
holes using first law of black hole thermodynamics and Hamilton-Jacobi
equation beyond the semiclassical approximation in Schwarzschild like
coordinate system and Painleve coordinate system. Kruglo${\rm v}^{24,25}$
proposed the quantum tunneling of boson near the event horizon of black hole
using Proca equation, WKB approximation and Feynman prescription. The emission
temperature of Schwarzschild background geometry which is the same as the
Hawking temperature corresponding to scalar particle is also obtained.
Following their work, many interesting results have been obtained i${\rm
n.}^{26-29}$

The study of Lorentz symmetry violation theory has been discussed during the
past decades and many researchers have proposed different gravity models
induced by Lorentz symmetry violatio${\rm n.}^{30-34}$ Refs. [35-37] proposed
the Lorentz symmetry violation in flat space time using Dirac equation and
ether like vectors $u^{\alpha}$. Since then, the Lorentz violation has been
extended to curved space time by choosing ether like vectors $u^{\alpha}$ so
that it could hold $u^{\alpha}u_{\alpha}=$ $\rm{constan{\rm t}.}^{38-42}$

The paper is outlined as follows : In section 2, the modified Hawking
temperature, heat capacity and change in entropy near the event horizon of
Kerr-de Sitter black hole are derived. In section 3, the modified surface
gravity and the Hawking temperature of nonstationary KdS black hole are
discussed using tortoise coordinate transformation in Lorentz violation theory
in curved space time. Some discussions and conclusions are given in section 4
and 5 respectively.

##  2 Modified Hawking temperature of Kerr-de Sitter black hole

The Kerr-de Sitter (KdS) solution indicates the space time geometry of a
rotating black hole with non-zero cosmological constant. The vacuum solution
of KdS black hole is well known Boyer-Lindquist coordinates
$(t,r,\theta,\phi)$ with geometrical unit $(c=G=1)$ which is given b${\rm
y}^{43}$

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

where

$\displaystyle
R^{2}=r^{2}+a^{2}\cos^{2}\theta,\,\,\,\,\,\Xi=1+\frac{1}{3}\Lambda a^{2},$ (3)
$\displaystyle\Delta_{\theta}=1+\frac{1}{3}\Lambda a^{2}\cos^{2}\theta,$ (4)
$\displaystyle\Delta=(r^{2}+a^{2})\Big{(}1-\frac{1}{3}\Lambda
r^{2}\Big{)}-2Mr.$ (5)

Eq. (1) represents rotating KdS black hole for cosmological constant
$\Lambda$. The KdS black hole is well defined in the region $-\infty\leq
t\leq\infty$, $0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$. $M$ and $a$ are
the mass and rotational parameter of KdS black hole. If $\frac{1}{\Lambda}\geq
M^{2}>a^{2}$, then $\Delta=0$ gives four distinct roots i.e.
$r_{+},r_{h},r_{-}$ and $r_{--}$ $(r_{+}>r_{h}>r_{-}>r_{--})$. The biggest
root $r_{+}$ denotes the location of cosmological horizon. $r_{h}$ and $r_{-}$
represent the location of the event horizon and Cauchy horizon respectively.
If $r=0$, $\theta=\frac{\pi}{2}$, then the other side of $r=0$, $r=r_{--}$ is
taken as another cosmological horizo${\rm n.}^{44}$ According to Ref. [45],
the event horizon of KdS black hole $r=r_{h}$ can be written as

$\displaystyle r_{h}$ $\displaystyle=$
$\displaystyle\frac{1}{\Xi}\left(1+\frac{4\Lambda
M^{2}}{3\beta^{2}\Xi}+...\right)\left(M+\sqrt{M^{2}-a^{2}\Xi}\right),$ (6)

where $\beta=\sqrt{1-\frac{\Lambda}{3}a^{2}}$. There is a frame dragging
effect near the event horizon of KdS black hole. Let $\phi=\varphi-\Omega t$
and $\Omega=-\frac{g_{03}}{g_{33}}$. Then Eq. (1) reduces to

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

where the angular velocity at the event horizon of KdS black hole is given as

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\frac{a}{r_{h}^{2}+a^{2}}.$ (9)

According to Refs. [46, 47], the surface gravity of KdS black hole at the
event horizon $r=r_{h}$ is derived as

$\displaystyle\kappa=\lim_{g_{00}\rightarrow
0}\Bigg{(}-\frac{1}{2}\sqrt{-\frac{g^{11}}{g_{00}}}\frac{dg_{00}}{dr}\Bigg{)}=\frac{(r_{h}-M-\frac{2}{3}\Lambda
r_{h}^{3}-\frac{1}{3}\Lambda a^{2}r_{h})}{\Xi(r_{h}^{2}+a^{2})}.$ (10)

The Hawking temperature of KdS black hole is connected with surface gravity
via $T_{H}=\frac{\kappa}{2\pi}$ as

$\displaystyle T_{H}=\frac{1}{2\pi}\Bigg{[}\frac{(r_{h}-M-\frac{2}{3}\Lambda
r_{h}^{3}-\frac{1}{3}\Lambda a^{2}r_{h})}{\Xi(r_{h}^{2}+a^{2})}\Bigg{]}.$ (11)

To discuss heat capacity near the event horizon of black hole, the mass
parameter KdS black hole might be obtained from $\Delta(r_{h})=0$ as

$\displaystyle M=\frac{r_{h}}{2}+\frac{a^{2}}{2r_{h}}-\frac{\Lambda
r_{h}^{3}}{6}-\frac{\Lambda a^{2}r_{h}}{6}.$ (12)

The heat capacity $(C_{h})$ of black hole is defined by

$\displaystyle C_{h}$ $\displaystyle=$ $\displaystyle\frac{\partial
M}{\partial T_{H}}=\Bigg{(}\frac{\partial M}{\partial
r_{h}}\Bigg{)}\Bigg{(}\frac{\partial r_{h}}{\partial T_{H}}\Bigg{)}.$ (13)

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

$\displaystyle C_{h}=\dfrac{\partial M}{\partial T_{H}}$ $\displaystyle=$
$\displaystyle\frac{2\pi\Xi(r_{h}^{2}+a^{2})^{2}[3r_{h}^{2}-3a^{2}-\Lambda
r_{h}^{2}(3r_{h}^{2}+a^{2})]}{3(a^{4}-r_{h}^{4})+4a^{2}r_{h}^{2}(3-2\Lambda
r_{h}^{2})-\Lambda r_{h}^{2}(a^{4}+3r_{h}^{4})}.$ (14)

The modified form of Hamilton-Jacobi equation in Lorentz theory in curved
space time is given b${\rm y}^{48}$

$\displaystyle(g^{\mu\nu}+\lambda
u^{\mu}u^{\nu})\partial_{\mu}I\partial_{\nu}I+m^{2}$ $\displaystyle=$
$\displaystyle 0.$ (15)

where $\lambda$ and $u^{\alpha}$ are the correction parameter and ether like
vectors respectively. As $\lambda$ tends to zero in the above equation, the
Lorentz violation theory is cancelled and the original Hamilton-Jacobi
equation in curved space time is obtained. The ether like vectors $u^{\alpha}$
are constant in the flat space time of the canonical coordinate system. The
ether like vectors $u^{\alpha}$ are not constant in curved space time. But we
can take the vectors $u^{\alpha}$ from curved space time that satisfies the
condition $u^{\alpha}u_{\alpha}={\rm constant}$. To investigate the change in
entropy of stationary KdS black hole, we can choose $u^{\alpha}$ from (4) that
satisfies $u^{\alpha}u_{\alpha}={\rm constant}$ and $u^{\alpha}$ are related
to coordinate system acquired by the space time. The expressions of
$u^{t},u^{r},u^{\theta}$ and $u^{\phi}$ are defined by

$\displaystyle u^{t}$ $\displaystyle=$
$\displaystyle\frac{c_{t}}{\sqrt{-g_{tt}}}=\frac{c_{t}\Xi\sqrt{\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta}}{\sqrt{\Delta\Delta_{\theta}R^{2}}},$ (16) $\displaystyle
u^{r}$ $\displaystyle=$
$\displaystyle\frac{c_{r}}{\sqrt{g_{rr}}}=c_{r}\sqrt{\frac{\Delta}{R^{2}}},$
(17) $\displaystyle u^{\theta}$ $\displaystyle=$
$\displaystyle\frac{c_{\theta}}{\sqrt{g_{\theta\theta}}}=c_{\theta}\sqrt{\frac{\Delta_{\theta}}{R^{2}}},$
(18) $\displaystyle u^{\phi}$ $\displaystyle=$
$\displaystyle\frac{c_{\phi}}{\sqrt{g_{\phi\phi}}}=\frac{c_{\phi}\Xi\sqrt{R^{2}}}{\sqrt{[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta}]\sin^{2}\theta},$ (19)

where $c_{t}$, $c_{r}$, $c_{\theta}$ and $c_{\phi}$ are arbitrary constants.
$u^{\alpha}$ satisfies the condition

$\displaystyle
u^{\alpha}u_{\alpha}=-c_{t}^{2}+c_{r}^{2}+c_{\theta}^{2}+c_{\phi}^{2}={\rm
constant}.$ (20)

Using Eqs. (12) and (4) in Eq. (11), the dynamical equation of scalar particle
with mass $m$ in stationary KdS black hole is obtained as

$\displaystyle g^{00}(\frac{\partial I}{\partial t})^{2}+g^{11}(\frac{\partial
I}{\partial r})^{2}+g^{22}(\frac{\partial
I}{\partial\theta})^{2}+g^{33}(\frac{\partial I}{\partial\phi})^{2}+\lambda
u^{\mu}u^{\nu}\partial_{\mu}I\partial_{\nu}I+m^{2}=0.$ (21)

It is known that the above equation involves the variables $t,r,\theta$ and
$\phi$. To separate the variables on the Hamilton principal functions $I$, we
can choose the action $I$ as follows

$\displaystyle I$ $\displaystyle=$ $\displaystyle-\omega
t+S(r,\theta)+j\phi+\delta,$ (22)

where $S(r,\theta)$, $\omega$ and $j$ are the generalized momentum, particle
energy and angular momentum along the $\phi$-axis respectively and $\delta$ is
a complex constant. Using Eq. (15) in Eq. (14), a quadratic equation in
$\frac{\partial S}{\partial r}$ is obtained as

$\displaystyle A\Bigg{(}\frac{\partial S}{\partial
r}\Bigg{)}^{2}+B\left(\frac{\partial S}{\partial r}\right)+C=0.$ (23)

Then the two roots of the above equation are given by

$\displaystyle S=\int\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}dr,$ (24)

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle g^{11}+\lambda u^{r}u^{r},$
(25) $\displaystyle B$ $\displaystyle=$ $\displaystyle 2\lambda
u^{r}u^{\theta}\left(\frac{\partial I}{\partial\theta}\right)-2\lambda
u^{t}u^{r}(\omega-j\Omega)+2\lambda u^{r}u^{\phi}j,$ (26) $\displaystyle C$
$\displaystyle=$ $\displaystyle(g^{00}+\lambda
u^{t}u^{t})(\omega-j\Omega)^{2}+(g^{22}+\lambda
u^{\theta}u^{\theta})\left(\frac{\partial
I}{\partial\theta}\right)^{2}+(g^{33}+\lambda u^{\phi}u^{\phi})j^{2}$ (28)
$\displaystyle-2\lambda
u^{t}(\omega-j\Omega)\Big{(}u^{\theta}\left(\frac{\partial
I}{\partial\theta}\right)+u^{\phi}j\Big{)}+2\lambda
u^{\theta}u^{\phi}j\left(\frac{\partial I}{\partial\theta}\right)+m^{2}.$

Applying residue theorem of complex analysis and Feynman prescription near the
event horizon of KdS black hole, the integration of Eq. (17) can be written as

$\displaystyle S_{\pm}=\frac{i\pi\Xi(r_{h}^{2}+a^{2})\left[\lambda
c_{t}c_{r}\pm\sqrt{1-\lambda c_{t}^{2}+\lambda
c_{r}^{2}}\right](w-j\Omega)}{2(1+\lambda
c_{r}^{2})\\{(1-\frac{\Lambda}{3}a^{2})r_{h}-M-\frac{2\Lambda
r_{h}^{3}}{3}\\}},$ (29)

where $S_{+}$ and $S_{-}$ are the outgoing particle and ingoing particle
respectively. The probabilities which cross the black hole near the event
horizon are given by

$\displaystyle\Gamma_{emission}$ $\displaystyle=$ $\displaystyle{\rm
exp}(-2{\rm ImI})={\rm exp}[-2({\rm ImS_{+}}+{\rm Im}\delta)]$ (30)

and

$\displaystyle\Gamma_{absorption}$ $\displaystyle=$ $\displaystyle{\rm
exp}(-2{\rm ImI})={\rm exp}[-2({\rm ImS_{-}}+{\rm Im}\delta)].$ (31)

There is a $100\%$ chance the ingoing particle to enter the KdS black hole
according to WKB approximation. This indicates that ${\rm ImS_{+}}=-{\rm
ImS}_{-}$. We calculate the probability of outgoing particle as

$\displaystyle\Gamma_{rate}=\frac{\Gamma_{emission}}{\Gamma_{absorption}}$
$\displaystyle=$ $\displaystyle{\rm
exp}\Bigg{[}-\frac{2\gamma\Xi\pi(r_{h}^{2}+a^{2})(\omega-j\Omega)}{\\{(1-\frac{\Lambda}{3}a^{2})r_{h}-M-\frac{2\Lambda
r_{h}^{3}}{3}\\}}\Bigg{]},$ (32)

where $\gamma=\frac{\sqrt{1-\lambda c_{t}^{2}+\lambda c_{r}^{2}}}{1+\lambda
c_{r}^{2}}$. Eq. (22) is similar to Boltzmann factor according to
semiclassical approximation. The Hawking temperature near the event horizon of
KdS black hole in Lorentz violation theory is given by

$\displaystyle T=\frac{\\{(1-\frac{\Lambda}{3}a^{2})r_{h}-M-\frac{2\Lambda
r_{h}^{3}}{3}\\}}{2\gamma\pi\Xi(r_{h}^{2}+a^{2})}.$ (33)

If $\lambda=0$ in the above equation, the Lorentz violation has been
cancelled. In such case Eq. (23) is consistent with Eq. (7). If $\gamma>1$ or
$\gamma<1$, the Hawking temperature decreases or increases due to the presence
of correction term $\lambda$ and ether like vectors $u^{\alpha}$.

The modified heat capacity at the event horizon of KdS black hole is obtained
as

$\displaystyle C^{{}^{\prime}}_{h}$ $\displaystyle=$
$\displaystyle\dfrac{\partial M}{\partial
T}=\frac{2\pi\gamma\Xi(r_{h}^{2}+a^{2})^{2}[3r_{h}^{2}-3a^{2}-\Lambda
r_{h}^{2}(3r_{h}^{2}+a^{2})]}{3(a^{4}-r_{h}^{4})+4a^{2}r_{h}^{2}(3-2\Lambda
r_{h}^{2})-\Lambda r_{h}^{2}(a^{4}+3r_{h}^{4})}.$ (34)

As $\gamma$ tends to unity, Eq. (24) is consistent with original heat capacity
given in Eq. (10). If $\gamma>1$ or $\gamma<1$, the heat capacity increases or
decreases near the event horizon of KdS black hole depending upon the choices
of correction term $\lambda$ and ether like vectors $u^{\alpha}$.

Using Eqs. (3) and (5) in Eq. (19) for the outgoing particle, we get

$\displaystyle{\rm ImS}$ $\displaystyle=$
$\displaystyle\frac{\gamma^{{}^{\prime}}}{2}\left[\frac{\frac{\pi
k_{1}^{2}k_{2}^{2}}{\Xi}}{\frac{\beta^{2}}{\Xi}k_{1}k_{2}-M-\delta}\omega+\frac{\pi\Xi
a^{2}}{\frac{\beta^{2}}{\Xi}k_{1}k_{2}-M-\delta}\omega-\frac{\pi\Xi
a}{\frac{\beta^{2}}{\Xi}k_{1}k_{2}-M-\delta}j\right],$ (35)

where

$\displaystyle\gamma^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\lambda c_{t}c_{r}+\sqrt{1-\lambda c_{t}^{2}+\lambda
c_{r}^{2}}}{1+\lambda c^{2}_{r}},$ (36) $\displaystyle k_{1}$ $\displaystyle=$
$\displaystyle\left(1+\frac{4\Lambda M^{2}}{3\beta^{2}\Xi}+...\right),$ (37)
$\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\left(M+\sqrt{M^{2}-a^{2}\Xi}\right),$ (38)
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{2\Lambda}{3\Xi^{3}}\left(1+\frac{4\Lambda
M^{2}}{3\beta^{2}\Xi}+...\right)^{3}\left(M+\sqrt{M^{2}-a^{2}\Xi}\right)^{3}.$
(39)

To obtain the biggest value of the integration, we ignore second order terms
of KdS black hole mass parameter at the numerator and denominator in Eq. (25).
Then we find as

$\displaystyle{\rm ImS}$ $\displaystyle=$
$\displaystyle\frac{\gamma^{{}^{\prime}}}{2}\left[\frac{\pi
k_{2}^{2}}{\beta^{2}\left[k_{2}-\frac{M\Xi}{\beta^{2}}\right]}\omega+\frac{\pi\Xi^{2}a^{2}}{\beta^{2}\left[k_{2}-\frac{M\Xi}{\beta^{2}}\right]}\omega-\frac{\pi\Xi^{2}a}{\beta^{2}\left[k_{2}-\frac{M\Xi}{\beta^{2}}\right]}j\right].$
(40)

Taking the self-gravitational interaction into account, the mass parameter KdS
black hole is allowed to fluctuate. If a black hole emits a particle $\omega$
and angular momentum $j$, the KdS black hole parameter will be $M-\omega$ and
$J-j$ respectively. To find the change in Bekenstein-Hawking entropy of KdS
black hole, the term $(1-\frac{\Xi}{\beta^{2}}){\rm M}$ is ignored. Then we
get

$\displaystyle{\rm
ImS}=\frac{\gamma^{{}^{\prime}}}{2}\left[\int^{\omega}_{0}\frac{\pi
k_{2}^{2}}{\beta^{2}\sqrt{M^{2}-a^{2}\Xi}}d\omega^{{}^{\prime}}+\int^{\omega}_{0}\frac{\pi\Xi^{2}a^{2}}{\beta^{2}\sqrt{M^{2}-a^{2}\Xi}}d\omega^{{}^{\prime}}-\int^{j}_{0}\frac{\pi\Xi^{2}a}{\beta^{2}\sqrt{M^{2}-a^{2}\Xi}}dj^{{}^{\prime}}\right].$
(41) (42)

Changing $M$ by $M-\omega$ and $j$ by $J-j$ and putting the value of $k_{2}$
in the above equation, we obtain

$\displaystyle{\rm ImS}$ $\displaystyle=$
$\displaystyle\frac{\gamma^{{}^{\prime}}}{2}\Bigg{[}\frac{-\pi}{\beta^{2}}\int^{M-\omega}_{M}\frac{k_{2}^{2}}{\sqrt{(M^{\prime}-\omega^{\prime})^{2}-a^{2}\Xi}}d(M^{\prime}-\omega^{{}^{\prime}})$
(45)
$\displaystyle-\frac{\pi\Xi^{2}a^{2}}{\beta^{2}}\int^{M-\omega}_{M}\frac{1}{\sqrt{(M^{\prime}-\omega^{\prime})^{2}-a^{2}\Xi}}d(M^{\prime}-\omega^{{}^{\prime}})$
$\displaystyle+\frac{\pi\Xi^{2}a}{\beta^{2}}\int^{J-j}_{j}\frac{1}{\sqrt{(M^{\prime}-\omega^{\prime})^{2}-a^{2}\Xi}}d(J-j^{\prime})\Bigg{]},$

where $J-j=(M-\omega)a$. The imaginary part of the action finally yields

$\displaystyle{\rm ImS}$ $\displaystyle=$
$\displaystyle\frac{-\gamma^{{}^{\prime}}\pi}{2\beta^{2}}\Bigg{[}\int^{M-\omega}_{M}\frac{2(M^{\prime}-\omega^{\prime})^{2}+2(M^{\prime}-\omega^{\prime})\sqrt{(M^{\prime}-\omega^{\prime})^{2}-a^{2}\Xi}}{\sqrt{(M^{\prime}-\omega^{\prime})^{2}-a^{2}\Xi}}d(M^{\prime}-\omega^{{}^{\prime}})$
(47)
$\displaystyle-\int^{M-\omega}_{M}\frac{a^{2}\Xi}{\sqrt{(M^{\prime}-\omega^{\prime})^{2}-a^{2}\Xi}}d(M^{\prime}-\omega^{{}^{\prime}})\Bigg{]}.$

Calculating the $\omega^{{}^{\prime}}$ integral, that gives

$\displaystyle{\rm ImS}$ $\displaystyle=$
$\displaystyle\frac{-\gamma^{{}^{\prime}}\pi}{2\beta^{2}}\Bigg{[}(M-\omega)\sqrt{(M-\omega)^{2}-a^{2}\Xi}+(M-\omega)^{2}-M\sqrt{M^{2}-a^{2}\Xi}-M^{2}\Bigg{]},$
(48) $\displaystyle=$
$\displaystyle\frac{-\gamma^{{}^{\prime}}\pi}{4\beta^{2}}\Bigg{[}\Bigg{(}(M-\omega)+\sqrt{(M-\omega)^{2}-a^{2}\Xi}\Bigg{)}^{2}-\Bigg{(}M+\sqrt{M^{2}-a^{2}\Xi}\Bigg{)}^{2}\Bigg{]},$
(49) $\displaystyle=$
$\displaystyle\frac{-\gamma^{{}^{\prime}}\pi}{2}(r_{f}^{2}-r_{i}^{2}).$ (50)

Using WKB approximation, the tunneling rate is obtained as

$\displaystyle\Gamma$ $\displaystyle\sim$ $\displaystyle exp(-2\rm ImS),$ (51)
$\displaystyle=$ $\displaystyle
exp[\gamma^{{}^{\prime}}\pi(r_{f}^{2}-r_{i}^{2})],$ (52) $\displaystyle=$
$\displaystyle exp[\gamma^{{}^{\prime}}\Delta S_{BH}],$ (53)

where $\gamma^{\prime}\Delta
S_{BH}=\gamma^{\prime}(S_{BH}(M-\omega)-S_{BH}(M))$ is the modified entropy of
KdS black hole in Lorentz violation theory.
$r_{i}=\frac{1}{\sqrt{2}\beta}\Bigg{[}M+\sqrt{M^{2}-a^{2}\Xi}\Bigg{]}$ and
$r_{f}=\frac{1}{\sqrt{2}\beta}\Bigg{[}(M-\omega)+\sqrt{(M-\omega)^{2}-a^{2}\Xi}\Bigg{]}$
are the locations of horizons before and after emission of particle. If
$\lambda=0$, the Lorentz violation theory has been cancelled, the original
change in Bekenstein-Hawking entropy is obtained. When
$\gamma^{{}^{\prime}}>1$, the change in Bekenstein-Hawking entropy increases
and $\gamma^{{}^{\prime}}<1$, the change in Bekenstein-Hawking entropy
decreases near the event horizon of KdS black hole. In above cases, the change
in Bekenstein-Hawking entropy depends on correction parameter $\lambda$ and
ether like vectors $u^{\alpha}$.

##  3 Nonstationary rotating KdS black hole

The metric of rotating nonstationary KdS black hole in retarded time
coordinate${\rm s}^{10}$$(u,r,\theta,\phi)$ is defined by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}\Xi^{2}}[\Delta_{\lambda}-\Delta_{\theta}a^{2}\sin^{2}\theta]du^{2}-\frac{R^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{2a}{R^{2}\Xi^{2}}[\Delta_{\theta}(r^{2}+a^{2})-\Delta_{\lambda}]\sin^{2}\theta
dud\phi$ (55) $\displaystyle+\frac{2}{\Xi}[du-a\sin^{2}\theta
d\phi]dr-\frac{1}{R^{2}\Xi^{2}}[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta_{\lambda}a^{2}\sin^{2}\theta]\sin^{2}\theta
d\phi^{2},$

where $R^{2},\Xi,\Delta_{\theta}$ are given in Eq. (2). The term
$\Delta_{\lambda}$ is given by

$\displaystyle\Delta_{\lambda}=r^{2}+a^{2}-2M(u)r-\frac{1}{3}\Lambda
r^{2}(r^{2}+a^{2}),$ (56)

where $M(u)$ is the mass of nonstationary KdS black hole. The location of
event horizon of stationary and nonstationary KdS black hole can be obtained
from null surface equation $F(u,r,\theta,\phi)=0.$ The expression of null
surface equation is given by

$\displaystyle g^{\mu\nu}\frac{\partial F}{\partial x^{\mu}}\frac{\partial
F}{\partial x^{\nu}}=0.$ (57)

The location of event horizon of stationary and nonstationary black holes can
be obtained from the null surface equation (35) using generalized tortoise
coordinate transformation. The space time geometry outside the event horizon
of nonstationary black hole can be described by the tortoise coordinate and in
such case, $r_{*}$ tends to positive infinity when tending to infinite point
and $r_{*}$ tends to negative infinity at the event horizon of black hole. To
study the Hawking radiation of nonstationary black hole, the tortoise
coordinate transformation is defined b${\rm y}^{49-54}$

$\displaystyle r_{*}$ $\displaystyle=$ $\displaystyle
r+\frac{1}{2\kappa(u_{0},\theta_{0},\phi_{0})}ln{\frac{r-r_{h}(u,\theta,\phi)}{r_{h}(u,\theta,\phi)}},$
(58) $\displaystyle u_{*}$ $\displaystyle=$ $\displaystyle
u-u_{0},\,\,\theta_{*}=\theta-\theta_{0},\,\,\phi_{*}=\phi-\phi_{0},$ (59)

where $u_{0},\theta_{0}$ and $\phi_{0}$ are the arbitrary constants under the
coordinate transformation. From the above equation, we get

$\displaystyle\frac{\partial}{\partial r}$ $\displaystyle=$
$\displaystyle\Big{[}{1+\frac{1}{2\kappa(r-r_{h})}}\Big{]}\frac{\partial}{\partial
r_{*}},$ (60) $\displaystyle\frac{\partial}{\partial u}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial u_{*}}-\frac{rr_{h,u}}{2\kappa
r_{h}(r-r_{h})}\frac{\partial}{\partial r_{*}},$ (61)
$\displaystyle\frac{\partial}{\partial\theta}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\theta_{*}}-\frac{rr_{h,\theta}}{2\kappa
r_{h}(r-r_{h})}\frac{\partial}{\partial r_{*}},$ (62)
$\displaystyle\frac{\partial}{\partial\phi}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\phi_{*}}-\frac{rr_{h,\phi}}{2\kappa
r_{j}(r-r_{h})}\frac{\partial}{\partial r_{*}},$ (63)

where $r_{h,u}=\frac{\partial r_{h}}{\partial u}$,
$r_{h,\theta}=\frac{\partial r_{h}}{\partial\theta}$ and
$r_{h,\phi}=\frac{\partial r_{h}}{\partial\phi}$. $r_{h,u}$ represents the
evaporation rate at the event horizon of KdS space time. If $\frac{\partial
r_{h}}{\partial u}>0$, the event horizon of KdS space time is expanded
(absorbing black hole ) and when $\frac{\partial r_{h}}{\partial u}<0$, the
event horizon of KdS space time is contracted.
$\kappa\equiv\kappa(u_{0},\theta_{0},\phi_{0})$ is taken as surface gravity of
KdS space time which depends on retarded time and angular coordinates. Using
Eqs. (33) and (36) in Eq. (35) and taking $r\rightarrow r_{h}$, the horizon
equation of nonstationary KdS black hole is derived as

$\displaystyle\frac{a^{2}\Xi^{2}\sin^{2}\theta
r^{2}_{h,u}}{\Delta_{\theta}}+2(r_{h}^{2}+a^{2})\Xi
r_{h,u}+\frac{2a\Xi^{2}r_{h,u}r_{h,\phi}}{\Delta_{\theta}}+2a\Xi r_{h,u}$ (64)
$\displaystyle+\Delta_{\lambda}(r_{h})+\Delta_{\theta}r^{2}_{h,\theta}+\frac{\Xi^{2}r^{2}_{h,\phi}}{\Delta_{\theta}\sin^{2}\theta}=0,$
(65)

where $\Delta_{\lambda}(r_{h})=r^{2}_{h}+a^{2}-2M(u)r_{h}-\frac{1}{3}\Lambda
r^{2}_{h}(r^{2}_{h}+a^{2})$. It is noted that the location of event horizon of
nonstationary black hole varies with retarded time $u=t-r_{*}$ and angular co-
ordinates $\theta,\phi$. From Eqs. (11) and (33), the dynamical equation of
scalar particle with mass $m$ in curved space time is obtained as

$\displaystyle g^{00}\Big{(}\frac{\partial I}{\partial
u}\Big{)}^{2}+2g^{01}\Big{(}\frac{\partial I}{\partial
u}\Big{)}\Big{(}\frac{\partial I}{\partial
r}\Big{)}+2g^{03}\Big{(}\frac{\partial S}{\partial
u}\Big{)}\Big{(}\frac{\partial
I}{\partial\phi}\Big{)}+g^{11}\Big{(}\frac{\partial I}{\partial
r}\Big{)}^{2}+2g^{13}\Big{(}\frac{\partial I}{\partial
r}\Big{)}\Big{(}\frac{\partial I}{\partial\phi}\Big{)}$ (66)
$\displaystyle+g^{22}\Big{(}\frac{\partial
I}{\partial\theta}\Big{)}^{2}+g^{33}\Big{(}\frac{\partial
I}{\partial\phi}\Big{)}^{2}+\lambda
u^{\mu}u^{\nu}\partial_{\mu}I\partial_{\nu}I+m^{2}=0.$ (67)

Since ether like vectors are not constant in curved space time, we can choose
${\rm u^{\alpha}}$ from nonstationary space time Eq. (33) so that we can make
$u^{\alpha}u_{\alpha}={\rm constant}$. The ether like vectors ${\rm
u^{\alpha}}$ are related to the properties of black hole and system of
coordinate adopted by the metric space. The ether like vectors ${\rm
u^{\alpha}}$ are choosen as

$\displaystyle u^{u}$ $\displaystyle=$
$\displaystyle\frac{k_{u}}{\sqrt{g_{uu}}}=\frac{k_{u}\sqrt{R^{2}}\Xi}{\sqrt{\Delta_{\lambda}-\Delta_{\theta}a^{2}\sin^{2}\theta}},$
(68) $\displaystyle u^{r}$ $\displaystyle=$
$\displaystyle\frac{k_{r}\sqrt{g_{uu}}}{g_{ur}}=\frac{k_{r}\sqrt{\Delta_{\lambda}-\Delta_{\theta}a^{2}\sin^{2}\theta}}{\sqrt{R^{2}}},$
(69) $\displaystyle u^{\theta}$ $\displaystyle=$
$\displaystyle\frac{k_{\theta}}{\sqrt{g_{\theta\theta}}}=k_{\theta}\sqrt{\frac{-\Delta_{\theta}}{R^{2}}},$
(70) $\displaystyle u^{\phi}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{\lambda}k_{\phi}}{\sqrt{g_{\phi\phi}}}=\frac{\Delta_{\lambda}k_{\phi}\sqrt{R^{2}}\Xi}{\sqrt{-[\Delta_{\theta}(r^{2}+a^{2})^{2}-\Delta_{\lambda}a^{2}\sin^{2}\theta]\sin^{2}\theta}},$
(71)

where $k_{u}$, $k_{r}$, $k_{\theta}$ and $k_{\phi}$ are arbitrary constants.
Using Eqs. (37) and (40) in Eq. (39), we get

$\displaystyle\frac{D}{E}\Big{(}\frac{\partial I}{\partial
r_{*}}\Big{)}^{2}+2\Big{(}\frac{\partial I}{\partial
u_{*}}\Big{)}\Big{(}\frac{\partial I}{\partial
r_{*}}\Big{)}+2\frac{F}{E}\Big{(}\frac{\partial I}{\partial
r_{*}}\Big{)}+2\kappa(r-r_{h})\frac{G}{E}=0,$ (72)

where

$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa(r-r_{h})r^{2}_{h}}\Big{[}(g^{00}+\lambda
u^{u}u^{u})r^{2}r^{2}_{h,u}-2r_{h}\Big{\\{}(g^{01}+\lambda
u^{u}u^{r})rr_{h,u}+(g^{13}$ (76) $\displaystyle+\lambda
u^{\phi}u^{r})rr_{h,\phi}+\lambda
u^{r}u^{\theta}rr_{h,\theta}\Big{\\}}\\{2k(r-r_{h})+1\\}+2(g^{03}+\lambda
u^{u}u^{\phi})r^{2}r_{h,u}r_{h,\phi}$ $\displaystyle+r^{2}_{h}(g^{11}+\lambda
u^{r}u^{r})\\{2\kappa(r-r_{h})+1\\}^{2}+(g^{22}+\lambda
u^{\theta}u^{\theta})r^{2}r^{2}_{h,\theta}$ $\displaystyle+(g^{33}+\lambda
u^{\phi}u^{\phi})r^{2}r^{2}_{h,\phi}+2\lambda
u^{\theta}r^{2}r_{h,\theta}(u^{u}r_{h,u}+u^{\phi}r_{h,\phi})\Big{]},$
$\displaystyle E$ $\displaystyle=$ $\displaystyle-(g^{00}+\lambda
u^{u}u^{u})\frac{rr_{h,u}}{r_{h}}+(g^{01}+\lambda
u^{u}u^{r})\\{2k(r-r_{h})+1\\}$ (78) $\displaystyle-(g^{03}+\lambda
u^{u}u^{\phi})\frac{rr_{h,\phi}}{r_{h}}-\frac{\lambda
u^{u}u^{\theta}rr_{h,\theta}}{r_{h}},$ $\displaystyle F$ $\displaystyle=$
$\displaystyle-(g^{03}+\lambda
u^{u}u^{\phi})\frac{rr_{h,u}p_{\phi}}{r_{h}}+\\{g^{13}p_{\phi}+\lambda
u^{r}(u^{\phi}p_{\phi}+u^{\theta}p_{\theta})\\}\\{2k(r-r_{h})+1\\}$ (81)
$\displaystyle-(g^{22}+\lambda
u^{\theta}u^{\theta})\frac{rr_{h,\theta}p_{\theta}}{r_{h}}-(g^{33}+\lambda
u^{\phi}u^{\phi})\frac{rr_{h,\phi}p_{\phi}}{r_{h}}-\frac{\lambda
u^{u}u^{\theta}rr_{h,u}p_{\theta}}{r_{h}}$ $\displaystyle-\frac{\lambda
u^{\theta}u^{\phi}rr_{h,\phi}p_{\theta}}{r_{h}}-\frac{\lambda
u^{\theta}u^{\phi}rr_{h,\theta}p_{\phi}}{r_{h}},$ $\displaystyle G$
$\displaystyle=$ $\displaystyle
g^{00}\omega^{2}-2g^{03}p_{\phi}\omega+g^{22}p^{2}_{\theta}+g^{33}p^{2}_{\phi}+\lambda
u^{u}u^{u}\omega^{2}-2\lambda u^{u}u^{\phi}\omega p_{\phi}+\lambda
u^{\theta}u^{\theta}p^{2}_{\theta}$ (83) $\displaystyle+\lambda
u^{\phi}u^{\phi}p^{2}_{\phi}-2\lambda u^{u}u^{\theta}\omega
p_{\theta}+2\lambda u^{\theta}u^{\phi}p_{\theta}p_{\phi}+m^{2}.$

To study the Hawking temperature, the action ${\rm I}$ in Eq. (15) can be
writen as ${\rm I}=-\omega u_{*}+I_{0}(r_{*},\theta_{*},\phi_{*})$, then we
get

$\displaystyle\frac{\partial I}{\partial u_{*}}=-\omega,\,\,\,\frac{\partial
I}{\partial\theta_{*}}=p_{\theta},\,\,\,\frac{\partial
I}{\partial\phi_{*}}=p_{\phi},$ (84)

where $\omega$ is the energy of emitted scalar particle. $p_{\theta}$ and
$p_{\phi}$ are the components of generalized momenta of scalar particle along
the angular coordinates $\theta$ and $\phi$ respectively. To obtain surface
gravity and Hawking temperature at the event horizon of nonstationary KdS
black hole, we assume that the coefficient of $\Big{(}\frac{\partial
I}{\partial r_{*}}\Big{)}^{2}$ approaches to unity as $r\rightarrow r_{h}$,
$u\rightarrow u_{0}$, $\theta\rightarrow\theta_{0}$ and
$\phi\rightarrow\phi_{0}$. From Eq. (41), an infinite limit of the form
$\frac{0}{0}$ is obtained near the event horizon of KdS black hole. Using
L’Hopital’s rule, the surface gravity is obtained as

$\displaystyle\kappa=\frac{r_{h}(1+2\Xi r_{h,u})-M-\frac{2}{3}\Lambda
r^{3}_{h}-\frac{\Lambda}{3}a^{2}r_{h}-r^{-1}_{h}\\{\Delta_{\lambda}+\Xi(r_{h}^{2}+a^{2})r_{h,u}+a\Xi
r_{h,\phi}\\}}{\\{\Xi(r_{h}^{2}+a^{2})+\frac{a^{2}\Xi^{2}\sin^{2}\theta
r_{h,u}}{\Delta_{\theta}}\\}(1+2r_{h,u})+Z+\lambda Y},$ (85) (86)

where $Z$ and $Y$ are given by

$\displaystyle Z$ $\displaystyle=$ $\displaystyle
2\Delta_{\theta}r^{2}_{h,\theta}+r_{h,\phi}\Big{(}\frac{4a\Xi^{2}r_{h,u}}{\Delta_{\theta}}+\frac{a\Xi^{2}}{\Delta_{\theta}}+\frac{2\Xi^{2}r_{h,\phi}}{\Delta_{\theta}\sin^{2}\theta}+2a\Xi\Big{)},$
(87) $\displaystyle Y$ $\displaystyle=$
$\displaystyle\frac{k^{2}_{u}\rho^{4}r_{h,u}\Xi^{2}}{\Delta_{\theta}a^{2}\sin^{2}\theta}+\rho^{2}k_{u}k_{r}\Xi-\frac{\rho^{2}k_{u}k_{\theta}\Xi
r_{h,\theta}}{a\sin\theta}.$ (88)

Using Eq. (43) in Eq. (41) and taking $r\rightarrow r_{h}$, then we obtain

$\displaystyle\Bigg{(}\frac{\partial I}{\partial
r_{*}}\Bigg{)}^{2}+2(\omega-\omega_{0})\frac{\partial I}{\partial r_{*}}=0.$
(89)

The value of chemical potential, $\omega_{0}$ near the event horizon of
nonstationary KdS black hole is

$\displaystyle w_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{g^{01}-g^{00}r_{h,u}-g^{03}r_{h,\phi}-\lambda
u^{u}u^{u}r_{h,u}+\lambda u^{r}u^{r}-\lambda u^{u}u^{\phi}r_{h,\phi}-\lambda
u^{u}u^{\theta}r_{h,\theta}}$ (92)
$\displaystyle\times[g^{13}p_{\phi}-g^{03}r_{h,u}p_{\phi}-g^{22}r_{h,\theta}p_{\theta}-g^{33}r_{h,\phi}p_{\phi}-\lambda
u^{u}r_{h,u}(u^{\phi}p_{\phi}+u^{\theta}p_{\theta})$
$\displaystyle-\lambda(u^{\theta}r_{h,\theta}+u^{\phi}r_{h,\phi})(u^{\theta}p_{\theta}+u^{\phi}p_{\phi})+\lambda
u^{r}(u^{\theta}p_{\theta}+u^{\phi}p_{\phi})].$

Eq. (47) represents the chemical potential of nonstationary KdS space time due
to tortoise coordinate transformation (36). The chemical potential, $w_{0}$
depends on the black hole mass, cosmological constant, generalized momenta of
scalar particle, retarded time, angular coordinates, correction parameter
$\lambda$ and ether like vectors $u^{\alpha}$. If $\lambda$ and $u^{\alpha}$
tend to zero, Eq. (47) is consistent with earlier literatures [10, 46]. From
the solution of Eq. (37), we obtain

$\displaystyle\frac{\partial I}{\partial
r_{*}}=\frac{[2\kappa(r-r_{h})+1](\omega-\omega_{0})\pm(\omega-\omega_{0})}{[2\kappa(r-r_{h})]}.$
(93)

It is observed that Eq. (48) has a singularity near the event horizon of KdS
black hole. Integrating Eq. (48) by applying residue theorem of complex
analysis and Feynman prescription, the imaginary part of the radial action $I$
is derived as

$\displaystyle{\rm
ImI\pm}=\frac{\pi}{2\kappa}[(\omega-\omega_{0})\pm(\omega-\omega_{0})],$ (94)

where ${\rm I_{+}}$ and ${\rm I_{-}}$ represent the outgoing scalar particle
and ingoing scalar particle at the event horizon of KdS black hole. Taking
outgoing and ingoing of scalar particle, the tunneling probability which
crosses at the event horizon of KdS black hole is calculated as

$\displaystyle\Gamma=\frac{\Gamma_{emission}}{\Gamma_{absorption}}={\rm
exp}[-\frac{(\omega-\omega_{0})}{T}].$ (95)

The modified Hawking temperature near the event horizon of nonstationary KdS
black hole due to Lorentz violation theory is given by

$\displaystyle T$ $\displaystyle=$ $\displaystyle T_{h}(1+\lambda H)^{-1}$
(96) $\displaystyle=$ $\displaystyle T_{h}(1-\lambda
H+\lambda^{2}H^{2}-\lambda^{3}H^{3}+...),$ (97)

where the value of $T_{h}$ and $H$ are

$\displaystyle T_{h}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\Big{[}\frac{r_{h}(1+2\Xi
r_{h,u})-M-\frac{2}{3}\Lambda
r^{3}_{h}-\frac{\Lambda}{3}a^{2}r_{h}}{\\{\Xi(r_{h}^{2}+a^{2})+\frac{a^{2}\Xi^{2}\sin^{2}\theta
r_{h,u}}{\Delta_{\theta}}\\}(1+2r_{h,u})+Z}$ (99)
$\displaystyle-\frac{r^{-1}_{h}\\{\Delta_{\lambda}+\Xi(r_{h}^{2}+a^{2})r_{h,u}+a\Xi
r_{h,\phi}\\}}{\\{\Xi(r_{h}^{2}+a^{2})+\frac{a^{2}\Xi^{2}\sin^{2}\theta
r_{h,u}}{\Delta_{\theta}}\\}(1+2r_{h,u})+Z}\Big{]}$

and

$\displaystyle
H=\frac{\frac{k^{2}_{u}\rho^{4}r_{h,u}\Xi^{2}}{\Delta_{\theta}a^{2}\sin^{2}\theta}+\rho^{2}k_{u}k_{r}\Xi-\frac{\rho^{2}k_{u}k_{\theta}\Xi
r_{h,\theta}}{a\sin\theta}}{\\{\Xi(r_{h}^{2}+a^{2})+\frac{a^{2}\Xi^{2}\sin^{2}\theta
r_{h,u}}{\Delta_{\theta}}\\}(1+2r_{h,u})+Z}.$ (100)

From Eqs. (52) and (53), it is known that the Hawking temperature near the
event horizon of KdS black hole is modified due to Lorentz violation theory.
The modified Hawking temperature $(T)$ of nonstationary KdS black hole depends
not only on the mass of the black hole but also on the properties of event
horizon, cosmological constant $\Lambda$, retarded time $u$, correction term
$\lambda$ and on the ether like vectors $u^{\alpha}$. If
$k_{u}\rho^{2}r_{h_{,u}}\Xi+\Delta_{\theta}a\sin\theta(a\sin\theta
k_{r}-k_{\theta}r_{h,\theta})=0$, then $H\longrightarrow 0$. In such case, our
result is consistent with the earlier literature${\rm s.}^{10,11,46}$

##  4 Discussion

First the line element of stationary KdS black hole is transformed into static
form using frame dragging effect given in Eq. (4).

Figure 1: Plot of original and modified Hawking temperature with radius of
event horizon, $r_{h}$ of KdS black hole. Here
$a=0.1,\Lambda=0.6,\lambda=0.005,c_{r}=1.5,c_{t}=0.4$.

Using modified form of Hamilton-Jacobi equation, Feynman prescription and WKB
approximation, the modified Hawking temperature, heat capacity and change in
Bekenstein-Hawking entropy are derived in Eq. (23), Eq. (24) and Eq. (31)
respectively. It is noted that both are dependent on correction term $\lambda$
and ether like vectors $u^{\alpha}$.

If $\gamma>1$, the modified Hawking temperature near the event horizon of KdS
black hole decreases and if $\gamma<1$, the modified Hawking temperature
increases in Eq. (23). If $\gamma=1$, the modified heat capacity (24)
approaches to original heat capacity (10).

If $\gamma>1$, the modified heat capacity increases and if $\gamma<1$, the
modified heat capacity is smaller than that of original heat capacity near the
event horizon of stationary KdS black hole.

The change in Bekenstein-Hawking entropy near the event horizon of KdS black
hole increases or decreases if $\gamma^{\prime}>1$ or $\gamma^{\prime}<1$.
When $\lambda=0$ and $c_{t}=c_{r}$, $\lambda\neq 0,$ the Lorentz violation has
been cancelled and the original change in Bekenstein-Hawking entropy near the
event horizon of KdS black hole is recovered.

The Hawking temperature of rotating nonstationary KdS black hole is also
investigated using Klein-Gordon equation, generalized tortoise coordinate
transformation and L’Hopital rule in Lorentz violation theory. According to
Damour and Ruffin${\rm i}^{6}$ and Sanna${\rm n}^{7}$, the traditional
coordinate transformation is given by

$\displaystyle r_{*}$ $\displaystyle=$ $\displaystyle
r+\frac{1}{2\kappa_{1}}{\rm ln}(r-r_{h}(u,\theta,\phi)),$ (101) $\displaystyle
u_{*}$ $\displaystyle=$ $\displaystyle
u-u_{0},\,\,\theta_{*}=\theta-\theta_{0},\,\,\phi_{*}=\phi-\phi_{0}.$ (102)

Using Eq. (54) in Eq. (41), the modified surface gravity and Hawking
temperature of nonstationary rotating KdS black hole are

$\displaystyle\kappa_{1}=\frac{r_{h}(1+2\Xi r_{h,u})-M-\frac{2}{3}\Lambda
r^{3}_{h}-\frac{\Lambda}{3}a^{2}r_{h}}{{\\{\Xi(r_{h}^{2}+a^{2})+\frac{a^{2}\Xi^{2}\sin^{2}\theta
r_{h,u}}{\Delta_{\theta}}\\}(1+2r_{h,u})+Z+\lambda Y}}$ (103)

and

$\displaystyle T_{1}=\frac{1}{2\pi}\frac{r_{h}(1+2\Xi
r_{h,u})-M-\frac{2}{3}\Lambda
r^{3}_{h}-\frac{\Lambda}{3}a^{2}r_{h}}{{\\{\Xi(r_{h}^{2}+a^{2})+\frac{a^{2}\Xi^{2}\sin^{2}\theta
r_{h,u}}{\Delta_{\theta}}\\}(1+2r_{h,u})+Z+\lambda Y}}.$ (104)

In Eqs. (51) and (56) , if we put $\lambda$ = $p_{\theta}$ = $p_{\phi}$ = $0$
and $k_{u}\rho^{2}r_{h_{,u}}\Xi+\Delta_{\theta}a\sin\theta(a\sin\theta
k_{r}-k_{\theta}r_{h,\theta})=0$, Eqs. (51) and (56) are concordant with the
literatur${\rm e}.^{46}$ Eq. (54) gives another chemical potential of
nonstationary rotating KdS black hole in Lorentz violation theory as

$\displaystyle w_{p}$ $\displaystyle=$
$\displaystyle\frac{1}{g^{01}-g^{00}r_{h,u}-g^{03}r_{h,\phi}-\lambda
u^{u}u^{u}r_{h,u}+\lambda u^{r}u^{r}-\lambda u^{u}u^{\phi}r_{h,\phi}-\lambda
u^{u}u^{\theta}r_{h,\theta}}$ (107)
$\displaystyle\times[g^{13}p_{\phi}-g^{03}r_{h,u}p_{\phi}-g^{22}r_{h,\theta}p_{\theta}-g^{33}r_{h,\phi}p_{\phi}-\lambda
u^{u}r_{h,u}(u^{\phi}p_{\phi}+u^{\theta}p_{\theta})$
$\displaystyle-\lambda(u^{\theta}r_{h,\theta}+u^{\phi}r_{h,\phi})(u^{\theta}p_{\theta}+u^{\phi}p_{\phi})+\lambda
u^{r}(u^{\theta}p_{\theta}+u^{\phi}p_{\phi})].$

It is observed that Eq. (57) is consistent with Eq. (47). The surface
gravities derived from Eq. (44) and (55) can be combined as

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle\kappa_{1}+\Xi_{1},$
(108)

where $\kappa$ and $\kappa_{1}$ represent the surface gravities of
nonstationary KdS black hole due to tortoise coordinate transformations (36)
and (54). $\Xi_{1}$ indicates the constant term due to tortoise coordinate
transformation (36) and its expression is

$\displaystyle\Xi_{1}$ $\displaystyle=$
$\displaystyle-\frac{r^{-1}_{h}\\{\Delta_{\lambda}+\Xi(r_{h}^{2}+a^{2})r_{h,u}+a\Xi
r_{h,\phi}\\}}{\\{\Xi(r_{h}^{2}+a^{2})+\frac{a^{2}\Xi^{2}\sin^{2}\theta
r_{h,u}}{\Delta_{\theta}}\\}(1+2r_{h,u})+Z+\lambda Y}.$ (109)

The rate of correction for Hawking temperature near the event horizon of
nonstationary KdS black hole is given by

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle-\frac{r^{-1}_{h}\\{\Delta_{\lambda}+\Xi(r_{h}^{2}+a^{2})r_{h,u}+a\Xi
r_{h,\phi}\\}}{r_{h}(1+2\Xi r_{h,u})-M-\frac{2}{3}\Lambda
r^{3}_{h}-\frac{\Lambda}{3}a^{2}r_{h}}.$ (110)

It is worth mentioning that the correction rate is independent of correction
term $\lambda$ and the ether like vectors $u^{\alpha}$ but depends on black
hole mass $M$, rotational parameter $a$, angular coordinate $\theta$,
cosmological constant $\Lambda$ and generalized momenta. For stationary KdS
black hole in the absence of Lorentz violation theory, Eq. (51) and (56)
reduce to the same Hawking temperature as

$\displaystyle
T_{H}=T_{1}=\frac{1}{2\pi}\Bigg{[}\frac{r_{h}-M-\frac{2}{3}\Lambda
r^{3}_{h}-\frac{\Lambda}{3}a^{2}r_{h}}{\Xi(r_{h}^{2}+a^{2})}\Bigg{]}.$ (111)

It is noted that for the stationary KdS space time, the different tortoise
coordinate transformations give the same Hawking temperature in the absence of
Lorentz violation theory which is exactly equal to the actual calculation
given in Eq. (7). From Eq. (58), $\kappa_{1}$ approaches to zero, the Hawking
temperature $T$ does not tend to zero due to extra term $\Xi_{1}$. If
$\Xi_{1}$ approaches to zero in Eq. (58), Eq. (44) is consistent with Eq.
(55). From Eqs. (47) and (57), we observe that the chemical potential derived
from different tortoise coordinate transformations are equal near the event
horizon of black hole. It can be concluded that the tortoise coordinate
transformation given in Eq. (36) is more suitable and accurate in the study of
modified surface gravity near the event horizon of nonstationary KdS space
time in Lorentz violation theory. From Eqs. (44) and (55), the different
surface gravities are obtained using the different tortoise coordinate
transformations near the event horizon of nonstationary KdS black hole in
Lorentz violation theory.

##  5 Conclusion

In this paper, the tunneling of scalar particle near the event horizon of
stationary KdS black hole is investigated using Klein-Gordon equation in
Lorentz violation theory, Feynman prescription and WKB approximation. Then the
corresponding Hawking temperature, heat capacity and change in Bekenstein-
Hawking entropy near the event horizon are derived. The Hawking temperature,
heat capacity and change in entropy are modified due to presence of correction
term $\lambda$ and ether like vectors $u^{\alpha}$.

The modified surface gravities of nonstationary rotating black hole are also
studied using different tortoise coordinate transformations in Lorentz
violation theory. Using null surface equation and tortoise coordinate
transformation, the horizon equation of nonstationary KdS black hole is
obtained. The modified surface gravities and the modified Hawking temperatures
are derived with the help of event horizon equation. It is known that the
modified Hawking temperature depends not only on the correction term $\lambda$
but also on the ether like vectors $u^{\alpha}$. If we use Eq. (36) in the
study of surface gravity and Hawking radiation of black hole, a constant term
$\Xi_{1}$ is seen to be appeared in the expressions of surface gravity and
Hawking temperature near the event horizon of KdS black hole. If $\Xi_{1}$
tends to zero, the two modified surface gravities and Hawking temperatures are
equal. If $\lambda$ and $\Xi_{1}$ approach to zero, the original surface
gravities near the event horizon of nonstationary black hole are recovered and
are concordant with the earlier literature${\rm s.}^{8,9,54}$ It is also seen
that the correction rate $\delta$ of Hawking temperature in Lorentz violation
theory does not depend on correction term $\lambda$ and the ether like vectors
$u^{\alpha}$. The different tortoise coordinate transformations yield the same
chemical potential at the event horizon of black hole but the values of
surface gravities and Hawking temperatures are different. This shows that Eq.
(36) is more reliable and accurate in the study of modified surface gravity
near the event horizon of nonstationary of KdS black hole. For the stationary
space time, the different methods yield the same Hawking temperature and the
change in Bekenstein-Hawking entropy in the absence of Lorentz violation
theory.

## Acknowledgments

The first author acknowledges the DST INSPIRE, New Delhi, India for providing
financial support (Grant No. IF190759).

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