11institutetext: Department of Physics, Eastern Mediterranean University,
Gazimagosa, North Cyprus, Mersin 10, Turkey 11email:
<EMAIL_ADDRESS>22email<EMAIL_ADDRESS>
# Hawking Radiation of Linear Dilaton Black Holes in Various Theories

H.Pasaoglu 11 I.Sakalli 22 Address for offprint requests

###### Abstract

Using the Damour-Ruffini-Sannan, the Parikh-Wilczek and the thin film brick-
wall models, we investigate the Hawking radiation of uncharged massive
particles from 4-dimensional linear dilaton black holes, which are the
solutions to Einstein-Maxwell-Dilaton, Einstein-Yang-Mills-Dilaton and
Einstein-Yang-Mills-Born-Infeld-Dilaton theories. Our results show that the
tunneling rate is related to the change of Bekenstein-Hawking entropy.
Contrary to the many studies in the literature, here the emission spectrum is
precisely thermal. This implies that the derived emission spectrum is not
consistent with the unitarity of the quantum theory, which would possibly lead
to the information loss.

###### Keywords:

Entropy, Linear dilaton black holes, Tunneling effect, Thin film brick-wall
model

††journal: Int J Theor Phys††: Head note that is usually deleted††offprints:
Offprints Assistant

##  1 Introduction

Obeying the laws of black hole mechanics Bardeen , Hawking Nature ; Commun
proved that a stationary black hole can emit particles from its event horizon
with a temperature proportional to the surface gravity. According to this
idea, the vacuum fluctuations near the horizon would produce a virtual
particle pair, similar to electron-positron pair creation in a constant
electric field. When a virtual particle pair is created just inside or outside
the horizon, the sign of its energy changes as it crosses the horizon. So
after one member of the pair has tunneled to the opposite side, the pair can
materialize with zero total energy. This discovery also announced the relation
between the triple subjects – the quantum mechanics, thermodynamics and the
gravitation. After this pioneering study of Hawking, many methods have been
proposed to calculate the Hawking radiation for the last three decades.

One of the commonly used methods is known as Damour-Ruffini-Sannan (DRS)
Damour ; Sannan  method. This method is applicable to any Hawking temperature
problem in which the asymptotic behaviors of the wave equation near the event
horizon are known.

In 2000, Parikh and Wilczek ParikhWilczek  proposed a method based on null
geodesics in order to clarify more the Hawking radiation via tunneling across
the event horizon. Namely, they treated the Hawking radiation as a tunneling
process, and used the WKB approximation to determine the correction spectrum
for the black hole’s Hawking radiation. In their study, it is supposed that
the barrier depends on the tunneling particle itself. The crucial point of
this method is not to violate the energy conservation during the process of
particle emission and to pass to an appropriate coordinate system at horizon.
In general, the tunneling process is not precisely a thermal effect and it
explains the modification of the black hole radiation spectrum in which it
leads to the unitarity in the quantum theory INTParikh ; GRGParikh ;
ArxivParikh .

Another possible method to study the statistical origin of the black hole
entropy is the brick-wall model initially proposed by t’Hooft Hooft . The
brick-wall model identifies the black hole entropy by the entropy of a thermal
gas of quantum field excitations outside the event horizon. Since then, this
method has been satisfactorily applied to many black hole geometries (see for
instance Liu , and the references therein). Although t’ Hooft made significant
contribution to clarify the understanding and calculating the entopy of the
black holes, there were some drawbacks in his model. Those drawbacks are
overcome by the improved form of the original brick-wall model, which is
called as thin film brick-wall model PRDLiZhao . The thin film brick-wall
model gives us acceptable and net physical meaning of the entropy calculation.
In summary, since the entropy calculated by the thin film brick-wall model is
just from a small region (thin film) near the horizon, this improved version
of the brick-wall model represents explicitly the correlation between the
horizon and the entropy. In this study, we obtain the ultraviolet cut-off
distance as 90$\beta,$ where $\beta$ is the Boltzmann factor.

Hawking described the black hole radiation as tunneling triggered by the
vacuum fluctuations near the horizon. His discovery, which treats the black
hole radiation as being pure thermal gave also rise to a new paradox in the
black hole physics – the information loss paradox. Although, Parikh and
Wilczek’s tunneling process ParikhWilczek  is a way to overcome the
information loss paradox in the Hawking radiation, the information might not
be conserved in some black hole geometries. For instance, if only the
tunneling process of the outer horizon of the Reissner-Nordström black hole is
considered QJiang ; Ren , it can be shown that the information loss is
possible. The similar violation in the conservation of information happens in
the $4$-dimensional linear dilaton black holes (LDBHs) in various theories,
and we will explain its reason by using the differences in entropies of the
black holes before and after the emission.

The paper is organized as follows: In section 2, a brief overview of the
$4$-dimensional LDBHs in Einstein-Maxwell-Dilaton (EMD), Einstein-Yang-Mills-
Dilaton (EYMD) and Einstein-Yang-Mills-Born-Infeld-Dilaton (EYMBID) theories,
which they have been recently employed in Sakalli  for calculating the Hawking
radiation via the method of semi-classical radiation spectrum is given. Next,
we apply the DRS method to find the temperature of the LDBHs and the tunneling
rate of the chargeless particles crossing the event horizon. Section 3 is
devoted to the calculation of the entropy of the horizon by using all those
methods mentioned above. As it is expected, they all conclude with the same
result. Finally, we draw our conclusions and discussions.

Throughout the paper, the units $G=c=$h $=k_{B}$=1 are used.

##  2 LDBHs, Calculation of Their Temperature and Tunneling Rate

The line-element of $N$-dimensional ($N\geq 4$) LDBHs, which are static
spherically symmetric solutions in various theories (EMD, EYMD and EYMBID)
have been recently summarized by Sakalli . However, throughout this paper we
restrict ourselves to the $4$-dimensional LDBHs and follow the notations of
Sakalli .

Consider a general class of static, spherically symmetric spacetime for the
LDBHs as

$ds^{2}=-fdt^{2}+\frac{dr^{2}}{f}+A^{2}rd\Omega^{2},$ (1)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}.$ Here, the metric
function $f$ is given by Sakalli

$f=\Sigma r(1-\frac{r_{+}}{r}),$ (2)

where $r_{+}$ is the radius of the event horizon. The coefficients $\Sigma$
and $A$ in the metric (1) take different values according to the concerned
theory.

Since the present form of the metric represent asymptotically non-flat
solutions, one should consider the quasi-local mass definition $M$ of the
metric (1). In Sakalli , the relationship between the horizon $r_{+}$ and the
mass $M$ is explicitly given as

$r_{+}=\frac{4M}{\Sigma A^{2}},$ (3)

In the EMD theory Sakalli ; Chan ; Clement , the coefficients $\Sigma$ and $A$
are found as

$\Sigma\rightarrow\Sigma_{EMD}=\frac{1}{\gamma^{2}}\text{ \ and \ \ \
}A\rightarrow A_{EMD}=\gamma\text{,}$ (4)

where $\gamma$ is a constant related to the electric charge of the black hole.
Meanwhile, one can match the metric (1) to the LDBH’s metric of Clément et.al.
Clement  by setting $\gamma\equiv r_{0}$. Next, if one considers the EYMD and
EYMBID theories Mazhari1 ; Mazhari2 , the coefficients in the metric (1)
become

$\Sigma\rightarrow\Sigma_{EYMD}=\frac{1}{2Q^{2}}\text{ \ and \ \ \
}A\rightarrow A_{EYMD}=\sqrt{2}Q,$ (5)

and

$\Sigma\rightarrow\Sigma_{EYMBID}=\frac{1}{Q_{C}^{2}}\left[1-\sqrt{1-\frac{Q_{C}^{2}}{Q^{2}}}\right]\text{
\ and \ }A\rightarrow
A_{EYMBID}=\sqrt{2}Q\left(1-\frac{Q_{C}^{2}}{Q^{2}}\right)^{\frac{1}{4}},$ (6)

where $Q$ and $Q_{C}$ are YM charge and the critical value of YM charge,
respectively. The existence of the metric (1) in EYMBID theory depends
strictly on the condition Mazhari2

$Q^{2}>Q_{C}^{2}=\frac{1}{4\tilde{\beta}^{2}},$ (7)

where $\tilde{\beta}$ is the Born-Infeld parameter. Meanwhile, it is not
necessary to say that values of $\Sigma$ in equations (4), (5) and (6) are
always positive.

By using the definition of the surface gravity Wald , we get

$\kappa=\lim_{r\rightarrow r_{+}}\frac{f^{\prime}(r)}{2}=\frac{\Sigma}{2}.$
(8)

Since the surface gravity (8) is positive, one can deduce that it is directed
towards the singularity. As a consequence, it is attractive and the matter can
only fall into the black hole. This horizon is a future horizon to an
observer, who is located outside of it.

In curved spacetime, a massive test scalar field $\Phi$ with mass $\mu$ obeys
the covariant Klein-Gordon (KG) equation, which is given by

$\frac{1}{\sqrt{-\det g}}\partial_{\mu}\left(\sqrt{-\det
g}g^{\mu\nu}\partial_{\nu}\Phi\right)-\mu^{2}\Phi=0,$ (9)

The massive scalar wave equation $\Phi$ in metric (1) can be separated as
$\Phi=Y(\theta,\varphi)\psi(t,r)$ in which the radical KG equation (9)
satisfies the following equation:

$\frac{\partial^{2}\psi}{\partial
t^{2}}+f(\frac{f}{r}+\Sigma)\frac{\partial\psi}{\partial
r}+f^{2}\frac{\partial^{2}\psi}{\partial
r^{2}}-f(\mu^{2}-\frac{l(l+1)}{r})\psi=0,$ (10)

where $l$ is the angular quantum number. In order to change equation (10) into
a standard wave equation at the horizon, we introduce the tortoise coordinate
transformation, which is obtained from

$dr_{\ast}=\frac{dr}{f},$ (11)

After making the straightforward calculation, we find an appropriate
$r_{\ast}$ as

$r_{\ast}=\frac{1}{2\kappa}\ln(r-r_{+}),$ (12)

Thus, one can transform the radical equation (10) into the following form

$\frac{\partial^{2}\psi}{\partial
t^{2}}-\frac{f}{r}\frac{\partial\psi}{\partial
r_{\ast}}-\Sigma\frac{\partial\psi}{\partial
r_{\ast}}+\Sigma\frac{\partial\psi}{\partial
r_{\ast}}-\frac{\partial^{2}\psi}{\partial
r_{\ast}^{2}}+f[\mu^{2}-\frac{l(l+1)}{r}]\psi=0,$ (13)

While $r\rightarrow r_{+}$ in which $f\rightarrow 0$, the transformed radical
equation (13) can be reduced to the following standard form of the wave
equation as

$\frac{\partial^{2}\psi}{\partial t^{2}}-\frac{\partial^{2}\psi}{\partial
r_{\ast}^{2}}=0,$ (14)

This form of the wave equation reveals that there are propagating waves near
the horizon. The solutions of equation (14), which give us the ingoing and
outgoing waves at the black hole horizon surface $r_{+}$ are

$\psi_{out}=\exp(-i\omega t+i\omega r_{\ast}),$ (15)

$\psi_{in}=\exp(-i\omega t-i\omega r_{\ast}),$ (16)

When we introduce the ingoing Eddington-Finkelstein coordinate,
$v=t+r_{\ast}$, the line-element (1) of the LDBHs becomes

$ds^{2}=-fdv^{2}+2dvdr+A^{2}rd\Omega^{2},$ (17)

The present form of the metric does not attribute a singularity to the
horizon, so that the ingoing wave equation behaves regularly at the horizon.
This yields the solutions of ingoing and outgoing waves at the horizon $r_{+}$
as follows

$\psi_{out}=e^{-i\omega v}e^{2i\omega r_{\ast}},$ (18)

$\psi_{in}=e^{-i\omega v},$ (19)

Now, we consider only the outgoing waves. Namely,

$\psi_{out}(r>r_{+})=e^{-i\omega v}(r-r_{+})^{\frac{{}^{i\omega}}{\kappa}},$
(20)

which has a singularity at the horizon $r_{+}$. Therefore, equation (20) can
only describe the outgoing particles outside the horizon and strictly cannot
describe the particles, which are inside the horizon. In other words, the
description of the particles’ behavior inside horizon has to be made as well.
To this end, the outgoing wave $\psi_{out}$ should be analytically extended
from outside to the interior of the black hole by the lower half complex
$r$-plane

$(r-r_{+})\rightarrow\left|r-r_{+}\right|e^{-i\pi}=(r_{+}-r)e^{-i\pi},$ (21)

We can derive the solution of outgoing wave inside the horizon as follows

$\psi_{out}(r<r_{+})=\psi_{out}^{{}^{\prime}}\left(r<r_{+}\right)e^{\frac{{}^{\omega\pi}}{\kappa}},$
(22)

where

$\psi_{out}^{{}^{\prime}}\left(r<r_{+}\right)=e^{-i\omega
v}(r_{+}-r)^{\frac{{}^{i\omega}}{\kappa}},$ (23)

According to the Damour-Ruffini-Sannan (DRS) Damour ; Sannan  method, it is
possible to calculate the emission rate. The total outgoing wave function can
be written in a uniform form

$\psi=N_{\omega}[\Theta(r-r_{+})\psi_{out}\left(r>r_{+}\right)+e^{\frac{\omega\pi}{\kappa}}\Theta(r_{+}-r)\psi_{out}^{{}^{\prime}}\left(r<r_{+}\right)],$
(24)

where $\Theta$ is the Heaviside step function and $N_{\omega}$ represents the
normalization factor. From the normalization condition

$\left(\psi,\psi\right)=\pm 1,$ (25)

we can obtain the resulting radiation spectrum of scalar particles

$N_{\omega}^{2}=\frac{\Gamma}{1-\Gamma}=\frac{1}{e^{\frac{\omega}{T}}-1},$
(26)

and read the temperature of the horizon as

$T=\frac{\kappa}{2\pi},$ (27)

In equation (26) $\Gamma$ symbolizes the emission or tunneling rate, which is
found by the following ratio

$\Gamma=\left|\frac{\psi_{out}\left(r>r_{+}\right)}{\psi_{out}\left(r<r_{+}\right)}\right|^{2}=e^{\frac{-^{2\pi\omega}}{\kappa}}.$
(28)

One can remark for this section that the resulting temperature (27) obtained
from the DRS method is in agreement with the statistical Hawking temperature
Wald  computed as usual by dividing the surface gravity by $2\pi$.

##  3 Entropy of the Horizon

In this section, we shall use three different methods in order to show that
they all lead to the same entropy result. We first employ the DRS method,
which is worked in detail and obtained remarkable results in the previous
section. The second method will be the Parikh-Wilczek method ParikhWilczek
describing the Hawking radiation as a tunneling process. Last method that is
also going to be used in the calculation of the entropy is the thin film
brick-wall model PRDLiZhao .

In the DRS method, the emission rate of outgoing particles is found as in
equation (28). Accordingly, the probability of emission can be modified into
QJiang ; Jiang  (and references therein)

$\Gamma=e^{{}^{-2\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{\kappa}}}=e^{-\int_{0}^{\omega}\frac{d\omega^{\prime}}{T}}=e^{\Delta
S_{BH}},$ (29)

where $\Delta S_{BH}$ is the difference of Bekenstein-Hawking entropies of the
LDBHs before and after the emission of the particle.

On the other hand, the novel study on the tunneling effect is designated by
Parikh-Wilczek method ParikhWilczek , which proposes an approach for
calculating the tunneling rate at which particles tunnel across the event
horizon. They treated Hawking radiation as a tunneling process, and used the
WKB method ArxivParikh . In classical limit, we can also find the tunneling
rate by applying WKB approximation. This relates the tunneling amplitude to
the imaginary part of the particle action at stationary phase and the
Boltzmann factor for emission at the Hawking temperature.

In the WKB approximation, the imaginary part of the amplitude for outgoing
positive energy particle which crosses the horizon outwards from initial
radius of the horizon $r_{in}$ to the final radius of the horizon $r_{out}$
could be expressed by

$\mathop{\mathrm{I}m}I=\mathop{\mathrm{I}m}\int_{r_{in}}^{r_{out}}p_{r}dr=\mathop{\mathrm{I}m}\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}dp_{r}^{\prime}dr,$
(30)

By using the standard quantum mechanics, the tunneling rate $\Gamma$ is given
in the WKB approximation as KrausWilczek1 ; KrausWilczek2 ,

$\Gamma\sim\exp(-2\mathop{\mathrm{I}m}I),$ (31)

Here we can consider the particle with energy $\omega$ as a shell of energy
and fix the total mass $M$ (quasi-local mass) and allow the hole mass to
fluctuate. Then the Hamilton’s equation of motion can be used to write
$dp_{r}=\frac{dH}{\dot{r}},$ and it can be noted that the horizon moves
inwards from $M$ to $M-\omega$ while a particle emits. Introducing
$H=M-\omega$ and inserting the value of the
$\dot{r}\equiv\frac{dr}{dv}=\frac{f}{2}$ obtained from the null geodesic
equation into (30), we obtain

$\mathop{\mathrm{I}m}\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}dp_{r}^{\prime}dr=\mathop{\mathrm{I}m}\int_{r_{in}}^{r_{out}}\int_{M}^{M-\omega}\frac{dr}{\dot{r}}dH=\mathop{\mathrm{I}m}\int_{0}^{\omega}\int_{r_{in}}^{r_{out}}\frac{2dr}{\Sigma(r-r_{+})}\left(-d\omega^{\prime}\right),$
(32)

The $r$-integral can be done by deforming the contour. The deformation of the
integral is based on an assumption that the contour semicircles the residue in
a clockwise fashion. In this way, one can obtain

$\mathop{\mathrm{I}m}I=2\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{\Sigma},$
(33)

So, the tunneling rate (31) is

$\Gamma\sim\exp(-2\mathop{\mathrm{I}m}I)=\exp\left(-4\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{\Sigma}\right)=\exp\left(\Delta
S_{BH}\right),$ (34)

Our result (34) is consistent with the results of the other works
ParikhWilczek ; Kerner ; Zhang ; Chen ; Li .

Now, we come to the stage to apply the thin film brick-wall model PRDLiZhao ,
which was based on the brick wall model proposed firstly by t’Hooft Hooft .
According to this model, the considered field outside the horizon is assumed
to be non-zero only in a thin film, which exists in a small region bordered by
$r_{+}+\varepsilon$ and $r_{+}+\varepsilon+\delta$. Here, $\varepsilon$ is the
ultraviolet cut-off distance and $\delta$ is the thickness of the thin film.
In summary, both $\varepsilon$ and $\delta$ are positive infinitesimal
parameters. This model treats the entropy as being associated with the field
in the considered small region in which the local thermal equilibrium and the
statistical laws are valid Tian . That is why one can work out the entropy of
the horizon by using this model.

If one redefines the massive test scalar field $\Phi$ as being
$\Phi=e^{-i\omega t}\psi(r)Y_{lm}(\theta,\phi)$ in the KG equation (9) and
considers its radial part only, the wave vector is found with the help of WKB
approximation as

$k^{2}=\frac{1}{\Sigma r(1-\frac{r_{+}}{r})}[\frac{\omega^{2}}{\Sigma
r(1-\frac{r_{+}}{r})}-(\mu^{2}+\frac{l(l+1)}{A^{2}r})],$ (35)

Using the quantum statistical mechanics, we calculate the free energy from

$F=\frac{-1}{\pi}\int_{0}^{\infty}d\omega\int_{r}dr\int_{l}(2l+1)\frac{k}{e^{\beta\omega}-1}dl,$
(36)

While integrating equation (36) with respect to $l,$ one should consider the
upper limit of integration such that $k^{2}$ remains positive, and the lower
limit becomes zero. Briefly, we get

$F\cong\frac{-2A^{2}}{3\pi\Sigma^{2}}\int_{0}^{\infty}\frac{d\omega}{e^{\beta\omega}-1}\int_{r}\frac{r}{(r-r_{+})^{2}}[\omega^{2}-\mu^{2}\Sigma(r-r_{+})]^{\frac{3}{2}}dr,$
(37)

where $\beta$ denotes the inverse of the temperature. In equation (37), the
integration with respect to $r$ is quite difficult. On the other hand, the
thin film brick-wall model imposes us to take only the free energy of a thin
layer near horizon of a black hole, and the integration with respect to $r$
must be limited in the region $r_{+}+\varepsilon\leq r\leq
r_{+}+\varepsilon+\delta.$ The natural result of this choice sets the
coefficient of $\mu^{2}$ to zero, and whence the integration of equation (37)
with respect to $\omega$ becomes very simple such that it can be easily found
as $\pi^{4}/15\beta^{4}$. Finally, the equation (37) reduces to

$F\cong\frac{-2\pi^{3}A^{2}}{45\beta^{4}\Sigma^{2}}\int_{r_{+}+\varepsilon}^{r_{+}+\varepsilon+\delta}\frac{r}{(r-r_{+})^{2}}dr,$
(38)

$\cong\frac{-2\pi^{3}A^{2}r_{+}}{45\beta^{4}\Sigma^{2}}\int_{r_{+}+\varepsilon}^{r_{+}+\varepsilon+\delta}\frac{dr}{(r-r_{+})^{2}},$
(39)

$F\cong\frac{-2\pi^{3}A^{2}r_{+}}{45\beta^{4}\Sigma^{2}}\frac{\delta}{\varepsilon(\delta+\varepsilon)},$
(40)

and we can get the entropy

$S_{BH}=\beta^{2}\frac{\partial
F}{\partial\beta}=[\frac{8\pi^{3}A^{2}r_{+}}{45\beta^{3}\Sigma^{2}}]\frac{\delta}{\varepsilon(\delta+\varepsilon)},$
(41)

Since the beta is the inverse of the temperature

$\beta=\frac{1}{T}=\frac{4\pi}{\Sigma},$ (42)

and if we select an appropriate cut-off distance $\varepsilon$ and thickness
of thin film $\delta$ to satisfy

$\frac{\delta}{\varepsilon(\delta+\varepsilon)}=90\beta,$ (43)

the total entropy of the horizon becomes

$S_{BH}=\frac{1}{4}A_{h},$ (44)

where $A_{h}$ is the area of the the black hole horizon, i.e. $A_{h}=4\pi
A^{2}r_{+}.$ The derivative of the entropy (44) with respect to $M$ is

$\frac{\partial S_{BH}}{\partial M}=\pi A^{2}\frac{\partial r_{+}}{\partial
M}=\frac{4\pi}{\Sigma},$ (45)

Getting the integral of $M$, equation (45) becomes to

$\Delta S_{BH}=\int_{M}^{M-\omega}\frac{\partial S_{BH}}{\partial
M^{\prime}}dM^{\prime}=4\pi\int_{M}^{M-\omega}\frac{dM^{\prime}}{\Sigma},$
(46)

After substituting $M^{\prime}=M-\omega^{\prime}$ into the above equation, we
obtain

$\Delta S_{BH}=-\int_{0}^{\omega}\frac{\partial S_{BH}}{\partial
M^{\prime}}d\omega^{\prime}=-4\pi\int_{0}^{\omega}\frac{d\omega^{\prime}}{\Sigma},$
(47)

One can easily see that equation (47) is nothing but the results obtained both
from Parikh-Wilczek method (34) and the DRS method (29).

On the other hand, for the LDBHs the change of the entropy before and after
the radiation is

$\Delta S_{BH}=S(M-\omega)-S(M)=-\frac{2\pi\omega}{\kappa}.$ (48)

Since equation (48) contains only $\omega$, we deduce that the spectrum is
precisely thermal. In other words, the thermal spectrum does not suggest the
underlying unitary theory, and whence we can understand that the conservation
of information is violated.

##  4 Discussion and Conclusion

In this paper, we have effectively utilized three different methods (the DRS
model, the Parikh-Wilczek model and the thin film brick wall model) to
investigate the Hawking radiation for massive 4-dimensional LDBHs in the EMD,
EYMD and EYMBID theories. By considering the DRS method, the tunneling
probability for an outgoing positive energy particle or simply the tunneling
rate is neatly found. Later on, it is shown that the tunneling rate found from
the DRS method can be expressed in terms of the difference of Bekenstein-
Hawking entropies $\Delta S_{BH}$ of the black holes. Beside this, the other
two methods i.e. the Parikh-Wilczek method and the thin film brick-wall model
attribute also the same $\Delta S_{BH}$ result. In the thin film brick-wall
model, the cut-off factor is found to be 90$\beta$, which is exactly same as
in the calculation of the entropy for the Schwarzschild black hole Liancheng .

On the other hand, the obtained$\ \Delta S_{BH}$ result shows us that the
emission spectrum is nothing but a pure thermal spectrum. This result is not
consistent with the unitarity principle of quantum mechanics. It also implies
the violation of the conservation of information in the LDBHs.

Finally, further application of the Hawking radiation of the charged massive
particles via different methods to the case of LDBHs in higher dimensions
Sakalli  may reveal more information compared to the present case. This will
be our next problem in the near future.

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