Plagiarism Detector v. 1020 -Originality Report: [611131]

Plagiarism Detector v. 1020 -Originality Report:
% 4 wrds: 208 https://www.researchgate.net/publication/257028366_Fermions_Tunneling_from_Plebanski -Demia…
% 2 wrds: 85 https://link.springer.com/content/pdf/10.1007%2Fs10773 -008-9881 -0.pdf
% 2 wrds: 85 https://link.springer.com/content/pdf/10.1007/s10773 -008-9881 -0.pdf
[Show other Sources:]Analyzed document: 9/22/2017 2:02:02 PM
"articol.doc"
Licensed to: Originality report generated by unregistered Demo version!
 
Warning: Demo Version -reports are
incomplete!
To get full version, please order the
software:
Relation chart:
Distribution graph:
Comparison Preset: Word -to-Word. Detected language: English
Topsources of plagiarism:Page1 of8
9/22/2017 file://C:\Users\crisd\Documents\Plagiarism Detector reports\originality report 22.9.2017 1 …

[Show other Sources:]
id:1 Plagiarism detected: 0.17% https://www.researchgate.net/public… + 2 more resources!
id:2 Plagiarism detected: 0.19% https://link.springer.com/article/1…Processed resources details:
95-Ok/8-Failed
Important notes:
Wikipedia: Google Books: Ghostwriting services: Anti-cheating:
[notdetected] [notdetected] [notdetected] [notdetected]
Excluded Urls:
Included Urls:
Detailed document analysis:
\documentclass[12pt]{article}
\textheight = 24truecm
\textwidth = 16truecm
\hoffset = -2truecm
\voffset = -2truecm
%\usepackage{amssymb,amsmath}
\usepackage{amssymb,amsmath}
\usepackage{amsfonts}
\usepackage{eufrak}
\usepackage{graphicx}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\DeclareMathAlphabet {\mathitbf}{OML}{cmm}{b}{it}
\DeclareMathOperator{\IM}{Im}
\begin{document}
\begin{center}
\null\vspace{2cm}
{\large {\bf
Tunneling of Charged and Magnetized Fermions from
a \\
Rotating Dyonic Taub -NUT Black Hole}}\\
\vspace{2cm}
Kausari Sultana\footnotetext{K. Sultana\\Department of Mathematics, Shahjalal University of Science \&
Technology, Sylhet, Bangladesh \\E -mail: $[anonimizat]$}
\end{center}
\vspace{3cm}
\centerline{\bf Abstract}
\baselineskip=18pt
\bigskip
We investigate
tunneling of charged and magnetized Dirac particles from
a rotating dyonic Taub -NUT (TN) black hole (BH) called the Kerr -Newman -Kasuya -Tub-NUT (KNKTN) BHPage2 of8
9/22/2017 file://C:\Users\crisd\Documents\Plagiarism Detector reports\originality report 22.9.2017 1 …

id:3 Plagiarism detected: 0.17% https://link.springer.com/article/1…
id:4 Plagiarism detected: 0.36% https://www.researchgate.net/public… + 3 more resources!
id:5 Plagiarism detected: 0.29% https://link.springer.com/article/1…
id:6 Plagiarism detected: 0.22% https://www.researchgate.net/public… + 2 more resources!
id:7 Plagiarism detected: 0.22% https://www.researchgate.net/public… + 3 more resources!
id:8 Plagiarism detected: 0.15% https://www.researchgate.net/public…
id:9 Plagiarism detected: 0.22% https://www.researchgate.net/public…endowed with electric as
well as magnetic
charges. We derive
the tunneling probability of outgoing charged particles by using the semiclassical WKB approximation to the
covariant Dirac equation and obtain the corresponding Hawking temperature. The emission spectrum
deviates from the purely thermal spectrum with the leading term exactly the
Boltzman factor, if
energy conservation and the backreaction of particles to the
spacetime are considered. The results provides a quantum -corrected radiation temperature depending on the BH
background and the radiation particle's energy, angular momentum, and
charges. The results are consistent
with those already available in literature.
\vspace{0.25cm}\\
\textbf{Keywords:} Dyonic black hole, Hawking radiation, Quantum Tunneling, Backreaction, NUT parameter.
\newpage
\section{Introduction}\label{sec1}
Hawking's discovery that a BH can radiate thermally \cite{1,2} has attracted lots of physicists' attention and many
papers have appeared to deeply discuss the quantum radiation of BHs via different methods \cite{3,4,5,6,7}.
Much interests have grown up in exploring quantum phenomenon of Hawking radiation
from BHs as a tunneling technique of emitting quantum
[Incomplete report data! Please purchase the software to get complete Report!]
 
Warning: Demo Version -reports are incomplete!
To get full version, please order the software:
particles. The tunneling rate with the WKB approximation takes the form: $\Gamma\varpropto\exp[ -2\IM{I}]$
with $I$ the classical action of the trajectory. Therefore, it is important for this tunneling method to calculate the
imaginary part of the action. There are two universal methods to compute particle's action. The first one is the
null geodesic method developed by Parikh and Wilczek (PW) \cite{8,9}, following the work of Kraus and WilczekPage3 of8
9/22/2017 file://C:\Users\crisd\Documents\Plagiarism Detector reports\originality report 22.9.2017 1 …

\cite{10}, in which the imaginary part of the action is regarded as the only contribution of the momentum $p_r$
of the emitted null $s$ -wave. The barrier is created by the outgoing particles themselves and a corrected
spectrum can be derived when self -gravitation of particles is taken into account. The second tunneling method,
called Hamilton -Jacobi (HJ) ansatz, is proposed by Srinivasan and Padmanabhan \cite{11,12} and has been
developed by Angheben et al. \cite{13}, which successfully derives the imaginary part of the action by solving
the HJ equation. In this method, the same conclusion as the first method can be drawn. Kerner and Mann \cite
{14} investigated quantum tunnelling methods for calculating temperatures of KN and TN BHs, using both the
null-geodesic and HJ methods. Subsequently, using the WKB approximation to the covariant Dirac equation,
Kerner and Mann (KM) developed the calculations of the tunneling process for the spin -1/2 particle's emission
from nonrotating BHs \cite{15} and investigated the Hawking temperature for the KN BH \cite{16}. This model
hasbeen extended to study the tunneling of charged and magnetized fermions from the RN \cite{17} and KN –
AdS \cite{18} BHs with magnetic charges. HJ method and KM tunneling approach have also been exploited in
investigating tunneling of scalar and Dirac particles from the TN -AdS BH \cite{19}. There are some works of
investigating the tunneling phenomenon for different BHs \cite{20,21,22,23,24} (including accelerating as well
as rotating \cite{25,26,27,28}) by employing the aforementioned methods. In this paper, we apply the KM
tunneling approach to calculate emission rate of charged and magnetized fermions from a rotating dyonic TN
BH namely the KNK BH generalized NUT or magnetic monopole parameter. This BH is a more general
background to be investigated. Conservation of energy and the particles' backreaction lead to the same
terminations as the previous works. The result can be treated as tunneling radiation at a quantum corrected
temperature, which is dependent on not only the BH background, but also the tunneling particle's energy,
angular momentum, and charges. The interest in the possibility of dyonic BHs has grown since magnetic
monopoles have been predicted in various extensions of the standard model of particle physics. The magnetic
monopole hypothesis in general relativity was put forward by Dirac \cite{29} relatively long ago. His ingenious
suggestion of existing magnetic monopole in nature was neglected due to the failure to detect such objects. In
recent years, however, the development of gauge theories has shed new light on it. The string theory \cite{30}
admits the existence of such objects. By exhausting the energies related to rotation and charges, the KNKTN
BH may reduce to the interesting TN BH, which plays an important role in the conceptional development of
general relativity and in the construction of brane solutions in string theory and M -theory \cite{31,32,33}.
According to Misner the TN spacetime has the interpretation of being a counter example to almost anything
\cite{34}. It has drawn a particular interest in recent years, because it plays the role in furthering our
understanding of the AdS/CFT correspondence \cite{14,35,36,37} and in this regard, the thermodynamics of
various TN solutions has become a subject of intense study. The Dirac field in TN background has been
analyzed \cite{38,39,40}. Of course, the existence of the closed time -like geodesics in the TN spacetime
violates the causality condition. Nevertheless, one can explore the half -closed time -like geodesics of Taub area
in the NUT area, so the naked singularity does exist. In the meantime, it admits no angular velocity and no
superradiation occurs at the event horizon. Hawking radiation from the TN BH was found to agree with Parikh
and Wilczek's result \cite{41}. The paper is organized as follows. In Sec 2, we describe the background
geometry of the KNKTN BH with magnetic charges and perform the dragging coordinate transformation. In Sec
3, we provide Dirac equation of charged and magnetized particles and compute the tunneling probability as well
as the corresponding temperature across the event horizon. A precise construction of the particles action has
also been done. In Sec 4, we calculate the emitting rate by considering backreaction of the radiation. Finally,
we give our concluding remarks in Sec 5. \section{Dyonic KNKTN Black Hole}\label{sec2} The metric of the
dyonic KNKTN BH in Boyer -Lindquist coordinates can be written as: \begin{displaymath} ds^2= -\frac{\Delta_r}
{\Sigma}(dt -\eta d\varphi)^2 +\frac{\Sigma}{\Delta_{r}}dr^2+\Sigma d\theta^2 +\frac{\sin^2\theta}{\Sigma}(adt –
\rho^2d\varphi)^2,\eqno(1) \end{displaymath} where $\Sigma=r^{2}+(n+a\cos\theta)^{2}$, $\eta=a\sin^{2}\theta –
2n\cos\theta$, $\Delta_r=\rho^2 -2(Mr+n^2)+Q^2+P^2$, $\rho^2=r^2+a^2+n^2$ and $M$, $a(=J/M)$, $Q$, $P$,
$n$ are respectively the mass, angular momentum per unit mass parameter, electric charge, magnetic charge
and NUT (magnetic mass) parameters of the BH. The presence of the NUT parameter makes the spacetime
asymptotically nonflat. In a recent work, Aliev \cite{42} interpreted the NUT parameter as generating a
„rotational effect''. The associated „specific angular momentum'' is $J_M=nM$, which we have incorporated in
the following analysis. The gauge potential associated with the metric eq(1) is \cite{43} \begin{eqnarray*}
&&\mathcal A=A_\mu+iB_\mu=\left[ -\frac{Qr}{\Sigma}(dt -\etad\varphi) -\frac{P(n+a\cos\theta)}{a\Sigma} (adt –
\rho^2d\varphi)\right]\\ &&\quad\quad\quad\quad\quad\quad\quad\quad +i\left[ -\frac{Pr}{\Sigma}(dt -\eta d\varphi)
+\frac{Q(n+a\cos\theta)}{a\Sigma} (adt-\rho^2d\varphi)\right]. \end{eqnarray*} In our analysis we write the
electric potential $A_\mu$ and the magnetic -like potential $B_\mu$ as \begin{displaymath} A_\mu= -\frac{Qr}
{\Sigma}(dt -\eta d\varphi),\quad B_\mu= -\frac{Pr}{\Sigma}(dt -\eta d\varphi),\eqno(2) \end{displaymath}
considering the appropriate limiting case. The event (outer) horizon $r_+$ and Cauchy (inner) horizon $r_ -$ of
the BH are given by $r_\pm=M\pm\sqrt{M^2 -Q^{2} -P^{2} -a^2+n^2}$. Since the outer infinite red -shift surface
obtained from ${\rm g}^{\mu\nu}\partial_\mu\partial_\nu f=0$ of the metric eq(1) doesn't coincide with the event
horizon, the geometrical optics limit cannot be used at the horizon and the semi -classical WKB approximate is
invalid there. In order to view the Hawking radiation of fermions, we need to choose one coordinate system in
which they will be coincident. We can perform this by either selecting the dragging coordinate systemPage4 of8
9/22/2017 file://C:\Users\crisd\Documents\Plagiarism Detector reports\originality report 22.9.2017 1 …

$(t,r,\theta)$ with $d\varphi= -\frac{{\rm g}_{03}}{{\rm g}_{33}}dt$, or introducing a new coordinate system
$(t,r,\theta,\chi)$. We adopt the latter one and set $\chi=\varphi -\Omega t$, where $\Omega= -\frac{{\rm g}_{03}}
{{\rm g}_{33}} =\frac{\rho^2a\sin^2\theta -\Delta_r\eta} {\rho^4\sin^2\theta -\Delta_r\eta^2}$ is the dragged
angular velocity of the BH. The metric eq(1) then takes the form \begin{displaymath} ds^{2}= -Fdt^2+G^{ -1}
dr^2+Hd\theta^2+Kd\chi^2,\eqno(3) \end{displaymath} where \begin{displaymath} F(r,\theta)=\frac
{\Delta_r\sin^2\theta (\rho^2 -a\eta)^2}{\Sigma(\rho^4\sin^2\theta -\Delta_r\eta^2)},\quad G(r,\theta)=\frac
{\Delta_r}{\Sigma},\nonumber \end{displaymath} \begin{displaymath} K(r,\theta)=\frac{1}{\Sigma}(\rho^4\sin^2
\theta -\Delta_r\eta^2),\quad H(r,\theta)=\Sigma,\eqno(4) \end{displaymath} with the corresponding gauge
potential of electric and magnetic fields \begin{displaymath} A_\mu= -\frac{Qr}{\Sigma}\cdot\frac{\rho^2\sin^2
\theta(\rho^2 -a\eta)}{\rho^4\sin^2\theta -\Delta_r\eta^2}dt +\frac{Qr\eta}{\Sigma}d\chi,\nonumber \end
{displaymath} \begin{displaymath} B_\mu= -\frac{Pr}{\Sigma}\cdot\frac{\rho^2\sin^2\theta(\rho^2 -a\eta)}{\rho^4
\sin^2\theta -\Delta_r\eta^2}dt +\frac{Pr\eta}{\Sigma}d\chi.\eqno(5) \end{displaymath} In metric eq(3), the outer
(inner) horizons coincide with the outer (inner) infinite red -shift surfaces. The Landau's condition of the
coordinate clock synchronization is also satisfied. So, we can investigate the radiation of fermions for this BH.
\section{Charged and Magnetized Particles Tunneling}\label{sec3} In order to study the tunneling of charged
and magnetized fermion of mass $\mu_o$ from a KNKTN BH, we consider the Dirac equation in covariant form
as \cite{28} \begin{displaymath} \gamma^{\mu}\left[i\hbar \left(\partial_{\mu}+\frac{i}{2}\Gamma^{\alpha\beta}
_\mu \Sigma_{\alpha\beta}\right)+q_{\rm e}A_\mu+q_{\rm m}B_\mu\right]\Psi+\mu_o\Psi=0,\eqno(6) \end
{displaymath} where $\Gamma^{\alpha\beta}_\mu={\rm g}^{\beta\gamma}\Gamma^\alpha_{\mu\gamma}$,
$\Sigma_{\alpha\beta}=\frac{1}{4}i[\gamma^{\alpha}, \gamma^{\beta}]$ and $\gamma^\mu$ marices satisfy
$\{\gamma^\mu,\gamma^\nu\}=2{\rm g}^{\mu\nu}I$. We choose the $\gamma^{\mu}$ matrices as \begin
{displaymath} \gamma^t=\frac{1}{\sqrt{F(r,\theta)}}\left(\begin{array}{rl}i\qquad 0\\0\quad -i\end{array}\right),
\gamma^r=\sqrt{G(r,\theta)}\left(\begin{array}{rl}0\quad \sigma^3\\\sigma^3\quad 0 \end{array}\right),\nonumber
\end{displaymath} \begin{displaymath} \gamma^{\theta}=\frac{1}{\sqrt{H(r,\theta)}} \left(\begin{array}{rl}0\quad
\sigma^{1}\\\sigma^{1}\quad 0 \end{array}\right), \gamma^{\chi}=\frac{1}{\sqrt{K(r,\theta)}} \left(\begin{array}{rl}0
\quad \sigma^{2}\\\sigma^{2}\quad 0 \end{array}\right),\eqno(7) \end{displaymath} where $\sigma^i$ $(i = 1, 2,
3)$ are Pauli matrices: \begin{displaymath} \sigma^{1}=\left(\begin{array}{rl}0\quad 1\\1\quad 0 \end{array}
\right),\quad \sigma^{2}=\left(\begin{array}{rl}0\quad -i\\i\qquad 0 \end{array}\right),\quad \sigma^{3}=\left(\begin
{array}{rl}1\qquad 0\\0\quad -1\end{array}\right).\eqno(8) \end{displaymath} Dirac matrices imply that
$[\gamma^\alpha,\gamma^\beta]=0$ or $ -[\gamma^\beta,\gamma^\alpha]$ according as $\alpha=\beta$ or
$\alpha\neq\beta$. Consequently, contribution of the term containing $\Sigma_{\alpha\beta}$ in eq(6) is zero.
The eigenvectors of $\sigma^3$ are denoted by $\xi_{\uparrow/\downarrow}$. Then measuring spin in the $r$ –
direction we have two spin states for the spinor wave function $\Psi$ (related to the particle's action): \emph
{spin -up} in $+$ve $r$ -direction and \emph{spin -down} in $ -$ve $r$ -direction. Accordingly, we assume the two
following ansatz for the spin -1/2 Dirac field \cite{15}: \begin{displaymath} \Psi_{\uparrow}=\left[\begin{array}{rl}
A\xi_{\uparrow} \\B\xi_{\uparrow} \end{array}\right] \exp\left[\frac{i}{\hbar} I_{\uparrow}\right] =[A,0,B,0]' \exp\left
[\frac{i}{\hbar} I_{\uparrow}\right],\eqno(9) \end{displaymath} \begin{displaymath} \Psi_{\downarrow}=\left[\begin
{array}{rl}C\xi_{\downarrow} \\D\xi_{\downarrow} \end{array}\right] \exp\left[\frac{i}{\hbar} I_{\downarrow}\right] =
[0,C,0,D]' \exp\left[\frac{i}{\hbar} I_{\downarrow}\right],\eqno(10) \end{displaymath} where the action of the
radiant spin particles $I_{\uparrow/\downarrow}$ and the wave modes represented by $A$, $B$, $C$, $D$ are
all functions of $(t,r,\theta,\chi)$. In this paper, we only analyze the spin -up case because the final result is the
same as the spin-down case, as can be presented by using the methods described below. Inserting eq(9) into
the Dirac Equation eq(6) and applying WKB approximation, we find \begin{displaymath} -\left(\frac{iA}{\sqrt{F
(r,\theta)}}(\partial_{t} I_{\uparrow} -q_{\rm e}A_t -q_{\rm m}B_t) +B\sqrt{G(r,\theta)}\partial_{r}I_{\uparrow}\right)
+\mu_oA=0,\eqno(11) \end{displaymath} \begin{displaymath} -B\left(\frac{1}{\sqrt{H(r,\theta)}}\partial_{\theta} I_
{\uparrow} +\frac{i}{\sqrt{K(r,\theta)}}(\partial_\chi I_{\uparrow} +q_{\rm e}A_\chi+q_{\rm m}B_\chi)\right)=0,\eqno
(12) \end{displaymath} \begin{displaymath} \left(\frac{iB}{\sqrt{F(r,\theta)}}(\partial_{t} I_{\uparrow} -q_{\rm e}A_t –
q_{\rm m}B_t) -A\sqrt{G(r,\theta)}\partial_{r}I_{\uparrow}\right) +\mu_oB=0,\eqno(13) \end{displaymath} \begin
{displaymath} -A\left(\frac{1}{\sqrt{H(r,\theta)}}\partial_{\theta} I_{\uparrow} +\frac{i}{\sqrt{K(r,\theta)}}
(\partial_\chi I_{\uparrow} +q_{\rm e}A_\chi+q_{\rm m}B_\chi)\right)=0.\eqno(14) \end{displaymath} It is quite
difficult to immediately determine the action. Nevertheless, taking into account the existence of time -like killing
vector $(\frac{\partial}{\partial t})^{a}$ and space -like killing vector $(\frac{\partial}{\partial \varphi})^{a}$ in the
stationary space time, we carry out the separation variable as \begin{displaymath} I_{\uparrow}= -\omega t+W(r)
+\tilde{j}\varphi+\Theta(\theta),\eqno(15) \end{displaymath} where $\omega$ is the energy and $\tilde{j}=\tilde{j}
(j,j_M)$ is the magnetic quantum number of the particle. Using eq.(15) in eq.(11) –eq.(14) and Taylor's
expansion of $F(r,\theta)$ near the horizon $r_+$, we obtain with $iA=B$, $iB=A$, \begin{displaymath} -B\left
(\frac{ -\omega+\omega_o}{\sqrt{(r -r_+)\partial_rF(r_+,\theta)}} +\sqrt{(r -r_+)\partial_rG(r_+,\theta)}(\partial_rW)
\right) +\mu_oA=0,\eqno(16) \end{displaymath} \begin{displaymath} -B\left(\frac{1}{\sqrt{H(r_+,\theta)}}\partial_
{\theta} \Theta +\frac{i}{\sqrt{K(r_+,\theta)}}\tilde{j} +q_{\rm e}A_\chi(r_+)+q_{\rm m}B_\chi(r_+))\right)=0,\eqno
(17) \end{displaymath} \begin{displaymath} A\left(\frac{ -\omega+\omega_o}{\sqrt{(r -r_+)\partial_rF(r_+,\theta)}} –
\sqrt{(r -r_+)\partial_rG(r_+,\theta)}(\partial_rW)\right) +\mu_oB=0,\eqno(18) \end{displaymath} \begin
{displaymath} -A\left(\frac{1}{\sqrt{H(r_+,\theta)}}\partial_{\theta} \Theta +\frac{i}{\sqrt{K(r_+,\theta)}}\tilde{j} +q_Page5 of8
9/22/2017 file://C:\Users\crisd\Documents\Plagiarism Detector reports\originality report 22.9.2017 1 …

{\rm e}A_\chi(r_+)+q_{\rm m}B_\chi(r_+))\right)=0,\eqno(19) \end{displaymath} where $\omega_o=\frac{\tilde{j}
a}{r_+^2+a^2+n^2} +\frac{q_{\rm e}Qr_+}{r^2_++a^2+n^2}+\frac{q_{\rm m}Pr_+}{r^2_++a^2+n^2}$. We deal
first with eq.(16) –eq.(19) for massless $(\mu_o= 0)$ case. Accordingly, eq.(16) and eq.(18) imply that \begin
{displaymath} W'(r)=W'_+(r)= -W'_-(r)=\bigg[\frac{(r_+^2+a^2+n^2) (\omega -\omega_o)} {(r-r_+)(r_+ -r_-)}
\bigg],\eqno(20) \end{displaymath} where $W_+$ corresponds to outward solutions and $W_ -$ corresponds to
the incoming solutions. Thus the particle's tunneling probability from inside to outside the horizon is \begin
{displaymath} \Gamma=\frac{\mbox{Prob[out]}}{\mbox{Prob[in]}} =\frac{\exp[ -2(\IM{W_+}+\IM{\Theta})]}{\exp[ -2
(\IM{W_ -}+\IM{\Theta})]}=\exp[ -4\IM{W_+}],\eqno(21) \end{displaymath} where $W_+$ is the integral of eq.(20).
There is a pole at the horizon, $r=r_+$, in eq.(20). After integrating around the pole, we find \begin{displaymath}
W_+=\bigg[\frac{\pi i(r_+^2+a^2+n^2)(\omega -\omega_o)} {r_+ -r_-}\bigg],\eqno(22) \end{displaymath} and
consequently, the resultant tunneling probability to leading order in $\hbar$ is given by \begin{displaymath}
\Gamma=\exp\bigg[\frac{ -4\pi(r_+^2+a^2+n^2)(\omega -\omega_o)} {r_+-r_-}\bigg]=\exp[ -2\pi(\omega –
\omega_o)/\kappa].\eqno(23) \end{displaymath} Hence, the Hawking temperature of the KNKTN BH is
recovered as \begin{displaymath} T=\frac{1}{\beta}=\frac{\kappa}{2\pi}, \quad \kappa=\frac{r_+ -r_-} {2
(r_+^2+a^2+n^2)}.\eqno(24) \end{displaymath} This is fully in consistence with that obtained by other method.
For $n=0$, it reduces to the temperature of the KN BH \cite{16}, which further reduces to the temperature of the
RN BH for $a=0$ \cite{17}. In absence of charge ($Q=0=P$), it exactly becomes the Hawking temperature of
the Schwarzschild BH \cite{25}. In the massive particles ($\mu_o\neq0$) case, adopting the same process, we
canfind the same temperature. Thus, when a BH radiates massless particles and massive particles, the
tunneling probability and Hawking temperature remain the same and are not related to the kind of particles. In
the case of the spin -down case, adopting the corresponding (spin -down) wave function and analyzing it again
as the analogous process, we can find the same result. The action $I_\uparrow$ in the spin -up case is obtained
by solving eq.(16) –eq.(19). For outgoing particles, eq.(16) gives on integration \begin{displaymath} W(r)=W_+
(r)\nonumber \end{displaymath} \begin{displaymath} \;\;\,\quad\quad=\int\frac{\mu_oA}{B\sqrt{(r -r_+)\partial_r G
(r_+,\theta)}}dr -\frac{ -\omega+\omega_o}{\sqrt{\partial_r F(r_+,\theta)\partial_rG(r_+,\theta)}}\ln(r -r_+),\eqno(25)
\end{displaymath} while for the incoming particles, eq.(18) yields \begin{displaymath} W(r)=W_ -(r)\nonumber
\end{displaymath} \begin{displaymath} \;\;\,\quad\quad=\int\frac{\mu_oB}{A\sqrt{(r -r_+)\partial_r G(r_+,\theta)}}
dr+\frac{ -\omega+\omega_o}{\sqrt{\partial_r F(r_+,\theta)\partial_rG(r_+,\theta)}}\ln(r -r_+).\eqno(26) \end
{displaymath} Equations eq.(17) and eq.(19) imply that \begin{displaymath} \Theta= -i\bigg[\tilde{j}\ln\tan\frac
{\theta}{2}+\frac{a -(q_{\rm e}Q+q_{\rm m}P)r_+}{r_+^2+a^2+n^2} (a\cos\theta+2n\ln\sin\theta)\bigg].\eqno(27)
\end{displaymath} Equations eq.(25) and eq.(27) with eq.(15) determine the outgoing massive particles' action.
This expression reduces to the massless particles' action for $\mu_o=0$. Likewise, one can find the action for
theingoing particle either massive or massless. \section{Backreaction in the Tunneling Process}\label{sec4} In
this section, we consider the emitting particles' backreaction on the spacetime. When a particle with energy
$\omega_i$, charge $q_{i{\rm e}}$, magnet $q_{i{\rm m}}$, and angular momentum $\tilde{j}_i$ tunnels out of
the BH, the parameters $M$, $Q$, $P$, $a$, $n$ should be substituted by $(M -\omega_i)$, $(Q -q_{i{\rm e}})$,
$(P-q_{i{\rm m}})$, $a_i=\frac{Ma -\tilde{j}_i}{M -\omega_i}$ and $n_i=\frac{Mn -\tilde{j}_i}{M -\omega_i}$,
respectively. Then, the emission rate is \begin{displaymath} \Gamma_i=\exp[ -2\pi(\omega_i -\omega_{i0})/
\kappa_i],\eqno(28) \end{displaymath} where \begin{displaymath} \omega_{i0}=\tilde{j}_i\Omega_{i+}+q_{i\rm e}
A_{i+}+q_{i\rm m}B_{i+}\nonumber \end{displaymath} \begin{displaymath} \;\;\,\quad=\frac{\tilde{j}_ia_i}{r_{i+}
^2+a_i^2+n_i^2}+\frac{q_{i\rm e}(Q-q_{i{\rm e}})r_{i+}}{r^2_{i+}+a_i^2+n_i^2}+\frac{q_{i\rm m}(P -q_{i{\rm m}})r_
{i+}}{r^2_{i+}+a_i^2+n_i^2},\nonumber \end{displaymath} \begin{displaymath} r_{i\pm}=(M -\omega_i)\pm\sqrt
{(M-\omega_i)^2 -a_i^2 -z_i^2+n_i^2},\nonumber \end{displaymath} \begin{displaymath} \;\,z_i^2=(Q -q_{i\rm e})
^2+(P -q_{i\rm m})^2,\quad\kappa_i=\frac{(r_{i+} -r_{i-})}{2(r_{i+}^2 +a_i^2+n_i^2)}.\eqno(29) \end{displaymath}
For emission of many particles and thinking that they radiate one by one, we get \begin{displaymath}
\Gamma=\prod_i\Gamma_i=\exp\left[\sum_i( -2\pi(\omega_i -\omega_{i0}))/\kappa_i\right].\eqno(30) \end
{displaymath} If the emission is regarded as a continuous procession, the sum in eq.(30) could be replaced by
integration \begin{displaymath} \Gamma=\exp[ -2\pi\int(d\omega' -\Omega'_+ d\tilde{j}' -A'_+dq'_{\rm e}-B'_+dq'_
{\rm m}) /\kappa'].\eqno(31) \end{displaymath} We consider the entropy $S=A/4=\pi(r_+^2+a^2+n^2)$ and
obtain the difference between the entropies of the horizon before and after the emission, $\Delta S=S_f -S_i$:
\begin{eqnarray} &&\Delta S=\pi\bigg[2(M -\omega)^2 -(Q-q_{\rm e})^2 -(P-q_{\rm m})^2+2n'^2\nonumber\\
&&\quad\quad\quad +2(M -\omega)\sqrt{(M -\omega)^2 -(Q-q_{\rm e})^2 -(P-q_{\rm m})^2 -a'^2+n'^2}\nonumber\\
&&\quad\quad\quad -2M^2+Q^2+P^2 -2n^2 -2M\sqrt{M^2 -Q^2-P^2-a^2+n^2}\bigg].\nonumber \hspace{2cm}
(32) \end{eqnarray} We find \begin{eqnarray} &&\frac{1}{2\pi}\frac{\partial(\Delta S)}{\partial\omega'}= -\frac{((M –
\omega')+\sqrt{(M -\omega')^2 -z'^2-a'^2+n'^2})^2}{\sqrt{(M -\omega')^2 -z'^2-a'^2+n'^2}}\nonumber\\
&&\quad\quad\quad\quad\quad\;\;\; -\frac{(M -\omega')a'^2+n'^2[(M -\omega') -2r'_+]}{ (M-\omega')\sqrt{(M –
\omega')^2 -z'^2-a'^2+n'^2}}\nonumber\\ &&\;\quad\quad\quad\quad\;\;= -2\frac{r'^2_++a'^2+n'^2}{r'_+ -r'_-}-\frac
{4n'^2r'_+}{(M -\omega')(r'_+ -r'_-)} \simeq -\frac{1}{\kappa'},\nonumber\\ &&\frac{1}{2\pi}\frac{\partial(\Delta S)}
{\partial q'_{\rm e}}=\frac{(Q -q'_{\rm e})((M -\omega')+\sqrt{(M -\omega')^2 -z'^2-a'^2+n'^2})}{\sqrt{(M -\omega')^2 –
z'^2-a'^2+n'^2}} =\frac{A'_+}{\kappa'},\nonumber\\ &&\frac{1}{2\pi}\frac{\partial(\Delta S)}{\partial q'_{\rm m}}
=\frac{(Q -q'_{\rm m})((M -\omega')+\sqrt{(M -\omega')^2 -z'^2-a'^2+n'^2})}{\sqrt{(M -\omega')^2 -z'^2-a'^2+n'^2}}
=\frac{B'_+}{\kappa'},\nonumber\\ &&\frac{1}{2\pi}\frac{\partial(\Delta S)}{\partial \tilde{j}'}=\frac{a'}{\sqrt{(M -Page6 of8
9/22/2017 file://C:\Users\crisd\Documents\Plagiarism Detector reports\originality report 22.9.2017 1 …

\omega')^2 -z'^2-a'^2+n'^2}} =\frac{\Omega'_+}{\kappa'},\nonumber \hspace{4cm} (33) \end{eqnarray} where
$a'=\big(\frac{Ma -\tilde{j}'}{M -\omega'}\big)$, $n'=\big(\frac{Mn -\tilde{j}'}{M -\omega'}\big)$ and $z'^2=(Q -q'_{\rm
e})^2+(P -q'_{\rm m})^2$. From eq.(31) and eq.(33), we deduce that the emission rate is connected with the
change in Bekenstein -Hawking entropy: \begin{displaymath} \Gamma=e^{\int d(\Delta S)}=e^{\Delta S}.\eqno
(34)\end{displaymath} Expanding the emission rate $\Gamma$ in $\omega$, $q_{\rm e}$, $q_{\rm m}$ and
$\tilde{j}$, one finds \begin{displaymath} \Gamma=\exp[ -\beta(\omega -\omega_0)+{\cal O}(\omega,q_{\rm e},q_
{\rmm},\tilde{j})^2].\eqno(35) \end{displaymath} It depicts that the emission rate in the tunneling approach, up to
first order in $\omega$, retrieves the Boltzmann factor of the form $\exp[ -\beta\omega]$ with $\beta$ the
inverse Hawking temperature. The higher order terms of $\omega$, $q_{\rm e}$, $q_{\rm m}$, $\tilde{j}$
describe self -interaction effects resulting from the energy conservation. They are a deviation from a purely
thermal spectrum. So, some information can be brought out of the BH with the corrected spectrum. This can
give an explanation to the information loss paradox. Equation eq.(35) can be put in the form \begin
{displaymath} \Gamma=\exp[ -\beta'(\omega -\omega_0)],\quad \beta'=\frac{1}{T'}=\beta\left[1 -\frac{{\cal O}
(\omega,q_{\rm e},q_{\rm m},\tilde{j})^2} {\beta(\omega -\omega_0)}\right],\eqno(36) \end{displaymath} where
$\beta^\prime$ can be treated as an inverse quantum -corrected temperature. This result goes over to the KN
BH case \cite{44} in the limit $n=0=P$. In quantum mechanics the emitting rate is given by $ \Gamma
(i\rightarrow f)=|A_{fi}|^2\delta_p$ where $A_{fi}$ is the amplitude for the tunneling action. The \emph{phase
space factor} $\delta_p$ is the average of the number of microstates of the initial state and the number of
microstates of the final state. Since the number of microstates of the initial and final states are the exponent of
the initial and final entropies, $\Gamma=e^{S_f}/e^{S_i}=e^{\Delta S}$. Manifestly, this is consistent with our
result. Thus, our result satisfies the underlying unitary theory in quantum mechanics and thereby provides a
might explanation to the BH information puzzle. \section{Concluding Remarks}\label{sec5} Our concern in this
study is to apply the KM fermion tunneling method \cite{15,16} to charged and magnetized fermion cases for a
rotating dyonic TN BH described by the KNKTN metric eq.(1). In this semiclassical method the horizon plays a
role of two way energy barrier for a pair of positive and negative energy particles. Although classically a particle
can only fall inside the event horizon, there exists a crossing of a particle's energy level near the event horizon
in the semiclassical approach. The energy of a negative energy particle can be larger than the energy of the
nearby positive energy level. As a result, it can travel across the forbidden region because of the quantum
tunneling effect and becomes a positive energy particle that can move out. In our investigation we have
therefore computed tunneling probabilities for both incoming as well as outgoing particles. We find the tunneling
probability of emission for the spin -up particle case at the event horizon and recover the corresponding
Hawking temperature. The result shows that the tunneling probability depends upon fermion's charges but not
upon its mass. The corresponding Hawking temperature depends upon mass, rotation and NUT parameters as
well as electric and magnetic charges of the BH. One can find the result for the spin -down particle case with the
equations of the spin -up case by only excluding a negative sign. The Hawking temperature implies that the
tunneling rates of the spin -up and spin -down particles are the same and it remains invariant as well for both
massive and massless cases \cite{15}. We also have calculated corrections to the fermion emission
temperature by computing corrections to the tunneling probability with fully taking into account conservation of
energy and considering self -gravitational interaction and backreaction of radiant particles. The result shows that
the tunneling probability of fermion is related to the change of the Bekenstein -Hawking entropy. We find that the
quantum -corrected radiation temperature is dependent on the BH background and the radiation particle's
energy, angular momentum and charges. The result implies that the emission spectrum is not purely thermal
anymore and the leading term is exactly the Boltzman factor. This is consistent with the previous works done by
using PW or HJ method. Our analysis gives as well results in agreement to that obtained by Damour -Ruffini's
method in the case of charged Dirac particles' Hawking radiation from a KN BH \cite{44}. The corrected
spectrum of emission can bring some information out of the BH. The underlying unitary theory may be satisfied
as well. In the limit $n=0=P$ our results reduce to the results of the KN BH \cite{16} and $a=0=n$ yields the
results of the RN BH \cite{17}. \begin{thebibliography}{99} \bibitem{1} S.W. Hawking, Nature \textbf{248} (1974)
30. \bibitem{2} S.W. Hawking, Commun. Math. Phys. \textbf{43} (1975) 199. \bibitem{3} T. Damour, R. Ruffini,
Phys. Rev. D \textbf{14} (1976) 332. \bibitem{4} G.W. Gibbons, S.W. Hawking, Phys. Rev. D \textbf{15} (1977)
2752. \bibitem{5} J.Y. Zhu, A.D. Bao, Z. Zhao, Int. J. Theor. Phys. \textbf{34} (1995) 2049. \bibitem{6} J.L. Jing,
Chin. Phys. \textbf{10} (2001) 234. \bibitem{7} S.Q. Wu, X. Cai, Nuovo Cimento B \textbf{115} (2000) 143.
\bibitem{8} M.K. Parikh, F. Wilczek, Phys. Rev. Lett. \textbf{85}, (2000) 5042. \bibitem{9} M.K. Parikh, Gen.
Relativ. Gravit. \textbf{36} (2004) 2419. \bibitem{10} P. Kraus, F. Wilczek, Nucl. Phys. B \textbf{433} (1995)
403. \bibitem{11} K. Srinivasan, T. Padmanabhan, Phys. Rev. D \textbf{60} (1999) 024007. \bibitem{12} S.
Shankaranarayanan, K. Srinivasan, T. Padmanabhan, Mod. Phys. Lett. A \textbf{16} (2001) 571. \bibitem{13}
M. Angheben, M. Nadalini, L. Vanzo, S. Zerbini, JHEP \textbf{0505} (2005) 014. \bibitem{14} R. Kerner, R.B.
Mann, Phys. Rev. D \textbf{73} (2006) 104010. \bibitem{15} R. Kerner, R.B. Mann, Class. Quant. Grav. \textbf
{25} (2008) 095014. \bibitem{16} R. Kerner, R.B. Mann, Phys. Lett. B \textbf{665} (2008) 277. \bibitem{17} Q.
Li, Y. -W. Han, Int. J. Theor. Phys. \textbf{47} (2008) 3248. \bibitem{18} X. -X. Zeng, Q. Li, J. -B.Hao, Int. J.
Theor. Phys. \textbf{48} (2009) 1090. \bibitem{19} X. -X. Zeng, Q. Li, Chin. Phys. B \textbf{18} (2009) 4716.
\bibitem{20} M. Sharif, W. Javed, J. Korean Phys. Soc. \textbf{57} (2010) 217. \bibitem{21} M. Sharif, W. Javed,Page7 of8
9/22/2017 file://C:\Users\crisd\Documents\Plagiarism Detector reports\originality report 22.9.2017 1 …

Astrophys. Space Sci. \textbf{337} (2012) 335. \bibitem{22} M. Sharif, W. Javed, J. Exp. Theor. Phys. \textbf
{114} (2012) 933. \bibitem{23} M. Sharif, W. Javed, J. Exp. Theor. Phys. \textbf{115} (2012) 782. \bibitem{24}
M. Sharif, W. Javed, Can. J. Phys. \textbf{91} (2013) 43. \bibitem{25} U.A. Gillani, M. Rehman, K. Saifullah,
JCAP \textbf{06} (2011) 016. \bibitem{26} U.A. Gillani, K. Saifullah, Phys. Lett. B \textbf{699} (2011) 15. \bibitem
{27} M. Rehman, K. Saifullah, JCAP \textbf{03} (2011) 001. \bibitem{28} M. Sharif, W. Javed, Gen. Relativ.
Gravit. \textbf{45} (2013) 1051. \bibitem{29} P.A.M. Dirac, Phys. Rev. \textbf{74} (1948) 817. \bibitem{30} S.
Mignemi, Phys. Rev. D \textbf{51} (1995) 934. \bibitem{31} S. Cherkis, A. Hashimoto, J. High Energy Phys.
\textbf{11} (2002) 036. \bibitem{32} R. Clarkson, A.M. Ghezelbash, R.B. Mann, J. High Energy Phys. \textbf{04}
(2004) 063. \bibitem{33} R. Clarkson, A.M. Ghezelbash, R.B. Mann, J. High Energy Phys. \textbf{08} (2004)
025. \bibitem{34} C.W. Misner, Taub -NUT space as a counter example to almost anything, In: Ehlers, J. (ed.)
Relativity Theory and Astrophysics 1. Lectures in Applied Mathematics, Am. Math. Soc., Providence, vol. 8
(1967) p. 160. \bibitem{35} S.W. Hawking, C.J. Hunter, D.N. Page, Phys. Rev. D \textbf{59} (1999) 044033.
\bibitem{36} A. Chamblin, R. Emparan, C.V. Johnson, R.C. Myers, Phys. Rev. D \textbf{59} (1999) 064010.
\bibitem{37} R.B. Mann, Phys. Rev. D \textbf{60} (1999) 104047. \bibitem{38} A. Comtet, P.A. Horv\'{a}thy,
Phys. Lett. B \textbf{349} (1995) 49. \bibitem{39} I.I. Cot\u{a}escu, M. Visinescu, Int. J. Mod. Phys. A \textbf{16}
(2001) 1743. \bibitem{40} I.I. Cot\u{a}escu, M. Visinescu, Phys. Lett. B \textbf{502} (2001) 229. \bibitem{41} D.
Chen, X. Zu, Acta Phys. Pol. B \textbf{39} (2008) 1329. \bibitem{42} A.N. Aliev, Phys. Rev. D \textbf{77} (2008)
044038. \bibitem{43} J.A. C\'{a}zares, arXiv:1211.6685v2 [hep -th] 25 Feb 2013. \bibitem{44} S. Zhou, W. Liu,
Phys. Rev. D \textbf{77} (2008) 104021. \end{thebibliography} \end{document}
Plagiarism Detector
Your right to know the authenticity!Page8 of8
9/22/2017 file://C:\Users\crisd\Documents\Plagiarism Detector reports\originality report 22.9.2017 1 …