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\begin{document}
\title{\textbf{Non-strictly black body spectrum from the tunnelling mechanism }}
\author{\textbf{Christian Corda}}
\maketitle
\begin{center}
Institute for Theoretical Physics and Advanced Mathematics Einstein-Galilei,
Via Santa Gonda 14, 59100 Prato, Italy
\par\end{center}
\begin{center}
\textit{E-mail address:} \textcolor{blue}{cordac.galilei@gmail.com}
\par\end{center}
\begin{abstract}
A modern and largely used approach to obtain Hawking radiation is
the tunnelling mechanism. However, in various papers in the literature,
the analysis concerned almost only to obtain the Hawking temperature
through a comparison of the probability of emission of an outgoing
particle with the Boltzmann factor.
In a interesting and well written paper, Banerjee and Majhi improved
the approach, by explicitly finding a black body spectrum associated
with black holes. On the other hand, this result, which has been obtained
by using a reformulation of the tunnelling mechanism, is in contrast
which the remarkable result by Parikh and Wilczek, that, indeed, found
a probability of emission which is compatible with a non-strictly
thermal spectrum.
By using our recent introduction of an effective state for a black
hole, here we solve such a contradiction, through a slight modification
of the analysis by Banerjee and Majhi. The final result will be a
non-strictly black body spectrum from the tunnelling mechanism.
\end{abstract}
\section{Introduction}
In recent years, the tunnelling mechanism has been an elegant and
largely used approach to obtain Hawking radiation \cite{key-1}, see
for example \cite{key-2}-\cite{key-6} and refs. within. A problem
on such an approach was that, in the cited and in other papers in
the literature, the analysis has been finalized almost only to obtain
the Hawking temperature through a comparison of the probability of
emission of an outgoing particle with the Boltzmann factor. The problem
was apparently solved in the interesting work \cite{key-7}, where,
through a reformulation of the tunnelling mechanism, a black body
spectrum associated with black holes has been found. In any case,
this result is in contrast which the remarkable result in \cite{key-2,key-3},
that, indeed, found a probability of emission which is compatible
with a non-strictly thermal spectrum.
By introducing an \emph{effective state}, we recently interpreted
black hole's quasi-normal modes in terms of quantum levels by finding
a natural connection between Hawking radiation and quasi-normal modes
\cite{key-8,key-9}. Here we show that the \emph{effective quantities}
permit also to solve the above cited contradiction, through a slight
modification of the analysis in \cite{key-7}. The final result will
be a non-strictly black body spectrum from the tunnelling mechanism.
For the sake of simplicity, in this paper we refer to the Schwarzschild
black hole and we work with $G=c=k_{B}=\hbar=\frac{1}{4\pi\epsilon_{0}}=1$
(Planck units).
\section{A review of the strictly thermal analysis}
We emphasize that in this Section we closely follow \cite{key-7}.
Let us consider a Schwarzschild black hole. The Schwarzschild line
element is (but see \cite{key-11} for clarifying historical notes
to this notion)
\begin{equation}
ds^{2}=-(1-\frac{2M}{r})dt^{2}+\frac{dr^{2}}{1-\frac{2M}{r}}+r^{2}(\sin^{2}\theta d\varphi^{2}+d\theta^{2}).\label{eq: Hilbert}
\end{equation}
The event horizon is defined by $r_{H}=2M$ \cite{key-7,key-10}.
As we want to discuss Hawking radiation like tunnelling, the radial
trajectory is relevant \cite{key-2,key-3,key-7}. Hence, we consider
only the $(r\lyxmathsym{\textminus}t)$ sector of the line element
(\ref{eq: Hilbert}) \cite{key-7}. Let us consider a Klein-Gordon
massless field $\varphi\:$ which obeys to \cite{key-7}
\begin{equation}
g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\varphi=0,\label{eq: Klein-Gordon}
\end{equation}
which, considering only the $(r\lyxmathsym{\textminus}t)\quad$sector,
becomes in the Schwarzschild spacetime (\ref{eq: Hilbert}) \cite{key-7}
\begin{equation}
-\frac{1}{1-\frac{2M}{r})}\partial_{t}^{2}\varphi+\frac{1}{4M}\partial_{r}\varphi+(1-\frac{2M}{r})\partial_{r}^{2}\varphi=0,\label{eq: reduced Klein-Gordon}
\end{equation}
where $\frac{1}{4M}\:$ is the black hole's surface gravity.
The standard WKB ansatz gives \cite{key-7}
\begin{equation}
\varphi(r,t)=\exp\left[-iS(r,t)\right].\label{eq: fi}
\end{equation}
$S(r,t)$ can be expanded as \cite{key-7}
\begin{equation}
S(r,t)=S_{0}(r,t)+\sum_{m=1}^{m=\infty}S_{m}(r,t).\label{eq: S}
\end{equation}
Inserting eq. (\ref{eq: S}) in eq. (\ref{eq: reduced Klein-Gordon})
and taking the semi-classical limit (the Planck constant $\rightarrow0$),
one gets \cite{key-7}
\begin{equation}
\partial_{t}S_{0}(r,t)=\pm(1-\frac{2M}{r})\partial_{R}S_{0}(r,t),\label{eq: de S}
\end{equation}
which is the well know semi-classical Hamilton-Jacobi equation \cite{key-4,key-7,key-12}.
Eq. (\ref{eq: de S}) can be also obtained starting from Dirac \cite{key-7,key-13}
or Maxwell equations \cite{key-7,key-14}. Eq. (\ref{eq: de S}) depends
also on imposing the chirality (holomorphic) condition on $\varphi\:$
with the WKB ansatz in eq. (\ref{eq: fi}), with the $+\;\lyxmathsym{\textminus}\;$
solutions standing for the left (right) modes \cite{key-15}.
Considering the timelike Killing vector of the stationary Schwarzschild
line element (\ref{eq: Hilbert}), one chooses an ansatz for $S_{0}(r,t)$
as \cite{key-7}
\begin{equation}
S_{0}(r,t)=\omega t+\tilde{S}_{0}(r).\label{eq: S con 0}
\end{equation}
$\omega\:$ in eq. (\ref{eq: S con 0}) is the conserved quantity
which corresponds to the timelike Killing vector \cite{key-7}. Inserting
eq. (\ref{eq: S con 0}) in eq. (\ref{eq: de S}) one gets a solution
for $\tilde{S}_{0}(r),$ that re-inserted in eq. (\ref{eq: S con 0})
gives \cite{key-7}
\begin{equation}
S_{0}(r,t)=\omega(t\pm r_{*}),\label{eq: S con 0 - 1}
\end{equation}
where $r_{*}\equiv\int\frac{dr}{1-\frac{2M}{r}}$. Now, one introduces
the sets of null tortoise coordinates defined as \cite{key-7}
\begin{equation}
\begin{array}{c}
u\equiv t-r_{*}\\
\\
v\equiv t+r_{*}.
\end{array}\label{eq: tortoise coordinates}
\end{equation}
These coordinates are defined inside and outside the event horizon
\cite{key-7}. If one expresses eq. (\ref{eq: S con 0 - 1}) in terms
of the tortoise coordinates (\ref{eq: tortoise coordinates}) and
then substitutes in eq. (4) , the right and left modes for both sectors
can be written down as \cite{key-7}
\begin{equation}
\begin{array}{c}
\left[\varphi^{\left(R\right)}\right]_{in}=\left[\exp\left(-i\omega u\right)\right]_{in};\quad\quad\left[\varphi^{\left(L\right)}\right]_{in}=\left[\exp\left(-i\omega v\right)\right]_{in}\\
\\
\left[\varphi^{\left(R\right)}\right]_{out}=\left[\exp\left(-i\omega u\right)\right]_{out}\quad\quad\left[\varphi^{\left(L\right)}\right]_{out}=\left[\exp\left(-i\omega v\right)\right]_{out}.
\end{array}\label{eq: modi fi}
\end{equation}
In the tunnelling framework, after the production of a virtual pair
of particles, one member of such a pair tunnels through the horizon
in a quantum mechanical way \cite{key-7}. The nature of the coordinates
changes while the particle crosses the horizon \cite{key-2,key-3,key-7}.
One takes into account this by using with Kruskal coordinates, which
are a type of coordinates viable on both sides of the horizon \cite{key-7}.
The Kruskal time coordinate $T\:$ and the Krsuskal space coordinate
$X\:$ are defined, inside and outside the horizon, as \cite{key-7,key-10,key-16}
\begin{equation}
\begin{array}{ccc}
T_{in}\equiv\left[\exp\left(\frac{r_{*_{in}}}{4M}\right)\right]\cosh\left(\frac{t_{in}}{4M}\right); & & X_{in}\equiv\left[\exp\left(\frac{r_{*_{in}}}{4M}\right)\right]\sinh\left(\frac{t_{in}}{4M}\right)\\
\\
T_{out}\equiv\left[\exp\left(\frac{r_{*_{out}}}{4M}\right)\right]\sinh\left(\frac{t_{out}}{4M}\right); & & X_{out}\equiv\left[\exp\left(\frac{r_{*_{out}}}{4M}\right)\right]\cosh\left(\frac{t_{out}}{4M}\right).
\end{array}\label{eq: in - out}
\end{equation}
The connection between the two sets of coordinates is given by \cite{key-7,key-10,key-16}
\begin{equation}
\begin{array}{c}
t_{in}\rightarrow t_{out}-2\pi iM\\
\\
r_{*_{in}}\rightarrow r_{*_{out}}+2\pi iM.
\end{array}\label{eq: in to out}
\end{equation}
Eq. (\ref{eq: in to out}) implies $T_{in}\rightarrow T_{out}\:$
and $T_{in}\rightarrow T_{out}$ \cite{key-7}. The relations which
connect the null coordinates defined inside and outside the horizon
can be obtained using the definition (\ref{eq: tortoise coordinates})
\cite{key-7}
\begin{equation}
\begin{array}{c}
u_{in}=t_{in}-r_{*_{in}}\rightarrow u_{out}-2\pi iM\\
\\
v_{in}=t_{in}+r_{*_{in}}\rightarrow v_{out}.
\end{array}\label{eq: u v in out}
\end{equation}
The two eqs. (\ref{eq: u v in out}) imply that the inside and outside
modes are connected by \cite{key-7}
\begin{equation}
\begin{array}{c}
\left[\varphi^{\left(R\right)}\right]_{in}=\exp\left(-4\pi M\omega\right)\left[\varphi^{\left(R\right)}\right]_{out}\\
\\
\left[\varphi^{\left(L\right)}\right]_{in}=\left[\varphi^{\left(L\right)}\right]_{out}.
\end{array}\label{eq: fi in out}
\end{equation}
In order to find the spectrum, one starts to consider $n\:$ non-interacting
virtual pairs created inside the black hole \cite{key-7}. The modes
defined in the first set of eq. (\ref{eq: modi fi}) represent each
pair \cite{key-7}. In that way, when observed from outside, the physical
state of the system can be written down by using the transformations
(14) \cite{key-7}
\begin{equation}
|\Psi>=N\sum_{n}|n_{in}^{(L)}>\otimes|n_{in}^{(R)}>\rightarrow N\sum_{n}\exp\left(-4\pi nM\omega\right)|n_{out}^{(L)}>\otimes|n_{out}^{(R)}>\label{eq: prodotto tensore}
\end{equation}
$N\:$ in eq. (15) is a normalization constant, while represents $|n_{out}^{(L)}>$
the number $n\:$ of left going modes \cite{key-7}. By applying the
normalization condition $<\Psi|\Psi>=1\:$ one gets \cite{key-7}
\begin{equation}
N=\left(\sum_{n}\exp\left(-8\pi nM\omega\right)\right)^{-\frac{1}{2}}.\label{eq: N}
\end{equation}
We recall that $n=0,1,2,3,....\:$ for bosons and $n=0,1\:$ for fermions
respectively \cite{key-7}. Hence, the two values of the normalization
constant are \cite{key-7}
\begin{equation}
\begin{array}{c}
N_{boson}=\left(1-\exp\left(-8\pi nM\omega\right)\right)^{\frac{1}{2}}\\
\\
N_{fermion}=\left(1+\exp\left(-8\pi nM\omega\right)\right)^{-\frac{1}{2}}.
\end{array}\label{eq: N bosons fermions}
\end{equation}
Thus, one writes down the (normalized) physical states of the system
for bosons and fermions as \cite{key-7}
\begin{equation}
\begin{array}{c}
|\Psi>_{boson}=\left(1-\exp\left(-8\pi nM\omega\right)\right)^{\frac{1}{2}}\sum_{n}\exp\left(-4\pi nM\omega\right)|n_{out}^{(L)}>\otimes|n_{out}^{(R)}>\\
\\
|\Psi>_{fermion}=\left(1+\exp\left(-8\pi nM\omega\right)\right)^{-\frac{1}{2}}\sum_{n}\exp\left(-4\pi nM\omega\right)|n_{out}^{(L)}>\otimes|n_{out}^{(R)}>
\end{array}\label{eq: physical states}
\end{equation}
Hereafter we focus the analysis only on bosons. In fact, for fermions
the analysis is identical \cite{key-7}. The density matrix operator
of the system is \cite{key-7}
\begin{equation}
\begin{array}{c}
\hat{\rho}_{boson}\equiv\Psi>_{boson}<\Psi|_{boson}\\
\\
=\left(1-\exp\left(-8\pi nM\omega\right)\right)\sum_{n,m}\exp\left[-4\pi(n+m)M\omega\right]|n_{out}^{(L)}>\otimes|n_{out}^{(R)}>_{boson}=tr\left[\hat{\rho}_{boson}^{(R)}\right]=\frac{1}{\exp\left(-8\pi nM\omega\right)-1},\label{eq: traccia}
\end{equation}
where the trace has been taken over all the eigenstates and the final
result has been obtained through a bit of algebra, see \cite{key-7}
for details. The result of eq. (\ref{eq: traccia}) is the well known
Bose-Einstein distribution. A similar analysis works also for fermions
\cite{key-7}, and one easily gets the well known Fermi-Dirac distribution
\begin{equation}
_{fermion}=\frac{1}{\exp\left(-8\pi nM\omega\right)+1},\label{eq: traccia 2}
\end{equation}
Both the distributions correspond to a black body spectrum with the
Hawking temperature \cite{key-1,key-7}
\begin{equation}
T_{H}\equiv\frac{1}{8\pi M}.\label{eq: Hawking temperature}
\end{equation}
\section{Non-thermal correction}
The result in \cite{key-7}, that we reviewed in previous Section,
is remarkable. In fact, we have see that, through a reformulation
of the tunnelling mechanism, one can found a black body spectrum associated
with black holes which is in perfect agreement with the famous original
result by Hawking \cite{key-1}. On the other hand, it is in contrast
with another remarkable result \cite{key-2,key-3}. In fact, the probability
of emission connected with the two distributions (21) and (22) is
given by \cite{key-1,key-2,key-3}
\begin{equation}
\Gamma\sim\exp(-\frac{\omega}{T_{H}}).\label{eq: hawking probability}
\end{equation}
But in \cite{key-2,key-3} a remarkable correction, through an exact
calculation of the action for a tunnelling spherically symmetric particle,
has been found, yielding
\begin{equation}
\Gamma\sim\exp[-\frac{\omega}{T_{H}}(1-\frac{\omega}{2M})].\label{eq: Parikh Correction}
\end{equation}
This important result, which is clearly in contrast with the result
in \cite{key-7}, that we reviewed in previous Section, enables a
correction, the additional term $\frac{\omega}{2M}$ \cite{key-2,key-3}.
The important difference is that the authors of \cite{key-7} did
not taken into due account the conservation of the energy, which generates
a dynamical instead of static geometry of the black hole \cite{key-2,key-3}.
In other words, the energy conservation forces the black hole to contract
during the process of radiation \cite{key-2,key-3}. Therefore, the
horizon recedes from its original radius, and, at the end of the emission,
the radius becomes smaller \cite{key-2,key-3}. The consequence is
that black holes do not strictly emit like black bodies \cite{key-2,key-3}.
It is important to recall that the tunnelling is a \emph{discrete}
instead of \emph{continuous} process \cite{key-8}. In fact, two different
\emph{countable} black hole's physical states must be considered,
the physical state before the emission of the particle and the physical
state after the emission of the particle \cite{key-8}. Thus, the
emission of the particle can be interpreted like a \emph{quantum}
\emph{transition} of frequency $\omega$ between the two discrete
states \cite{key-8}. In the language of the tunnelling mechanism,
a trajectory in imaginary or complex time joins two separated classical
turning points \cite{key-2,key-3}. Another important consequence
is that the radiation spectrum is also discrete \cite{key-8}. Let
us clarify this important issue in a better way. At a well fixed Hawking
temperature and the statistical probability distribution (\ref{eq: Parikh Correction})
are continuous functions. On the other hand, the Hawking temperature
in (\ref{eq: Parikh Correction}) varies in time with a character
which is \emph{discrete}. In fact, the forbidden region traversed
by the emitting particle has a \emph{finite} size \cite{key-3}. Considering
a strictly thermal approximation, the turning points have zero separation.
Therefore, it is not clear what joining trajectory has to be considered
because there is not barrier \cite{key-3}. The problem is solved
if we argue that the forbidden finite region from $r_{initial}=2M\:$
to $r_{final}=2(M\lyxmathsym{\textminus}\omega)\:$ that the tunnelling
particle traverses works like barrier \cite{key-3}. Thus, the intriguing
explanation is that it is the particle itself which generates a tunnel
through the horizon \cite{key-3}.
A good way to take into due account the dynamical geometry of the
black hole during the emission of the particle is to introduce the
black hole's \emph{effective state}. By introducing the \emph{effective
temperature }\cite{key-8,key-9}
\begin{equation}
T_{E}(\omega)\equiv\frac{2M}{2M-\omega}T_{H}=\frac{1}{4\pi(2M-\omega)},\label{eq: Corda Temperature}
\end{equation}
one re-writes eq. (\ref{eq: Corda Temperature}) in a Boltzmann-like
form similar to the original probability found by Hawking
\begin{equation}
\Gamma\sim\exp[-\beta_{E}(\omega)\omega]=\exp(-\frac{\omega}{T_{E}(\omega)}),\label{eq: Corda Probability}
\end{equation}
\noindent where $\exp[-\beta_{E}(\omega)\omega]$ is the \emph{effective
Boltzmann factor,} with $\beta_{E}(\omega)\equiv\frac{1}{T_{E}(\omega)}$
\cite{key-8,key-9}. Hence, the effective temperature replaces the
Hawking temperature in the equation of the probability of emission
\cite{key-8,key-9}. Let us discuss the physical interpretation. In
various fields of science, we can takes into account the deviation
from the thermal spectrum of an emitting body by introducing an effective
temperature. It represents the temperature of a black body that would
emit the same total amount of radiation\emph{. }We introduced the
concept of effective temperature in the black hole's physics in \cite{key-8,key-9}.
$T_{E}(\omega)$ depends on the energy-frequency of the emitted radiation
and the ratio $\frac{T_{E}(\omega)}{T_{H}}=\frac{2M}{2M-\omega}$
represents the deviation of the radiation spectrum of a black hole
from the strictly thermal feature \cite{key-8,key-9}.
The introduction of $T_{E}(\omega)$ permits the introduction of others
\emph{effective quantities}. In fact, let us consider the initial
mass of the black hole \emph{before} the emission, $M$, and the final
mass of the hole \emph{after} the emission, $M-\omega$ respectively
\cite{key-8,key-9}. The \emph{effective mass }and the \emph{effective
horizon} of the black hole \emph{during} its contraction, i.e. \emph{during}
the emission of the particle, are defined as \cite{key-8,key-9}
\begin{equation}
M_{E}\equiv M-\frac{\omega}{2},\mbox{ }r_{E}\equiv2M_{E}.\label{eq: effective quantities}
\end{equation}
\noindent The above effective quantities are average quantities\emph{
}\cite{key-8,key-9}. \emph{$r_{E}$ }is the average of the initial
and final horizons and \emph{$M_{E}\:$ }is the average of the initial
and final masses \cite{key-8,key-9}. Therefore, \emph{$T_{E}\:$
}is the inverse of the average value of the inverses of the initial
and final Hawking temperatures (\emph{before} the emission $T_{H\mbox{ initial}}=\frac{1}{8\pi M}$,
\emph{after} the emission $T_{H\mbox{ final}}=\frac{1}{8\pi(M-\omega)}$
respectively) \cite{key-8,key-9}. Thus, the Hawking temperature \emph{has
a discrete character in time}.
We stress that the introduction of the effective temperature does
not degrade the importance of the Hawking temperature. Indeed, as
the Hawking temperature changes with a discrete behavior in time,
it is not clear which value of such a temperature has to be associated
to the emission of the particle. Has one to consider the value of
the Hawking temperature \emph{before} the \emph{emission} or the value
of the Hawking temperature after the emission? The answer is that
one must consider an \emph{intermediate} value, the effective temperature,
which is the inverse of the average value of the inverses of the initial
and final Hawking temperatures. In a certain sense, it represents
the value of the Hawking temperature \emph{during} the emission. $T_{E}(\omega)$
takes into account the non-strictly thermal character of the radiation
spectrum and the non-strictly continuous character of subsequent emissions
of Hawking quanta.
Therefore, one can define two further effective quantities. The \emph{effective
Schwarzschild line element }is given by
\begin{equation}
ds^{2}\equiv-(1-\frac{2M_{E}}{r})dt^{2}+\frac{dr^{2}}{1-\frac{2M_{E}}{r}}+r^{2}(\sin^{2}\theta d\varphi^{2}+d\theta^{2}),\label{eq: Hilbert effective}
\end{equation}
and, consequently, the \emph{effective} \emph{surface gravity} is
defined as\emph{ }$\frac{1}{4M_{E}}.$ Thus, the effective line element
(29) takes into account the \emph{dynamical} geometry of the black
hole during the emission of the particle. Now, one can replace eq.
(\ref{eq: reduced Klein-Gordon}) for the $(r\lyxmathsym{\textminus}t)\:$
sector with the \emph{effective equation}
\begin{equation}
-\frac{1}{1-\frac{2M_{E}}{r})}\partial_{t}^{2}\varphi+\frac{1}{4M_{E}}\partial_{r}\varphi+(1-\frac{2M_{E}}{r})\partial_{r}^{2}\varphi=0.\label{eq: reduced Klein-Gordon effective}
\end{equation}
In analogous way, putting $r_{*_{E}}\equiv\int\frac{dr}{1-\frac{2M_{E}}{r}}$
the two eqs. (9) are replaced by the \emph{effective tortoise coordinates}
\begin{equation}
\begin{array}{c}
u\equiv t-r_{*_{E}}\\
\\
v\equiv t+r_{*_{E}}.
\end{array}\label{eq: tortoise coordinates effective}
\end{equation}
Clearly, if one follows step by step the analysis in Section 2, at
the end obtains the correct physical states for boson and fermions
as
\begin{equation}
\begin{array}{c}
|\Psi>_{boson}=\left(1-\exp\left(-8\pi nM_{E}\omega\right)\right)^{\frac{1}{2}}\sum_{n}\exp\left(-4\pi nM_{E}\omega\right)|n_{out}^{(L)}>\otimes|n_{out}^{(R)}>\\
\\
|\Psi>_{fermion}=\left(1+\exp\left(-8\pi nM_{E}\omega\right)\right)^{-\frac{1}{2}}\sum_{n}\exp\left(-4\pi nM_{E}\omega\right)|n_{out}^{(L)}>\otimes|n_{out}^{(R)}>
\end{array}\label{eq: physical states-1}
\end{equation}
and the correct distributions as
\begin{equation}
\begin{array}{c}
_{boson}=\frac{1}{\exp\left(-8\pi nM_{E}\omega\right)-1}=\frac{1}{\exp\left[-4\pi n\left(M-\omega\right)\omega\right]-1}\\
\\
_{fermion}=\frac{1}{\exp\left(-8\pi nM_{E}\omega\right)+1}=\frac{1}{\exp\left[-4\pi n\left(M-\omega\right)\omega\right]+1},
\end{array}\label{eq: final distributions}
\end{equation}
which represent the distributions associated to the probability of
emission (25).
We recall that, this deviation from strict thermality is consistent
with unitarity \cite{key-3,key-8,key-9} and has profound implications
for the black hole information puzzle because arguments that information
is lost during black hole's evaporation rely in part on the assumption
of strict thermal behavior of the spectrum \cite{key-3,key-8,key-9,key-17}.
In other words, the process of black hole's evaporation should be
unitary, information should be preserved and the underlying quantum
gravity theory should be unitary too.
\section{Conclusion remarks}
In the remarkable paper \cite{key-7} the tunnelling approach on Hawking
radiation has been improved by explicitly finding a black body spectrum
associated with black holes. But a problem is this result, which has
been obtained by using a reformulation of the tunnelling mechanism,
is in contrast which the other remarkable result in \cite{key-2,key-3},
that, indeed, found a probability of emission which is compatible
with a non-strictly thermal spectrum.
By using our recent introduction of an effective state for a black
hole \cite{key-8,key-9} in this paper we solved such a contradiction,
through a slight modification of the analysis in \cite{key-7}. The
final result consists in a non-strictly black body spectrum from tunnelling
mechanism.
\section{Acknowledgements}
It is a pleasure to thank Hossein Hendi, Erasmo Recami and Ram Gopal
Vishwakarma, and, in addition, my students Reza Katebi and Nathan
Schmidt, for useful discussions on black hole physics.
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