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Connected with the paper "Effective temperature for black holes" by the same author, published in JHEP 1108, 101 (2011).
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Effective temperature, black hole's entropy, quantum levels
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\begin{document}
\title{\textbf{Black hole's quantum levels}}
\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}
By introducing a black hole's \emph{effective temperature, }which
takes into account both of the non-strictly thermal and non-strictly
continuous characters of Hawking radiation, we recently re-analyzed
black hole's quasi-normal modes and interpreted them naturally in
terms of quantum levels for emissions of particles. After a careful
review of previous results, in this work we improve such an analysis
by removing an approximation that we implicitly used in our previous
work and by obtaining the corrected expressions for the formulas of
the horizon's area quantization and the number of quanta of area and
hence also for Bekenstein-Hawking entropy, its sub-leading corrections
and the number of micro-states, i.e. quantities which are fundamental
to realize unitary quantum gravity theory, like functions of the quantum
``overtone'' number $e$ (\textit{emission}) and, in turn, of the
black hole's quantum excited level. Another approximation concerning
the maximum value of $e$ is also corrected.
We also consider quasi-normal modes in terms of quantum levels for
\emph{absorptions} too, in addition to our previous analysis which
considered quasi-normal modes naturally associated to Hawking radiation
and hence to emissions only. In that case, the above cited quantities
result to be functions of the quantum ``overtone'' number $a$ (\emph{absorption}).
In this way, the whole black hole's quantum spectrum, for both of
absorption and emission is obtained.
Previous results in the literature are re-obtained in the limit of
very large ``overtone'' numbers $e$ and $a$ and of very small
quasi-normal mode's frequency.
The results of this paper are very important on the route to realize
the underlying unitary quantum gravity theory. In fact, black holes
are considered natural theoretical laboratories to test such a theory
which has to match the semi-classical results of the present paper.
\end{abstract}
\section{Introduction}
Parikh and Wilczek \cite{key-1,key-2} have shown that Hawking radiation's
spectrum \cite{key-3} cannot be strictly thermal. Such a non-strictly
thermal character implies that the spectrum is also not strictly continuous
and thus generates a natural correspondence between Hawking radiation
and black hole's quasi-normal modes \cite{key-4}. This issue endorses
the idea that, in an underlying unitary quantum gravity theory, black
holes result in highly excited states.
We recently used this key point to re-analyze the spectrum of black
hole's quasi-normal modes by introducing a black hole's \emph{effective
temperature }\cite{key-4,key-5}\emph{.} Our analysis changed the
physical understanding of such a spectrum and enabled a re-examination
of various results in the literature which realized important modifies
on quantum physics of black holes \cite{key-4,key-5}. In particular,
the formula of the horizon's area quantization and the number of quanta
of area were modified becoming functions of the quantum emission's
``overtone'' number $e$ \cite{key-4,key-5}. Consequently, Bekenstein-Hawking
entropy, its sub-leading corrections and the number of micro-states,
i.e. quantities which are fundamental to realize the underlying unitary
quantum gravity theory, were also modified \cite{key-4,key-5}. They
became functions of the emission's quantum ``overtone'' number $e$
too, while previous results in the literature were re-obtained in
the very large $e$ limit \cite{key-4,key-5}.
The analysis in \cite{key-4,key-5} permitted to naturally interpret
quasi-normal modes in terms of quantum levels in the case of \emph{emission}
(Hawking radiation).
Here we remove an approximation that was implicitly used in our previous
works \cite{key-4,key-5}, by obtaining the corrected expressions
for the cited formulas like functions of $e.$ Another approximation
concerning the maximum value of $e$ is also corrected.
The analysis is further improved by considering quasi-normal modes
in terms of quantum levels for \emph{absorption} too. In this way,
the whole black hole's quantum spectrum, for both of absorptions and
emissions are obtained.
In the case of absorptions, Bekenstein-Hawking entropy, its sub-leading
corrections and the number of micro-states result to be functions
of the quantum ``overtone'' number $a$ (\emph{absorption}).
Previous results in the literature are now re-obtained in the limit
of very large quantum ``overtone'' numbers $e$ and $a,$ and very
small quasi-normal mode's frequency.
\section{Effective temperature, Hawking radiation and quasi-normal modes:
a review}
\noindent Analyzing Hawking radiation \cite{key-3} as tunneling,
Parikh and Wilczek showed that the radiation spectrum cannot be strictly
thermal \cite{key-1,key-2}. Parikh released an intriguing physical
interpretation of this fundamental issue by discussing the existence
of a secret tunnel through the black hole's horizon \cite{key-1}.
The energy conservation implies that the black hole contracts during
the process of radiation \cite{key-1,key-2}. Thus, the horizon recedes
from its original radius to a new, smaller radius \cite{key-1,key-2}.
The consequence is that black holes cannot strictly emit thermally
\cite{key-1,key-2}. This is consistent with unitarity \cite{key-1}
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-1,key-2,key-6}.
\noindent Working with $G=c=k_{B}=\hbar=\frac{1}{4\pi\epsilon_{0}}=1$
(Planck units), the probability of emission is \cite{key-1,key-2,key-3}
\begin{equation}
\Gamma\sim\exp(-\frac{\omega}{T_{H}}),\label{eq: hawking probability}
\end{equation}
\noindent where $T_{H}\equiv\frac{1}{8\pi M}$ is the Hawking temperature
and $\omega$ the energy-frequency of the emitted radiation.
\noindent Parikh and Wilczek released a remarkable correction, due
to an exact calculation of the action for a tunneling spherically
symmetric particle, which yields \cite{key-1,key-2}
\noindent
\begin{equation}
\Gamma\sim\exp[-\frac{\omega}{T_{H}}(1-\frac{\omega}{2M})].\label{eq: Parikh Correction}
\end{equation}
\noindent This important result, which takes into account the conservation
of energy, enables a correction, the additional term $\frac{\omega}{2M}$
\cite{key-1,key-2}.
\noindent In various frameworks of physics and astrophysics the deviation
from the thermal spectrum of an emitting body is taken into account
by introducing an \emph{effective temperature }which represents the
temperature of a black body that would emit the same total amount
of radiation \cite{key-4,key-5}. The effective temperature can be
introduced for black holes too \cite{key-4,key-5}. It depends from
the energy-frequency of the emitted radiation and is defined as \cite{key-4,key-5}
\noindent
\begin{equation}
T_{E}(\omega)\equiv\frac{2M}{2M-\omega}T_{H}=\frac{1}{4\pi(2M-\omega)}.\label{eq: Corda Temperature}
\end{equation}
\noindent Then, eq. (\ref{eq: Parikh Correction}) can be rewritten
in Boltzmann-like form \cite{key-4,key-5}
\noindent
\begin{equation}
\Gamma\sim\exp[-\beta_{E}(\omega)\omega]=\exp(-\frac{\omega}{T_{E}(\omega)}),\label{eq: Corda Probability}
\end{equation}
\noindent where $\beta_{E}(\omega)\equiv\frac{1}{T_{E}(\omega)}$
and $\exp[-\beta_{E}(\omega)\omega]$ is the \emph{effective Boltzmann
factor} appropriate for an object with inverse effective temperature
$T_{E}(\omega)$ \cite{key-4,key-5}. 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-4,key-5}. If $M$ is
the initial mass of the black hole \emph{before} the emission, and
$M-\omega$ is the final mass of the hole \emph{after} the emission
\cite{key-2,key-4,key-5}, eqs. (\ref{eq: Parikh Correction}) and
(\ref{eq: Corda Temperature}) enable the introduction of the \emph{effective
mass }and of the \emph{effective horizon} \cite{key-4,key-5}
\begin{equation}
M_{E}\equiv M-\frac{\omega}{2},\mbox{ }r_{E}\equiv2M_{E}\label{eq: effective quantities}
\end{equation}
\noindent of the black hole \emph{during} the emission of the particle,
i.e. \emph{during} the contraction's phase of the black hole \cite{key-4,key-5}.
The \emph{effective quantities $T_{E},$ $M_{E}$ }and\emph{ $r_{E}$
}are average quantities. \emph{$M_{E}$ }is the average of the initial
and final masses, \emph{$r_{E}$ }is the average of the initial and
final horizons and \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 $T_{H\mbox{ final}}=\frac{1}{8\pi(M-\omega)}$) \cite{key-4}.
Notice that the analyzed process is \emph{discrete} rather than \emph{continuous}
\cite{key-4}. In fact, the black hole's state before the emission
of the particle and the black hole's state after the emission of the
particle are different countable black hole's physical states separated
by an \emph{effective state} which is characterized by the effective
quantities \cite{key-4}. Hence, the emission of the particle can
be interpreted like a \emph{quantum} \emph{transition} of frequency
$\omega$ between the two discrete states \cite{key-4}. The tunneling
visualization is that whenever a tunneling event works, two separated
classical turning points are joined by a trajectory in imaginary or
complex time \cite{key-1,key-4}.
\noindent In \cite{key-4,key-5} we have shown that the correction
to the thermal spectrum is also very important for the physical interpretation
of black hole's quasi-normal modes, which, in turn, results very important
to realize unitary quantum gravity theory as black holes are considered
theoretical laboratories for developing such an ultimate theory and
their quasi-normal modes are the best and most natural candidates
for an interpretation in terms of quantum levels \cite{key-4,key-5,key-7}.
\noindent The intriguing idea that black hole's quasi-normal modes
carry important information about black hole's area quantization is
due to the remarkable works by Hod \cite{key-8,key-9}. Hod's original
proposal found various objections over the years \cite{key-7,key-10}
which have been answered in a good way by Maggiore \cite{key-7},
who refined Hod's conjecture. Quasi-normal modes are also believed
to probe the small scale structure of the spacetime \cite{key-11}.
\noindent The quasi-normal frequencies are usually labelled as $\omega_{nl},$
where $l$ is the angular momentum quantum number \cite{key-4,key-5,key-7,key-12}.
For each $l$ ($l$$\geq2$ for gravitational perturbations), there
is a countable sequence of quasi-normal modes, labelled by the ``overtone''
number $n$ ($n=1,2,...$) \cite{key-4,key-5,key-7}. For large $n$
the quasi-normal frequencies of the Schwarzschild black hole become
independent of $l$ having the structure \cite{key-4,key-5,key-7,key-12}
\noindent
\begin{equation}
\begin{array}{c}
\omega_{n}=\ln3\times T_{H}+2\pi i(n+\frac{1}{2})\times T_{H}+\mathcal{O}(n^{-\frac{1}{2}})=\\
\\
=\frac{\ln3}{8\pi M}+\frac{2\pi i}{8\pi M}(n+\frac{1}{2})+\mathcal{O}(n^{-\frac{1}{2}}).
\end{array}\label{eq: quasinormal modes}
\end{equation}
\noindent This result was originally obtained numerically in \cite{key-13,key-14},
while an analytic proof was given later in \cite{key-15,key-16}.
\noindent The spectrum of black hole's quasi-normal modes can be analysed
in terms of superposition of damped oscillations, of the form \cite{key-4,key-5,key-7}
\begin{equation}
\exp(-i\omega_{I}t)[a\sin\omega_{R}t+b\cos\omega_{R}t]\label{eq: damped oscillations}
\end{equation}
\noindent with a spectrum of complex frequencies $\omega=\omega_{R}+i\omega_{I}.$
A damped harmonic oscillator $\mu(t)$ is governed by the equation
\cite{key-4,key-5,key-7}
\begin{equation}
\ddot{\mu}+K\dot{\mu}+\omega_{0}^{2}\mu=F(t),\label{eq: oscillatore}
\end{equation}
\noindent where $K$ is the damping constant, $\omega_{0}$ the proper
frequency of the harmonic oscillator, and $F(t)$ an external force
per unit mass. If $F(t)\sim\delta(t),$ i.e. considering the response
to a Dirac delta function, the result for $\mu(t)$ is a superposition
of a term oscillating as $\exp(i\omega t)$ and of a term oscillating
as $\exp(-i\omega t)$, see \cite{key-7} for details. Then, the behavior
(\ref{eq: damped oscillations}) is reproduced by a damped harmonic
oscillator, through the identifications \cite{key-4,key-5,key-7}
\noindent
\begin{equation}
\begin{array}{ccc}
\frac{K}{2}=\omega_{I}, & & \sqrt{\omega_{0}^{2}-\frac{K}{4}^{2}}=\omega_{R},\end{array}\label{eq: identificazioni}
\end{equation}
\noindent which gives
\begin{equation}
\omega_{0}=\sqrt{\omega_{R}^{2}+\omega_{I}^{2}}.\label{eq: omega 0}
\end{equation}
\noindent In \cite{key-7} it has been emphasized that the identification
$\omega_{0}=\omega_{R}$ is correct only in the approximation $\frac{K}{2}\ll\omega_{0},$
i.e. only for very long-lived modes. For a lot of black hole's quasi-normal
modes, for example for highly excited modes, the opposite limit can
be correct. Maggiore \cite{key-7} used this observation to re-examine
some aspects of quantum physics of black holes that were discussed
in previous literature assuming that the relevant frequencies were
$(\omega_{R})_{n}$ rather than $(\omega_{0})_{n}$.
\noindent A problem concerning attempts to associate quasi-normal
modes to Hawking radiation was that ideas on the continuous character
of Hawking radiation did not agree with attempts to interpret the
frequency of the quasi-normal modes \cite{key-15}. In fact, the discrete
character of the energy spectrum (\ref{eq: quasinormal modes}) should
be incompatible with the spectrum of Hawking radiation whose energies
are of the same order but continuous \cite{key-15}. Actually, the
issue that Hawking radiation is not strictly thermal and, as we have
shown, it has discrete rather than continuous character, removes the
above difficulty \cite{key-4}. In other words, the discrete character
of Hawking radiation permits to interpret the quasi-normal frequencies
$\omega_{nl}$ in terms of energies of physical Hawking quanta too
\cite{key-4}. In fact, quasi-normal modes are damped oscillations
representing the reaction of a black hole to small, discrete perturbations
\cite{key-4,key-5,key-7,key-8,key-9}. A discrete perturbation can
be the capture of a particle which causes an increase in the horizon
area \cite{key-7,key-8,key-9}. Hence, if the emission of a particle
which causes a decrease in the horizon area is a discrete rather than
continuous process, it is quite natural to assume that it is also
a perturbation which generates a reaction in terms of countable quasi-normal
modes \cite{key-4}. This natural correspondence between Hawking radiation
and black hole's quasi-normal modes permits to consider quasi-normal
modes in terms of quantum levels not only for absorbed energies like
in \cite{key-7,key-8,key-9}, but also for emitted energies like in
\cite{key-4,key-5}. This issue endorses the idea that, in an underlying
unitary quantum gravity theory, black holes can be considered highly
excited states \cite{key-4,key-5,key-7}.
\noindent The introduction of the effective temperature $T_{E}(\omega)$
can be applied to the analysis of the spectrum of black hole's quasi-normal
modes \cite{key-4,key-5}. Another key point is that eq. (\ref{eq: quasinormal modes})
is an approximation as it has been derived with the assumption that
the black hole's radiation spectrum is strictly thermal. To take into
due account the deviation from the thermal spectrum in eq. (\ref{eq: Parikh Correction})
one has to substitute the Hawking temperature $T_{H}$ with the effective
temperature $T_{E}$ in eq. (\ref{eq: quasinormal modes}) \cite{key-4,key-5}.
Therefore, the correct expression for the quasi-normal frequencies
of the Schwarzschild black hole, which takes into account the non-strictly
thermal behavior of the radiation spectrum is \cite{key-4,key-5}
\noindent
\begin{equation}
\begin{array}{c}
\omega_{e}=\ln3\times T_{E}(\omega_{e})+2\pi i(e+\frac{1}{2})\times T_{E}(\omega_{e})+\mathcal{O}(e^{-\frac{1}{2}})=\\
\\
=\frac{\ln3}{4\pi\left[2M-(\omega_{0})_{e}\right]}+\frac{2\pi i}{4\pi\left[2M-(\omega_{0})_{e}\right]}(e+\frac{1}{2})+\mathcal{O}(e^{-\frac{1}{2}}).
\end{array}\label{eq: quasinormal modes corrected}
\end{equation}
Notice that in eq. (\ref{eq: quasinormal modes corrected}) and in
the following the quantum ``overtone'' number for the emission's
levels is labelled $e$ (\textit{emission}) differently from \cite{key-4,key-5}
where it was labelled $n$. This is because here we will introduce
a second quantum ``overtone'' number $a$ (\emph{absorption}) for
the absorption's levels.
\noindent The important result (\ref{eq: quasinormal modes corrected})
can be explained as follows \cite{key-4,key-5}. Quasi-normal modes
are frequencies of the radial spin-j perturbations $\phi$ of the
four-dimensional Schwarzschild background which are governed by the
following master differential equation \cite{key-15,key-16}
\noindent
\begin{equation}
\left(-\frac{\partial^{2}}{\partial x^{2}}+V(x)-\omega^{2}\right)\phi.\label{eq: diff.}
\end{equation}
\noindent By introducing the Regge-Wheeler potential ($j=2$ for gravitational
perturbations) eq. (\ref{eq: diff.}) is treated as a Schrodinger
equation \cite{key-15,key-16}
\noindent
\begin{equation}
V(x)=V\left[x(r)\right]=\left(1-\frac{2M}{r}\right)\left(\frac{l(l+1)}{r^{2}}-\frac{6M}{r^{3}}\right).\label{eq: Regge-Wheeler}
\end{equation}
\noindent The relation between the Regge-Wheeler ``tortoise'' coordinate
$x$ and the radial coordinate $r$ is \cite{key-15,key-16}
\noindent
\begin{equation}
\begin{array}{c}
x=r+2M\ln\left(\frac{r}{2M}-1\right)\\
\\
\frac{\partial}{\partial x}=\left(1-\frac{2M}{r}\right)\frac{\partial}{\partial r}.
\end{array}\label{eq: tortoise}
\end{equation}
\noindent In \cite{key-15}, Motl derived eq. (\ref{eq: quasinormal modes})
with a rigorous analytical calculation which starts from eqs. (\ref{eq: diff.})
and (\ref{eq: Regge-Wheeler}) and satisfies purely outgoing boundary
conditions both at the horizon ($r=2M$) and in the asymptotic region
($r=\infty$). In order to take into due account the conservation
of energy, one has to substitute the original black hole's mass $M$
in eqs. (\ref{eq: diff.}) and (\ref{eq: Regge-Wheeler}) with the
effective mass of the contracting black hole defined in eq. (\ref{eq: effective quantities})
\cite{key-4,key-5}.
\noindent Hence, eqs. (\ref{eq: Regge-Wheeler}) and (\ref{eq: tortoise})
are replaced by the \emph{effective equations }\cite{key-4,key-5}
\noindent
\begin{equation}
V(x)=V\left[x(r)\right]=\left(1-\frac{2M_{E}}{r}\right)\left(\frac{l(l+1)}{r^{2}}-\frac{6M_{E}}{r^{3}}\right)\label{eq: effettiva 1}
\end{equation}
\noindent and
\begin{equation}
\begin{array}{c}
x=r+2M_{E}\ln\left(\frac{r}{2M_{E}}-1\right)\\
\\
\frac{\partial}{\partial x}=\left(1-\frac{2M_{E}}{r}\right)\frac{\partial}{\partial r}.
\end{array}\label{eq: effettiva 2}
\end{equation}
\noindent By realizing step by step the same rigorous analytical calculation
in \cite{key-13}, but starting from eqs. (\ref{eq: diff.}) and (\ref{eq: effettiva 1})
and satisfying purely outgoing boundary conditions both at the effective
horizon ($r_{E}=2M_{E}$) and in the asymptotic region ($r=\infty$),
the final result will be, obviously and rigorously, eq. (\ref{eq: quasinormal modes corrected})
\cite{key-4,key-5}.
\noindent An intuitive, elegant interpretation is the following \cite{key-4,key-5}.
The imaginary part of (\ref{eq: quasinormal modes}) can be easily
understood \cite{key-16}. The quasi-normal frequencies determine
the position of poles of a Green's function on the given background,
and the Euclidean black hole solution converges to a thermal circle
at infinity with the inverse temperature $\beta_{H}=\frac{1}{T_{H}}$
\cite{key-16}. Thus, the spacing of the poles in eq. (\ref{eq: quasinormal modes})
coincides with the spacing $2\pi iT_{H}$ expected for a thermal Green's
function \cite{key-16}. But, if one considers the deviation from
the thermal spectrum it is natural to assume that the Euclidean black
hole solution converges to a \emph{non-thermal} circle at infinity
\cite{key-4,key-5}. Therefore, it is straightforward the replacement
\cite{key-4,key-5}
\noindent
\begin{equation}
\beta_{H}=\frac{1}{T_{H}}\rightarrow\beta_{E}(\omega)=\frac{1}{T_{E}(\omega)},\label{eq: sostituiamo}
\end{equation}
\noindent which takes into account the deviation of the radiation
spectrum of a black hole from the strictly thermal feature. In this
way, the spacing of the poles in eq. (\ref{eq: quasinormal modes corrected})
coincides with the spacing \cite{key-4,key-5}
\noindent
\begin{equation}
2\pi iT_{E}(\omega)=2\pi iT_{H}(\frac{2M}{2M-\omega}),\label{eq: spacing}
\end{equation}
\noindent expected for a \emph{non-thermal} Green's function (a dependence
on the frequency is present) \cite{key-4,key-5}.
\noindent By using the new expression (\ref{eq: quasinormal modes corrected})
for the frequencies of quasi-normal modes, one defines \cite{key-4,key-5}
\begin{equation}
\begin{array}{ccc}
m_{0}\equiv\frac{\ln3}{4\pi[2M-(\omega_{0})_{e}]}, & & p_{e}\equiv\frac{2\pi}{4\pi[2M-(\omega_{0})_{e}]}(e+\frac{1}{2}).\end{array}\label{eq: definizioni}
\end{equation}
\noindent Then, eq. (\ref{eq: omega 0}) is re-written in the enlightening
form \cite{key-4,key-5}
\begin{equation}
(\omega_{0})_{e}=\sqrt{m_{0}^{2}+p_{e}^{2}}.\label{eq: enlightening}
\end{equation}
\noindent These results improve eqs. (8) and (9) in \cite{key-7}
as the new expression (\ref{eq: quasinormal modes corrected}) for
the frequencies of quasi-normal modes takes into account that the
radiation spectrum is not strictly thermal. For highly excited modes
one gets \cite{key-4,key-5}
\noindent
\begin{equation}
(\omega_{0})_{e}\approx p_{e}=\frac{2\pi}{4\pi[2M-(\omega_{0})_{e}]}(e+\frac{1}{2}).\label{eq: non equalmente spaziati}
\end{equation}
Thus, differently from \cite{key-7}, levels are \emph{not} equally
spaced even for highly excited modes \cite{key-4,key-5}. Indeed,
there are deviations due to the non-strictly thermal behavior of the
spectrum (black hole's effective temperature depends on the energy
level).
\noindent Using eq. (\ref{eq: definizioni}), one can re-write eq.
(\ref{eq: enlightening}) as \cite{key-4,key-5}
\noindent
\begin{equation}
(\omega_{0})_{e}=\frac{1}{4\pi[2M-(\omega_{0})_{e}]}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}},\label{eq: enlightening 2}
\end{equation}
\noindent which is easily solved giving \cite{key-4,key-5}
\begin{equation}
(\omega_{0})_{e}=M\pm\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}}.\label{eq: radici}
\end{equation}
\noindent As a black hole cannot emit more energy than its total mass,
the physical solution is the one obeying $(\omega_{0})_{e}