Content-Type: multipart/mixed; boundary="-------------1210260436345"

This is a multi-part message in MIME format.

---------------1210260436345
Content-Type: text/plain; name="12-129.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="12-129.comments"

Connected with the paper "Effective temperature for black holes" by the same author, published in JHEP 1108, 101 (2011).
---------------1210260436345
Content-Type: text/plain; name="12-129.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="12-129.keywords"

Effective temperature, black hole's entropy, quantum levels
---------------1210260436345
Content-Type: application/x-tex; name="Quantum levels.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="Quantum levels.tex"

%% LyX 2.0.2 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage{color}
\usepackage{amssymb}
\usepackage{babel}
\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 $(\omega_{0})_{e}<M$ \cite{key-4,key-5}
\begin{equation}
(\omega_{0})_{e}=M-\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}}.\label{eq: radice fisica}
\end{equation}


\noindent The interpretation is of a particle quantized with anti-periodic
boundary conditions on a circle of length \cite{key-4,key-5} 
\begin{equation}
L=\frac{1}{T_{E}(\omega_{0})_{e}}=4\pi\left(M+\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}}\right),\label{eq: lunghezza cerchio}
\end{equation}
 i.e. the length of the circle depends on the overtone number $e.$
Maggiore \cite{key-7} found a particle quantized with anti-periodic
boundary conditions on a circle of length $L=8\pi M.$ Our correction
takes into account the conservation of energy, i.e. the additional
term $\frac{\omega}{2M}$ in Eq. (\ref{eq: Parikh Correction}) \cite{key-4,key-5}. 

As $(\omega_{0})_{e}$ has to be a real number (an emitted energy),
we need also

\begin{equation}
M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}\geq0\label{eq: need}
\end{equation}
 

in eq. (\ref{eq: radice fisica}). The expression (\ref{eq: need})
is solved giving a maximum value for the emission's overtone number
$e$ \cite{key-4}

\begin{equation}
e\leq e_{max}=2\pi^{2}\left(\sqrt{16M^{4}-(\frac{\ln3}{\pi})^{2}}-1\right),\label{eq: n max}
\end{equation}


corresponding to $(\omega_{0})_{e_{max}}=M.$ Again, a black hole
cannot emit more energy than its total mass. Thus, the countable sequence
of quasi-normal modes for emitted energies is not infinity although
$e$ can be very large \cite{key-4}. By restoring ordinary units
in eq. (\ref{eq: n max}), one gets, for example, $e_{max}\sim10^{79}$
for a a black hole's mass of order 10 solar masses \cite{key-4}.

\noindent Various important consequences on the quantum physics of
black holes arise from the above approach \cite{key-4,key-5}. Let
us start with the \emph{area quantization}. 

\noindent Bekenstein \cite{key-17} showed that the area quantum of
the Schwarzschild black hole is $\triangle A=8\pi$ (notice that the
\emph{Planck length} $l_{p}=1.616\times10^{-33}\mbox{ }cm$ is equal
to one in Planck units). By using properties of the spectrum of Schwarzschild
black hole's quasi-normal modes, Hod found a different numerical coefficient
\cite{key-8,key-9}. Hod's analysis started by the observation that,
as for the Schwarzschild black hole the \emph{horizon area} $A$ is
related to the mass through the relation $A=16\pi M^{2},$ a variation
$\triangle M$ in the mass generates a variation

\noindent 
\begin{equation}
\triangle A=32\pi M\triangle M\label{eq: variazione area}
\end{equation}
 

\noindent in the area. 

\noindent An important criticism by Maggiore \cite{key-7} on Hod's
conjecture is that only transitions from the ground state (i.e. a
black hole which is not excited) to a state with large $a$ (or vice
versa) have been considered by Hod (notice that Maggiore and Hod considered
absorptions rather than emissions \cite{key-7,key-8,key-9}, hence
here we re-label the quantum ``overtone'' number $a$ (\emph{absorption})).
Actually, Bohr's correspondence principle strictly holds only for
transitions from $a$ to $a'$ where both $a,a'\gg1$ \cite{key-7}
and it is also legitimate to consider such transitions \cite{key-7}.
Thus, Maggiore suggested that $(\omega_{0})_{a}$ should be used rather
than $(\omega_{R})_{a}$ \cite{key-7}, re-obtaining the original
Bekenstein's result, i.e. $\triangle A=8\pi$. In any case, Maggiore's
result can be also improved if one takes into account the deviation
from the strictly thermal feature in eq. (\ref{eq: Parikh Correction}),
i.e. by using eq. (\ref{eq: quasinormal modes corrected}) rather
than eq. (\ref{eq: quasinormal modes}) \cite{key-4,key-5}. From
eq. (\ref{eq: radice fisica}) one sees that an emission involving
$e$ and $e-1$  gives a variation of energy 
\begin{equation}
\triangle M_{e}=(\omega_{0})_{e-1}-(\omega_{0})_{e}=-f_{e}(M,e)\label{eq: variazione}
\end{equation}


\noindent where we have defined \cite{key-4,key-5}

\noindent 
\begin{equation}
\begin{array}{c}
f_{e}(M,e)\equiv\\
\\
\equiv\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}}-\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}}.
\end{array}\label{eq: f(M,n)}
\end{equation}


\noindent The sign in (\ref{eq: variazione}), i.e. negative, is the
same of eq. (29) of \cite{key-4} and it is different with respect
to the correspondent eq. (30) in \cite{key-5} because here and in
\cite{key-4} we considered an emission while in \cite{key-5} we
considered an absorption.

\noindent By considering an absorption which generates a transition
from an unexcited black hole to a black hole with very large $a$,
Hod assumed \emph{Bohr's correspondence principle} to be valid for
large $a$ and enabled a semi-classical description even in absence
of a full unitary quantum gravity theory \cite{key-8,key-9}. Thus,
from eq. (\ref{eq: quasinormal modes}), the minimum quantum which
can be involved in the transition is $\triangle M_{a}=\omega=\frac{\ln3}{8\pi M}.$
This gives $\triangle A=4\ln3.$ The presence of the numerical factor
$4\ln3$ stimulated possible connections with loop quantum gravity
\cite{key-18}. 

\noindent Combining eqs. (\ref{eq: variazione area}) and (\ref{eq: variazione})
one gets \cite{key-4,key-5}

\noindent 
\begin{equation}
\triangle A=32\pi M\triangle M_{e}=-32\pi M\times f_{e}(M,e).\label{eq: area quantum}
\end{equation}


\noindent For very large $e$ (but we recall that $e\leq e_{max}$,
see eq. (\ref{eq: n max})) one obtains \cite{key-4,key-5}

\noindent 
\begin{equation}
\begin{array}{c}
f_{e}(M,e)\approx\\
\\
\approx\sqrt{M^{2}-\frac{1}{2}(e-\frac{1}{2})}-\sqrt{M^{2}-\frac{1}{2}(e+\frac{1}{2})}\approx\frac{1}{4M},
\end{array}\label{eq: circa}
\end{equation}


\noindent and eq. (\ref{eq: area quantum}) becomes $\triangle A\approx-8\pi$
which is the original result of Bekenstein for the area quantization
(a part a sign because we consider an emission rather than an absorption).
Then, only in the very large $e$ limit the levels are approximately
equally spaced \cite{key-4,key-5}. Indeed, for smaller $e$ there
are deviations, see eq. (\ref{eq: non equalmente spaziati}).

\noindent Important consequences on entropy and micro-states arise
from the above analysis \cite{key-4,key-5}.

\noindent Let us assume that, for large $e$, the horizon area is
quantized \cite{key-4,key-5,key-7} with a quantum $|\triangle A|=\alpha,$
where $\alpha=32\pi M\cdot f_{e}(M,e)$ for us \cite{key-4}, $\alpha=8\pi$
for Bekenstein \cite{key-17} and Maggiore \cite{key-7}, $\alpha=4\ln3$
for Hod \cite{key-8,key-9} and Dreyer \cite{key-18}. The total horizon
area must be $A=N|\triangle A|=N\alpha$ where the integer $N$ is
the number of quanta of area. Our approach gives \cite{key-4,key-5}

\noindent 
\begin{equation}
N=\frac{A}{|\triangle A|}=\frac{16\pi M^{2}}{\alpha}=\frac{16\pi M^{2}}{32\pi M\cdot f_{e}(M,e)}=\frac{M}{2f_{e}(M,e)}.\label{eq: N}
\end{equation}
 The famous formula of Bekenstein-Hawking entropy \cite{key-3,key-19,key-20}
now becomes \cite{key-4,key-5}

\noindent 
\begin{equation}
S_{BH}=\frac{A}{4}=8\pi NM|\triangle M_{e}|=8\pi NM\cdot f_{e}(M,e).\label{eq: Bekenstein-Hawking}
\end{equation}


\noindent Thus, we get the important result that Bekenstein-Hawking
entropy is a function of the quantum overtone number $e$.

\noindent In the very large $e$ limit eq. (\ref{eq: circa}) gives
$f_{e}(M,e)\rightarrow\frac{1}{4M}$ and the standard result \cite{key-7,key-21,key-22,key-23}

\noindent 
\begin{equation}
S_{BH}\rightarrow2\pi N\label{eq: entropia circa}
\end{equation}


\noindent is re-obtained \cite{key-4,key-5}.

\noindent On the other hand, it is a common and general belief that
there is no reason to expect that Bekenstein-Hawking entropy will
be the whole answer for a correct unitary quantum gravity theory \cite{key-4,key-5,key-24}.
For a better understanding of black hole's entropy one needs to go
beyond Bekenstein-Hawking entropy and identify the sub-leading corrections
\cite{key-4,key-5,key-24}. The quantum tunnelling approach can be
used to obtain the sub-leading corrections to the second order approximation
\cite{key-25}. One gets that the black hole's entropy contains three
parts: the usual Bekenstein-Hawking entropy, the logarithmic term
and the inverse area term \cite{key-25}

\noindent 
\begin{equation}
S_{total}=S_{BH}-\ln S_{BH}+\frac{3}{2A}.\label{eq: entropia totale}
\end{equation}


\noindent In fact, if one wants to satisfy the unitary quantum gravity
theory the logarithmic and inverse area terms are requested \cite{key-25}.
Apart from a coefficient, this correction to the black hole's entropy
is consistent with the one of loop quantum gravity \cite{key-25},
where the coefficient of the logarithmic term has been rigorously
fixed at $\frac{1}{2}$ \cite{key-25,key-26}. The correction (\ref{eq: Bekenstein-Hawking})
to Bekenstein-Hawking entropy permits to re-write eq. (\ref{eq: entropia totale})
as \cite{key-4,key-5}

\noindent 
\begin{equation}
S_{total}=8\pi NM\cdot f_{e}(M,e)-\ln\left[8\pi NM\cdot f_{e}(M,e)\right]+\frac{3}{64\pi NM\cdot f_{e}(M,e)}\label{eq: entropia totale 2}
\end{equation}


\noindent that in the very large $e$ limit becomes \cite{key-4,key-5}

\noindent 
\begin{equation}
S_{total}\rightarrow2\pi N-\ln2\pi N+\frac{3}{16\pi N}.\label{eq: entropia totale approssimata}
\end{equation}


\noindent These results imply that at level $N$ the black hole has
a number of micro-states \cite{key-4,key-5}

\noindent 
\begin{equation}
g(N)\propto\exp\left\{ 8\pi NM\cdot f_{e}(M,e)-\ln\left[8\pi NM\cdot f_{e}(M,e)\right]+\frac{3}{64\pi NM\cdot f_{e}(M,e)}\right\} ,\label{eq: microstati}
\end{equation}


\noindent that in the very large $e$ limit reads \cite{key-4,key-5}

\noindent 
\begin{equation}
g(N)\propto\exp\left[2\pi N-\ln\left(2\pi N\right)+\frac{3}{16\pi N}\right].\label{eq: microstati circa}
\end{equation}



\section{Removing some implicit simplifications}


\subsection{Varying mass of the black hole}

Actually, in previous Section, and hence also in \cite{key-4,key-5},
we used an implicit simplification. In this Subsection we improve
the analysis by removing such a simplification and by releasing the
correct results.

In fact, we note that in an emission from the ground state to a state
with large $e-1$ the mass of the black hole changes from $M$ to 

\begin{equation}
M_{e-1}\equiv M-(\omega_{0})_{e-1}\label{eq: me-1}
\end{equation}
 

and in the transition from state with $e-1$ to the state with $e$
it again changes from $M_{e-1}$ to

\begin{equation}
M_{e}\equiv M-(\omega_{0})_{e-1}+\triangle M_{e},\label{eq: me}
\end{equation}
 which, by using eq. (\ref{eq: variazione}), becomes 
\begin{equation}
\begin{array}{c}
M_{e}=M-(\omega_{0})_{e-1}-f_{e}(M,e)=\\
\\
=M-(\omega_{0})_{e-1}+(\omega_{0})_{e-1}-(\omega_{0})_{e}=M-(\omega_{0})_{e}.
\end{array}\label{eq: me 2}
\end{equation}


By considering eq. (\ref{eq: radice fisica}), eqs. (\ref{eq: me-1})
and (\ref{eq: me 2}) read

\begin{equation}
M_{e-1}=\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}}\label{eq: me 3}
\end{equation}


and 

\begin{equation}
M_{e}=\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}}.\label{eq: me 4}
\end{equation}


This implies that, if one uses eqs. (\ref{eq: radice fisica}) and
eq. (\ref{eq: me 3}), eq. (\ref{eq: area quantum}) has to be correctly
re-written as

\begin{equation}
\begin{array}{c}
\triangle A_{e-1}\equiv32\pi M_{e-1}\triangle M_{e}=-32\pi M_{e-1}\times f_{e}(M,e)=\\
\\
32\pi\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}}\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}}+\\
\\
-32\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}\right).
\end{array}\label{eq: area quantum e}
\end{equation}


This equation is very important as it gives the area quantum of an
excited black hole for an emission from the level $e-1$ to the level
$e$ in function of the quantum number $e$ and of the initial black
hole's mass.

In previous Section and in \cite{key-4,key-5}, we implicitly used
the simplification $(\omega_{0})_{e-1}\ll M$, i.e. the energy associated
to the quasi-normal frequency is much less than the mass-energy of
the black hole. Clearly, in that case the correction given by eq.
(\ref{eq: area quantum e}) results nonessential as one can neglect
the difference between the initial mass $M$ and the mass of the excited
black hole $M_{e-1}$, but it becomes very important when $(\omega_{0})_{e-1}\lesssim M$
i.e. in the last stages of the black hole's evaporation. In that case,
further corrections on formulas in previous Section and in \cite{key-4,key-5}
are needed. 

Putting $A_{e-1}\equiv16\pi M_{e-1}^{2}$, the formulas of the number
of quanta of area and of the Bekenstein-Hawking entropy become 

\noindent 
\begin{equation}
\begin{array}{c}
N_{e-1}\equiv\frac{A_{e-1}}{|\triangle A_{e-1}|}=\frac{16\pi M_{e-1}^{2}}{\alpha}=\frac{16\pi M_{e-1}^{2}}{32\pi M_{e-1}\cdot f_{e}(M,e)}=\frac{M_{e-1}}{2f_{e}(M,e)}=\\
\\
=\frac{\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}}}{2\left(\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}}-\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}}\right)}
\end{array}\label{eq: Ne}
\end{equation}


and 

\noindent 
\begin{equation}
\begin{array}{c}
\left(S_{BH}\right)_{e-1}\equiv\frac{A_{e-1}}{4}=8\pi N_{e-1}M_{e-1}|\triangle M_{e}|=8\pi N_{e-1}M_{e-1}\cdot f_{e}(M,e)=\\
\\
=4\pi\left(M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}\right).
\end{array}\label{eq: Bekenstein-Hawking e}
\end{equation}


respectively.

The formula of the total entropy that takes into account the sub-leading
corrections to Bekenstein-Hawking entropy becomes

\noindent 
\begin{equation}
\begin{array}{c}
\left(S_{total}\right)_{e-1}=8\pi N_{e-1}M_{e-1}\cdot f_{e}(M,e)-\ln\left[8\pi N_{e-1}M_{e-1}\cdot f_{e}(M,e)\right]+\frac{3}{64\pi N_{e-1}M_{e-1}\cdot f_{e}(M,e)}=\\
\\
=4\pi\left(M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}\right)+\\
\\
-\ln\left[4\pi\left(M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}\right)\right]+\frac{3}{32\pi\left(M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}\right)}
\end{array}\label{eq: entropia totale 3}
\end{equation}


which, in turn, implies that at level $N_{e-1}$ the black hole has
a number of micro-states 
\begin{equation}
\begin{array}{c}
g(N_{e-1})\propto\exp\left\{ 8\pi N_{e-1}M_{e-1}\cdot f_{e}(M,e)-\ln\left[8\pi N_{e-1}M_{e-1}\cdot f_{e}(M,e)\right]+\frac{3}{64\pi N_{e-1}M_{e-1}\cdot f_{e}(M,e)}\right\} =\\
\\
=\exp\{4\pi\left[M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}\right]+\\
\\
-\ln\left[4\pi\left(M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}\right)\right]+\frac{3}{32\pi\left(M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e-\frac{1}{2})^{2}}\right)}\}.
\end{array}\label{eq: microstati e}
\end{equation}


Again, all these corrections, which represent the correct formulas
of an excited black hole for an emission from the level $e-1$ to
the level $e$ in function of the quantum number $e$ and of the initial
black hole's mass, result very important in the last stages of the
black hole's evaporation, when $(\omega_{0})_{e-1}\lesssim M$.

Indeed, when $(\omega_{0})_{e-1}\ll M$ formulas (\ref{eq: area quantum}),
(\ref{eq: N}), (\ref{eq: entropia totale 2}) and (\ref{eq: microstati}),
are a good approximation. When both of the constrains $e\gg1$ and
$(\omega_{0})_{e}\ll M$ hold, formulas (\ref{eq: entropia circa}),
(\ref{eq: entropia totale approssimata}) and (\ref{eq: microstati circa})
can be used. They are in full agreement with previous results in the
literature \cite{key-7,key-24,key-25,key-26}.


\subsection{A note on black hole's remnants}

We recall that, by using the Generalized Uncertainty Principle, Adler,
Chen and Santiago \cite{key-27} have shown that the total evaporation
of a black hole is prevented in exactly the same way that the Uncertainty
Principle prevents the hydrogen atom from total collapse. In fact,
the collapse is prevented, not by symmetry, but by dynamics, as the
\emph{Planck distance} and the \emph{Planck mas}s are approached \cite{key-27}.
That important result implies that eq. (\ref{eq: need}) has to be
slightly modified, becoming (the Planck mass is equal to $1$ in Planck
units)

\begin{equation}
M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(e+\frac{1}{2})^{2}}\geq1.\label{eq: need 1}
\end{equation}


By solving eq. (\ref{eq: need 1}) one gets a different value of the
maximum value for the emission's overtone number $e$ 

\begin{equation}
e\leq e_{max}=2\pi^{2}\left(\sqrt{16(M^{2}-1)^{2}-(\frac{\ln3}{\pi})^{2}}-1\right).\label{eq: n max 1}
\end{equation}


By restoring ordinary units in eq. (\ref{eq: n max 1}), one gets
again $e_{max}\sim10^{79}$ for a a black hole's mass of order 10
solar masses, i.e. the order of magnitude of $e_{max}$ remains the
same. The difference becomes important only for very micro black holes,
i.e. when the original black hole's mass is of the order of the Planck
mass. 


\section{The absorption spectrum}

Now, we further improve the analysis by computing the absorption spectrum
of the Schwarzschild black hole.

In case of an absorption the expression for the quasi-normal frequencies
of the black hole results slightly modified with respect to eq. (\ref{eq: quasinormal modes corrected}),
which concerns the emission's frequencies, becoming 
\begin{equation}
\begin{array}{c}
\omega_{a}=\ln3\times T_{E}(\omega_{a})+2\pi i(a+\frac{1}{2})\times T_{E}(\omega_{a})+\mathcal{O}(a^{-\frac{1}{2}})=\\
\\
=\frac{\ln3}{4\pi\left[2M+(\omega_{0})_{a}\right]}+\frac{2\pi i}{4\pi\left[2M+(\omega_{0})_{a}\right]}(a+\frac{1}{2})+\mathcal{O}(a^{-\frac{1}{2}}),
\end{array}\label{eq: quasinormal modes corrected 2}
\end{equation}


where now $a$ is the quantum ``overtone'' number for the \emph{absorption's
levels}.

In fact, in case of an absorption the effective temperature (\ref{eq: Corda Temperature})
can be easily re-defined as 
\begin{equation}
T_{E}(\omega)\equiv\frac{2M}{2M+\omega}T_{H}=\frac{1}{4\pi(2M+\omega)}.\label{eq: Corda Temperature-1}
\end{equation}


In order to proof the correctness of eq. (\ref{eq: quasinormal modes corrected 2})
one can use the same discussion in Section 2 by using eqs. from (\ref{eq: diff.})
to (\ref{eq: effettiva 2}), but re-defining the effective mass\emph{
}and of the effective horizon as 
\begin{equation}
M_{E}\equiv M+\frac{\omega}{2},\mbox{ }r_{E}\equiv2M_{E}.\label{eq: effective quantities 2}
\end{equation}


Again, if one realizes 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 ($r_{E}=2M_{E}$ with $M_{E}$ given now by eq. (\ref{eq: effective quantities 2}))
and in the asymptotic region ($r=\infty$), the final result will
be, obviously and rigorously, eq. (\ref{eq: quasinormal modes corrected 2}).
Clearly\emph{ $M_{E}$ }is again the average of the initial and final
masses, \emph{$r_{E}$ }is again the average of the initial and final
horizons and \emph{$T_{E}$ }is again the inverse of the average value
of the inverses of the initial and final Hawking temperatures. Now
it is $T_{H\mbox{ initial}}=\frac{1}{8\pi M}$ \emph{before} the absorption
and $T_{H\mbox{ final}}=\frac{1}{8\pi(M+\omega)}$ \emph{after} the
absorption. In other words, \emph{$M_{E}$ }and $r_{E}$ are the effective
mass\emph{ }and of the effective horizon of the black hole \emph{during}
the absorption of the particle, i.e. \emph{during} the expansion's
phase of the black hole. 

The intuitive interpretation is the same of the quasi-normal frequencies
in case of emissions. The deviation of the spectrum of black hole's
quasi-normal modes from the strictly thermal feature implies that
the spacing of the poles in eq. (\ref{eq: quasinormal modes corrected 2})
coincides with the spacing 

\noindent 
\begin{equation}
2\pi iT_{E}(\omega)=2\pi iT_{H}(\frac{2M}{2M+\omega}),\label{eq: spacing 2}
\end{equation}


\noindent expected for a \emph{non-thermal} Green's function. In fact,
a dependence on the frequency is present in eq. (\ref{eq: spacing 2})
too.

\noindent By using the expression (\ref{eq: quasinormal modes corrected 2})
for the absorption's frequencies of quasi-normal modes, one re-writes
eqs. (\ref{eq: definizioni}) and (\ref{eq: enlightening}) as 
\begin{equation}
\begin{array}{ccc}
m_{0}\equiv\frac{\ln3}{4\pi[2M+(\omega_{0})_{a}]}, &  & p_{a}\equiv\frac{2\pi}{4\pi[2M+(\omega_{0})_{a}]}(a+\frac{1}{2}).\end{array}\label{eq: definizioni 2}
\end{equation}


\noindent and 
\begin{equation}
(\omega_{0})_{a}=\sqrt{m_{0}^{2}+p_{a}^{2}}.\label{eq: enlightening 3}
\end{equation}


\noindent respectively. For highly excited modes we find now 

\noindent 
\begin{equation}
(\omega_{0})_{a}\approx p_{a}=\frac{2\pi}{4\pi[2M+(\omega_{0})_{a}]}(a+\frac{1}{2}).\label{eq: non equalmente spaziati 2}
\end{equation}
 Hence, levels are not equally spaced even for absorption's highly
excited modes but there are deviations due to the non-strictly thermal
behavior of the spectrum. In other words, the energy's conservation
is taken into due account for absorptions too.

\noindent Using eq. (\ref{eq: definizioni 2}),  one can re-write
eq. (\ref{eq: enlightening 3}) as 

\noindent 
\begin{equation}
(\omega_{0})_{a}=\frac{1}{4\pi[2M+(\omega_{0})_{a}]}\sqrt{(\ln3)^{2}+4\pi^{2}(a+\frac{1}{2})^{2}}.\label{eq: enlightening 4}
\end{equation}


The solution of eq. (\ref{eq: enlightening 4}) is

\noindent 
\begin{equation}
(\omega_{0})_{a}=-M\pm\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a+\frac{1}{2})^{2}}}.\label{eq: radici a}
\end{equation}


Clearly, a black hole cannot absorb negative energies which are greater
than its mass in absolute value. Hence, the physical solution is the
one obeying $(\omega_{0})_{a}<|M|$, i.e. 
\begin{equation}
(\omega_{0})_{a}=\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a+\frac{1}{2})^{2}}}-M.\label{eq: phys. sol. a}
\end{equation}


\noindent The interpretation is of a particle quantized with anti-periodic
boundary conditions on a circle of length

\noindent 
\begin{equation}
L=\frac{1}{T_{E}(\omega_{0})_{a}}=4\pi\left(M+\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a+\frac{1}{2})^{2}}}\right),\label{eq: lunghezza cerchio a}
\end{equation}
 i.e. the length of the circle depends on the absorption's ``overtone''
number $a.$ 

\noindent From eq. (\ref{eq: phys. sol. a}) one sees that an absorption
involving $a$ and $a-1$  gives a variation of energy 
\begin{equation}
\triangle M_{a}=(\omega_{0})_{a}-(\omega_{0})_{a-1}=f_{a}(M,a)\label{eq: variazione a}
\end{equation}


\noindent where we have defined 

\noindent 
\begin{equation}
\begin{array}{c}
f_{a}(M,a)\equiv\\
\\
\equiv\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a+\frac{1}{2})^{2}}}-\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}}.
\end{array}\label{eq: f(M,n) a}
\end{equation}


We note that in an absorption from the ground state to a state with
large $a-1$ the mass of the black hole changes from $M$ to 

\begin{equation}
M_{a-1}\equiv M+(\omega_{0})_{a-1}\label{eq: ma-1}
\end{equation}
 

and in the absorption from state with $a-1$ to the state with $a$
it again changes from $M_{a-1}$ to

\begin{equation}
M_{a}\equiv M+(\omega_{0})_{a-1}+\triangle M_{a},\label{eq: ma}
\end{equation}
 which, by using eq. (\ref{eq: variazione a}), becomes 
\begin{equation}
\begin{array}{c}
M_{a}=M+(\omega_{0})_{a-1}+f_{a}(M,a)=\\
\\
=M+(\omega_{0})_{a-1}-(\omega_{0})_{a-1}+(\omega_{0})_{a}=M+(\omega_{0})_{a}.
\end{array}\label{eq: ma 2}
\end{equation}


By considering eq. (\ref{eq: phys. sol. a}), eqs. (\ref{eq: ma-1})
and (\ref{eq: ma 2}) read

\begin{equation}
M_{a-1}=\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}}\label{eq: ma 3}
\end{equation}


and 

\begin{equation}
M_{a}=\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a+\frac{1}{2})^{2}}}.\label{eq: ma 4}
\end{equation}


By using eqs. (\ref{eq: phys. sol. a}) and eq. (\ref{eq: ma 3}),
the equation of the area quantum (\ref{eq: variazione area}) becomes
\begin{equation}
\begin{array}{c}
\triangle A_{a-1}\equiv32\pi M_{a-1}\triangle M_{a}=32\pi M_{a-1}\times f_{a}(M,a)=\\
\\
32\pi\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}}\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a+\frac{1}{2})^{2}}}+\\
\\
-32\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}\right).
\end{array}\label{eq: area quantum a}
\end{equation}


Eq. (\ref{eq: area quantum a}) gives the area quantum of an excited
black hole for an absorption from the level $a-1$ to the level $a$
in function of the quantum number $a$ and of the initial black hole's
mass.

Putting $A_{a-1}\equiv16\pi M_{a-1}^{2}$, the formula of the number
of quanta of area is 
\begin{equation}
\begin{array}{c}
N_{a-1}\equiv\frac{A_{a-1}}{\triangle A_{a}}=\frac{16\pi M_{a-1}^{2}}{\alpha}=\frac{16\pi M_{a-1}^{2}}{32\pi M_{a-1}\cdot f_{a}(M,e)}=\frac{M_{a-1}}{2f_{a}(M,a)}=\\
\\
=\frac{\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}}}{2\left(\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a+\frac{1}{2})^{2}}}-\sqrt{M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}}\right)}.
\end{array}\label{eq: Na}
\end{equation}


Hence, we obtain the formula of the Bekenstein-Hawking entropy for
quantum levels in case of absorption 
\begin{equation}
\begin{array}{c}
\left(S_{BH}\right)_{a-1}\equiv\frac{A_{a-1}}{4}=8\pi N_{a-1}M_{a-1}\triangle M_{a}=8\pi N_{a-1}M_{a-1}\cdot f_{a}(M,a)=\\
\\
=4\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}\right).
\end{array}\label{eq: Bekenstein-Hawking a}
\end{equation}
The formula of the total entropy that takes into account the sub-leading
corrections to Bekenstein-Hawking entropy becomes

\noindent 
\begin{equation}
\begin{array}{c}
\left(S_{total}\right)_{a-1}=8\pi N_{a-1}M_{a-1}\cdot f_{a}(M,a)-\ln\left[8\pi N_{a-1}M_{a-1}\cdot f_{a}(M,a)\right]+\frac{3}{64\pi N_{a-1}M_{a-1}\cdot f_{a}(M,a)}=\\
\\
=4\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}\right)+\\
\\
-\ln\left[4\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}\right)\right]+\frac{3}{32\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}\right)}
\end{array}\label{eq: entropia totale 3 a}
\end{equation}


which, in turn, implies that at level $N_{a-1}$ in case of absorption
the black hole has a number of micro-states 
\begin{equation}
\begin{array}{c}
g(N_{a-1})\propto\exp\left\{ 8\pi N_{a-1}M_{a-1}\cdot f_{a}(M,a)-\ln\left[8\pi N_{a-1}M_{a-1}\cdot f_{a}(M,a)\right]+\frac{3}{64\pi N_{a-1}M_{a-1}\cdot f_{a}(M,a)}\right\} =\\
\\
=\exp\{4\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}\right)+\\
\\
-\ln\left[4\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}\right)\right]+\frac{3}{32\pi\left(M^{2}+\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(a-\frac{1}{2})^{2}}\right)}\}.
\end{array}\label{eq: microstati a}
\end{equation}


In the limit $(\omega_{0})_{a-1}\ll M$ (the energy associated to
the quasi-normal mode is much less than the mass-energy of the black
hole) we get $M_{a-1}\approx M,$ $A_{a-1}\approx A,$ $N_{a-1}\approx N,$
and eqs. (\ref{eq: Bekenstein-Hawking a}), (\ref{eq: entropia totale 3 a})
and (\ref{eq: microstati a}) read 
\begin{equation}
\left(S_{BH}\right)_{a-1}\approx\frac{A}{4}=8\pi NM\triangle M_{a}=8\pi NM\cdot f_{a}(M,a),\label{eq: Bekenstein-Hawking a 1}
\end{equation}
\begin{equation}
\left(S_{total}\right)_{a-1}\approx8\pi NM\cdot f_{a}(M,a)-\ln\left[8\pi NM\cdot f_{a}(M,a)\right]+\frac{3}{64\pi NM\cdot f_{a}(M,a)},\label{eq: entropia totale 3 a-1}
\end{equation}


and 
\begin{equation}
g(N_{a-1})\approx\exp\left\{ 8\pi NM\cdot f_{a}(M,a)-\ln\left[8\pi NM\cdot f_{a}(M,a)\right]+\frac{3}{64\pi NM\cdot f_{a}(M,a)}\right\} ,\label{eq: microstati a-1}
\end{equation}


respectively. On the other hand, if $a$ is enough large we get 
\begin{equation}
\begin{array}{c}
f_{a}(M,a)\approx\\
\\
\approx\sqrt{M^{2}+\frac{1}{2}(a+\frac{1}{2})}-\sqrt{M^{2}+\frac{1}{2}(a-\frac{1}{2})}\approx\frac{1}{4M}
\end{array}\label{eq: circa a}
\end{equation}


and eqs. (\ref{eq: Bekenstein-Hawking a 1}), (\ref{eq: entropia totale 3 a-1})
and (\ref{eq: microstati a-1}) reduce to 
\begin{equation}
\left(S_{BH}\right)_{a-1}\approx2\pi N,\label{eq: entropia circa-1}
\end{equation}
\begin{equation}
\left(S_{total}\right)_{a-1}\approx2\pi N-\ln\left(2\pi N\right)+\frac{3}{16\pi N}\label{eq: entropia totale approssimata-1}
\end{equation}
and 
\begin{equation}
g(N_{a-1})\approx\exp\left[2\pi N-\ln\left(2\pi N\right)+\frac{3}{16\pi N}\right],\label{eq: microstati circa-1}
\end{equation}


respectively, which, in turn, are in perfect agreement with (\ref{eq: entropia circa}),
(\ref{eq: entropia totale approssimata}) and (\ref{eq: microstati circa})
and with previous results in the literature \cite{key-7,key-24,key-25,key-26}.


\section{Conclusion remarks}

In this paper we analyzed black hole's quasi-normal modes in terms
of quantum levels following the idea that, in an underlying unitary
quantum gravity theory, black holes result in highly excited states.
By using the concept of effective temperature, we took into account
the important issue that quasi-normal modes' spectrum is not strictly
thermal in both of emission and absorption. The obtained results look
particularly intriguing as important modifies on quantum physics of
black holes have been realized. In fact, the formula of the horizon's
area quantization and the number of quanta of area result functions
of the quantum ``overtone'' number $e$ in case of emission and
of the quantum ``overtone'' number $a$ in case of absorption. 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, become functions of
the quantum ``overtone'' numbers too, $e$ for emission and $a$
for absorption. In other words, the cited important quantities result
to depend on the excited quantum state of the black hole in both of
the cases of emission and absorption.

An approximation concerning the maximum value of $e$ has been also
corrected.

It is important to emphasize that previous results in the literature
were re-obtained in the very large $e,a$ limits when the energy associated
to the quasi-normal frequency is much less than the mass-energy of
the black hole. This issue confirms the correctness of the analysis
in this paper which improves previous approximations. 
\begin{thebibliography}{10}
\bibitem{key-1}M. K. Parikh, Gen. Rel. Grav. 36, 2419 (2004, First
Award at Gravity Research Foundation).

\bibitem[2]{key-2}M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85,
5042 (2000).

\bibitem[3]{key-3}S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).

\bibitem[4]{key-4}C. Corda, arXiv:1205.5251 (2012, Honorable Mention
at Gravity Research Foundation, to appear in a Special Issue of Int.
Journ. Mod. Phys. D).

\bibitem[5]{key-5}C. Corda, JHEP 1108, 101 (2011). 

\bibitem[6]{key-6}S. W. Hawking, Phys. Rev. D 14, 2460 (1976).

\bibitem[7]{key-7}M. Maggiore, Phys. Rev. Lett. 100, 141301 (2008).

\bibitem[8]{key-8}S. Hod, Phys. Rev. Lett. 81 4293 (1998).

\bibitem[9]{key-9}S. Hod, Gen. Rel. Grav. 31, 1639 (1999, Fifth Award
at Gravity Research Foundation).

\bibitem[10]{key-10}R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys.
83, 793-836 (2011).

\bibitem[11]{key-11}M. Casals, Ad. C. Ottewill, arXiv:1112.2695v2
{[}gr-qc{]} (2011).

\bibitem[12]{key-12}T. Padmanabhan, Class. Quant. Grav. 21, L1 (2004).

\bibitem[13]{key-13}H. P. Nollert, Phys. Rev. D 47, 5253 (1993).

\bibitem[14]{key-14}N. Andersson, Class. Quant. Grav. 10, L61 (1993).

\bibitem[15]{key-15}L. Motl, Adv. Theor. Math. Phys. 6, 1135 (2003).

\bibitem[16]{key-16}L. Motl and A. Neitzke, Adv. Theor. Math. Phys.
7, 307 (2003).

\bibitem[17]{key-17}J. D. Bekenstein, Lett. Nuovo Cim. 11, 467 (1974). 

\bibitem[18]{key-18}O. Dreyer, Phys. Rev. Lett. 90, 081301 (2003). 

\bibitem[19]{key-19}J. D. Bekenstein, Nuovo Cim. Lett. 4, 737 (1972). 

\bibitem[20]{key-20}J. D. Bekenstein, Phys. Rev. D7, 2333 (1973). 

\bibitem[21]{key-21}A. Barvinsky and G. Kunstatter, arXiv:gr-qc/9607030v1
(1996). 

\bibitem[22]{key-22}A. Barvinsky S. Das and G. Kunstatter, Class.
Quant. Grav. 18, 4845 (2001). 

\bibitem[23]{key-23}D. Kothawala, T. Padmanabhan, S. Sarkar, Phys.
Rev. D 78, 104018 (2008). 

\bibitem[24]{key-24}S. Shankaranarayanan, Mod. Phys. Lett. A 23,
1975-1980 (2008). 

\bibitem[25]{key-25}J. Zhang, Phys. Lett. B 668, 353-356 (2008). 

\bibitem[26]{key-26}A. Ghosh, P. Mitra, Phys. Rev. D 71, 027502 (2005). 

\bibitem[27]{key-27}R. J. Adler, P. Chen and D. I. Santiago, Gen.
Rel. Grav. 3, 2101-2108 (2001, Third Award at Gravity Research Foundation).\end{thebibliography}

\end{document}
---------------1210260436345--
