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Black hole evaporation; Information loss paradox
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\begin{document}
\title{\textbf{Time dependent Schroedinger equation for black holes: No
information loss}}
\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}
Introducing a black hole's \emph{effective temperature}, we obtained
an Honorable Mention in the 2012 Gravity Research Foundation Essay
Competition interpreting black hole's quasi-normal modes naturally
in terms of quantum levels. This permitted us to show that quantities
which are fundamental to realize the underlying quantum gravity theory,
like Bekenstein-Hawking entropy, its sub-leading corrections and the
number of microstates, are function of the black hole's excited state,
i.e. of the black hole's quantum level, which is symbolized by the
quantum \textquotedblleft{}overtone\textquotedblright{} number $n$
that denotes the countable sequence of quasi-normal modes. Here, we
improve the analysis by finding a fundamental equation that directly
connects the probability of emission of an Hawking quantum with the
two emission levels which are involved in the transition. That equation
permits us to interpret the correspondence between Hawking radiation
and black hole quasi-normal modes in terms of a \emph{time dependent
Schroedinger system}. In such a system, the quasi-normal modes energies,
which are also the total energies emitted by the black hole in correspondence
of the various quantum levels, represent the eigenvalues of the unperturbed
Hamiltonian of the system and the Hawking quanta represent the energy
transitions among the eigenvalues, which correspond to a perturbed
Hamiltonian $\propto\delta(t)$. In this way, we explicitly write
down a \emph{time dependent Schroedinger equation }for the system
composed by Hawking radiation and black hole quasi-normal modes. The
states of the correspondent \emph{Schroedinger wave-function }can
be written in terms of a \emph{unitary} evolution matrix instead of
a density matrix. Thus, they result to be \emph{pure} states instead
of mixed ones. Hence, we conclude with the non-trivial consequence
that information comes out in black hole's evaporation. This issue
is also a confirmation of the assumption by 't Hooft that Schroedinger
equations can be used universally for all dynamics in the universe,
further endorsing the conclusion that black hole evaporation must
be information preserving.
\end{abstract}
The introduction of a black hole's \emph{effective temperature }\cite{key-1,key-2}\emph{,
}which takes into account both of the non-strictly thermal \cite{key-3,key-4}
and non-strictly continuous \cite{key-1} characters of Hawking radiation
\cite{key-5}, recently permitted us to re-analyse black hole's quasi-normal
modes and to interpret them naturally in terms of quantum levels for
emissions of particles \cite{key-1,key-2}. Here, we further improve
the analysis. A fundamental equation that directly connects the probability
of emission of an Hawking quantum with the two emission levels which
are involved in the transition is found. That equation enables to
interpret the correspondence between black hole quasi-normal modes
and Hawking radiation in terms of a time dependent \emph{Schroedinger
system.} The quasi-normal modes energies, which are also the total
energies emitted by the black hole in correspondence of the various
quantum levels, represent the eigenvalues of the unperturbed Hamiltonian
of the system and the Hawking quanta represent the energy transitions
among the eigenvalues corresponding to a perturbed Hamiltonian $\propto\delta(t)$.
Then, we can write down a \emph{time dependent Schroedinger equation
}for the system composed by Hawking radiation and black hole's quasi-normal
modes. The states of the correspondent \emph{Schroedinger wave-function}
are written in terms of a \emph{unitary} evolution matrix instead
of a density matrix. Thus, they result to be \emph{pure} states instead
of mixed ones. The conclusion implies the non-trivial consequence
that there is not information loss in black hole evaporation, in agreement
with the use of Feynman sum over histories \cite{key-6}, with the
results of string theory \cite{key-7}, and with the recent result
based on the Kerr-Schild formalism in Kerr-Newman black holes \cite{key-8}.
Hence, the final conclusion is that black hole evaporation must be
information preserving, confirming the assumption by 't Hooft that
Schroedinger equations can be used universally for all dynamics in
the universe \cite{key-24}.
\noindent Working with $G=c=k_{B}=\hbar=\frac{1}{4\pi\epsilon_{0}}=1$
(Planck units), the strictly thermal approximation gives the probability
of emission \cite{key-1}-\cite{key-5}
\begin{equation}
\Gamma\sim\exp(-\frac{\omega}{T_{H}}),\label{eq: hawking probability}
\end{equation}
\noindent where $\omega$ is the energy-frequency of the emitted particle
and $T_{H}\equiv\frac{1}{8\pi M}$ the Hawking temperature. A more
precise computation in the tunnelling framework, which takes into
account the energy conservation, i.e. the contraction of the black
hole which enables a back reaction due to the varying geometry, gives
a remarkable correction \cite{key-3,key-4}
\begin{equation}
\Gamma\sim\exp[-\frac{\omega}{T_{H}}(1-\frac{\omega}{2M})],\label{eq: Parikh Correction}
\end{equation}
where the additional term $\frac{\omega}{2M}\:$ is present \cite{key-3,key-4}.
By introducing the \emph{effective temperature }\cite{key-1,key-2}
\begin{equation}
T_{E}(\omega)\equiv\frac{2M}{2M-\omega}T_{H}=\frac{1}{4\pi(2M-\omega)},\label{eq: Corda Temperature}
\end{equation}
we can re-write eq. (\ref{eq: Corda Temperature}) in a Boltzmann-like
form similar to eq. (\ref{eq: hawking probability})
\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 $\beta_{E}(\omega)\equiv\frac{1}{T_{E}(\omega)}$.
In other words, the effective temperature replaces the Hawking temperature
in the equation of the probability of emission \cite{key-1,key-2}.
The physical interpretation is the following. In various fields of
Science, one takes into account the deviation from the thermal spectrum
of an emitting body by introducing an effective temperature which
represents the temperature of a black body that would emit the same
total amount of radiation\emph{. }We introduced the concept of effective
temperature also in the black hole's physics \cite{key-1,key-2}.
The effective temperature 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-1,key-2}. The introduction
of the effective temperature enables the introduction of others \emph{effective
quantities}. Following \cite{key-1,key-2}, 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, $M-\omega$. We can introduce
the \emph{effective mass }and the \emph{effective horizon}
\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} its contraction, i.e. \emph{during}
the emission of the particle \cite{key-1,key-2}. Such effective quantities
are average quantities \cite{key-1,key-2}. In fact, \emph{$r_{E}$
}is the average of the initial and final horizons while \emph{$M_{E}$
}is the average of the initial and final masses \cite{key-1,key-2}.
The effective temperature \emph{$T_{E}\:$ }is the inverse of the
average value of the inverses of the initial and final Hawking temperatures
(\emph{before} the emission we have $T_{H\mbox{ initial}}=\frac{1}{8\pi M}$,
\emph{after} the emission we have $T_{H\mbox{ final}}=\frac{1}{8\pi(M-\omega)}$
respectively) \cite{key-1,key-2}. It is important to recall that
the tunnelling is a \emph{discrete} instead of \emph{continuous} process
\cite{key-1}. In fact, we have two different \emph{countable} black
hole's physical states, the black hole's state before the emission
of the particle and the black hole's state after the emission of the
particle \cite{key-1}. 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-1}. In the language of
the tunnelling approach, a trajectory in imaginary or complex time
joins two separated classical turning points \cite{key-1,key-3}.
The consequence is that the radiation spectrum is also discrete \cite{key-1}.
It is better to clarify this important issue in details. If we consider
a well fixed Hawking temperature, the statistical probability distribution
(\ref{eq: Parikh Correction}) is a continuous function. On the other
hand, the Hawking temperature in (\ref{eq: Parikh Correction}) varies
in time with a character which is \emph{discrete}. The reason is that
the forbidden region which the emitting particle traverses has a \emph{finite}
size \cite{key-3}. That is exactly the reason because the effective
temperature (\ref{eq: Corda Temperature}) has been introduced. Considering
a strictly thermal approximation, the turning points have zero separation
and it is not clear what joining trajectory has to be considered as
there is not barrier \cite{key-3}. The problem is solved when one
argues 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}. In other words,
the intriguing explanation is that it is the particle itself which
generates a secret tunnel through the black hole horizon \cite{key-3}.
The physical consequences are that the spectrum is not strictly thermal
\cite{key-3,key-4} and the Hawking temperature \emph{has a discrete
character in time}. Hence, also the statistical probability distribution
(\ref{eq: Parikh Correction}) is \emph{discrete in time}. We stress
that the emitted energies are not only discrete, but also \emph{countable},
as it has been shown in \cite{key-9,key-10}. Interesting proposals
on the non strictly continuous character of Hawking radiation can
be found in the literature \cite{key-11,key-12}. It is straightforward
that discrete energy spectra rather than continuous ones are in general
associated to quantum systems of finite size \cite{key-11}. In the
tunnelling framework, the dynamics responsible of the discrete character
of the spectrum refer not only to the finite region enclosed by the
horizon of the black hole like in \cite{key-11}, but in particular
to the finite size of the forbidden region that is traversed by the
tunnelling particle \cite{key-3}. The discrete character of the spectrum
is also very important for the physical interpretation of black hole's
quasi-normal modes \cite{key-1,key-2}. The remarkable idea that black
hole's quasi-normal modes can release information about the area quantization
arises from the works \cite{key-13,key-14}. The original results
in \cite{key-13,key-14} found various objections \cite{key-15,key-16},
which have been well addressed in \cite{key-15}, where the initial
proposal in \cite{key-13,key-14} has been refined. Black hole's quasi-normal
frequencies are labelled as $\omega_{nl},$ where $l$ is the angular
momentum quantum number ($l$$\geq2$ for gravitational perturbations)
and $n$ ($n=1,2,...$) is the quantum ``overtone'' number which
denotes a countable sequence of quasi-normal modes for each $l$ \cite{key-1,key-2,key-13,key-14,key-15,key-16}.
The quasi-normal modes of the Schwarzschild black hole become independent
of $l\:$ for large $n$, and, in strictly thermal approximation,
their countable sequence is \cite{key-1,key-2,key-13,key-14,key-15,key-16}
\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 The quasi-normal frequencies (\ref{eq: quasinormal modes})
are interpreted like superposition of the damped oscillations \cite{key-1,key-2,key-15}
\begin{equation}
\exp(-i\omega_{I}t)[a\sin\omega_{R}t+b\cos\omega_{R}t]\label{eq: damped oscillations}
\end{equation}
\noindent which have a spectrum of complex frequencies $\omega=\omega_{R}+i\omega_{I}.$
The equation governing a damped harmonic oscillator $\mu(t)$ is \cite{key-1,key-2,key-15}
\begin{equation}
\ddot{\mu}+K\dot{\mu}+\omega_{0}^{2}\mu=F(t),\label{eq: oscillatore}
\end{equation}
\noindent where $\omega_{0}$ is the proper frequency of the harmonic
oscillator, $F(t)$ an external force per unit mass and $K$ the damping
constant. If one assumes that the external force per unit mass scales
like a Dirac delta function, i.e.
\noindent
\begin{equation}
F(t)\propto\delta(t),\label{eq: delta dirac}
\end{equation}
\noindent $\mu(t)$ results to be a superposition of a term oscillating
as $\exp(i\omega t)$ and of a term oscillating as $\exp(-i\omega t)$,
see \cite{key-15} for details. Then, through the identifications
\cite{key-1,key-2,key-15}
\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 give
\begin{equation}
\omega_{0}=\sqrt{\omega_{R}^{2}+\omega_{I}^{2}},\label{eq: omega 0}
\end{equation}
\noindent a damped harmonic oscillator reproduces the behavior (\ref{eq: damped oscillations}).
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 \cite{key-1,key-2,key-15}. For highly excited modes, which
represent a lot of black hole's quasi-normal modes, the opposite limit
is the correct one. By using this observation, 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 $(\omega_{0})_{n}$ can be re-examined in a correct way \cite{key-1,key-2,key-15}.
\noindent We recall that ideas on the continuous character of Hawking
radiation did not agree with attempts to interpret the frequency of
the discrete quasi-normal modes (\ref{eq: quasinormal modes}), preventing
to associate quasi-normal modes to Hawking radiation \cite{key-17}
as the discrete behavior of the energy spectrum (\ref{eq: quasinormal modes})
results incompatible with the spectrum of Hawking radiation whose
energies are of the same order but continuous \cite{key-17}. On the
other hand, the non-strictly thermal behavior which also implies,
as we have shown above, the non-strictly continuous character of Hawking
radiation, removes the above difficulty \cite{key-1}. Thus, the discrete
behavior of Hawking radiation permits to interpret the quasi-normal
frequencies (\ref{eq: quasinormal modes}) also like energies of physical
Hawking quanta \cite{key-1}. Quasi-normal modes represent the reaction
of a black hole to small, discrete perturbations in terms of damped
oscillations \cite{key-1,key-2,key-13,key-14,key-15}. The capture
of a particle which causes an increase in the horizon area is a type
of discrete perturbation \cite{key-13,key-14,key-15}. Thus, it is
very natural to assume that the emission of a particle which causes
a decrease in the horizon area is also a perturbation which generates
a reaction in terms of countable quasi-normal modes as it is a discrete
instead of continuous process \cite{key-1}. Based on such a natural
correspondence between Hawking radiation and black hole's quasi-normal
modes, one can consider quasi-normal modes in terms of quantum levels
for emitted energies too \cite{key-1,key-2}. This important point
is in agreement with the idea that black holes can be considered in
terms of highly excited states in the underlying quantum gravity theory
\cite{key-1,key-2,key-15}. When one enables the correspondence between
black hole's quasi-normal frequencies and the emission and/or the
absorption of particles, the validity of Bohr Correspondence Principle
\cite{key-18} is implicitly assumed. That Principle states that \textquotedblleft{}transition
frequencies at large quantum numbers should equal classical oscillation
frequencies\textquotedblright{} \cite{key-13,key-14}. We applied
the concept of effective temperature to the analysis of physical interpretation
of the spectrum of black hole's quasi-normal modes in \cite{key-1,key-2}.
Another key point of the analysis was that eq. (\ref{eq: quasinormal modes})
is an approximation as it implies the assumption that the black hole's
radiation spectrum is strictly thermal. The deviation from the thermal
spectrum in eq. (\ref{eq: Parikh Correction}) is taken into due account
when one replaces the Hawking temperature $T_{H}$ with the effective
temperature $T_{E}$ in eq. (\ref{eq: quasinormal modes}) \cite{key-1,key-2}.
In that way, the correct expression for the quasi-normal modes of
the Schwarzschild black hole becomes \cite{key-1,key-2}
\begin{equation}
\begin{array}{c}
\omega_{n}=\ln3\times T_{E}(|\omega_{n}|)+2\pi i(n+\frac{1}{2})\times T_{E}(|\omega_{n}|)+\mathcal{O}(n^{-\frac{1}{2}})=\\
\\
=\frac{\ln3}{4\pi(2M-|\omega_{n}|)}+\frac{2\pi i}{4\pi(2M-|\omega_{n}|)}(n+\frac{1}{2})+\mathcal{O}(n^{-\frac{1}{2}}).
\end{array}\label{eq: quasinormal modes corrected}
\end{equation}
The correct derivation of eq. (\ref{eq: quasinormal modes corrected})
can be found in\cite{key-1,key-2} (precise details of that derivation
are in the Appendix of \cite{key-25}. Here we recall its physical
interpretation. The imaginary part of (\ref{eq: quasinormal modes})
is explained as it follows. As on the given background the quasi-normal
modes determine the position of poles of a Green's function, the Euclidean
black hole solution converges to a thermal circle at infinity with
the inverse temperature $\beta_{H}=\frac{1}{T_{H}}$ \cite{key-19}.
Hence, 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-19}. Indeed, as the spectrum is not strictly thermal,
one naturally assumes that the Euclidean black hole solution converges
to a \emph{non-thermal} circle at infinity \cite{key-1,key-2}. Thus,
the replacement \cite{key-1,key-2}
\noindent
\begin{equation}
\beta_{H}=\frac{1}{T_{H}}\rightarrow\beta_{E}(\omega)=\frac{1}{T_{E}(\omega)},\label{eq: sostituiamo}
\end{equation}
\noindent takes into due account the non-strictly thermal feature
of the radiation spectrum of a black hole. Then, the spacing of the
poles in eq. (\ref{eq: quasinormal modes corrected}) coincides with
the spacing \cite{key-1,key-2}
\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 \cite{key-1,key-2}.
In fact a dependence on the frequency is now present \cite{key-1,key-2}.
The physical solution for the absolute values of the frequencies (\ref{eq: quasinormal modes corrected})
has been found in \cite{key-1,key-2}
\noindent
\begin{equation}
\begin{array}{c}
(\omega_{0})_{n}\equiv E_{n}=|\omega_{n}|=M-\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}}\\
\\
\simeq M-\sqrt{M^{2}-\frac{1}{2}(n+\frac{1}{2})}.
\end{array}\label{eq: radice fisica}
\end{equation}
$(\omega_{0})_{n}\:$ is interpreted like the total energy emitted
by the black hole at that time \cite{key-1,key-2}. Now, we show that
the energy emitted in an arbitrary transition $n\rightarrow m$, with
$m>n$, is proportional to the effective temperature associated to
the transition. Considering an emission from the ground state to a
state with large $n\:$ the mass of the black hole changes from $M\:$
to
\begin{equation}
M_{n}\equiv M-E_{n}.\label{eq: me-1}
\end{equation}
On the other hand, in the transition from the state with $n$ to the
state with $m>n$ the mass of the black hole changes again from $M_{n}\:$
to
\begin{equation}
M_{m}\equiv M-E_{n}+\Delta E_{n\rightarrow m}=M-E_{m},\label{eq: me}
\end{equation}
as it is $\Delta E_{n\rightarrow m}\equiv E_{m}-E_{n}$ \cite{key-1,key-2}.
Considering eq. (\ref{eq: radice fisica}), eqs. (\ref{eq: me-1})
and (\ref{eq: me}) read
\begin{equation}
M_{n}=\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}}\label{eq: me 3}
\end{equation}
and
\begin{equation}
M_{m}=\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(m+\frac{1}{2})^{2}}}.\label{eq: me 4}
\end{equation}
Now, we set
\begin{equation}
\begin{array}{c}
\Delta E_{n\rightarrow m}\equiv E_{m}-E_{n}=\\
\\
=\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(n+\frac{1}{2})^{2}}}-\sqrt{M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(m+\frac{1}{2})^{2}}}=\\
\\
=M_{n}-M_{m}=K\left[T_{E}\right]_{n\rightarrow m},
\end{array}\label{eq: differenza radici fisiche}
\end{equation}
where $\left[T_{E}\right]_{n\rightarrow m}$ is the effective temperature
associated to the transition $n\rightarrow m$, and $M_{n}$ and $M_{m}$
are given by eq. (\ref{eq: me}). Let us see if there are values of
the constant $K$ for which eq. (\ref{eq: differenza radici fisiche})
is satisfied. We recall that
\begin{equation}
\left[T_{E}\right]_{n\rightarrow m}=\frac{K}{4\pi\left(M_{n}+M_{m}\right)},\label{eq: temperatura efficace di transizione}
\end{equation}
as the effective temperature is the inverse of the average value of
the inverses of the initial and final Hawking temperatures. Thus,
eq. (\ref{eq: differenza radici fisiche}) can be rewritten as
\begin{equation}
=M_{n}^{2}-M_{m}^{2}=\frac{K}{4\pi}.\label{eq: differenza radici fisiche 2}
\end{equation}
By using eqs. (\ref{eq: me 3}) and (\ref{eq: me 4}), for large $m$
and $n$ eq. (\ref{eq: differenza radici fisiche 2}) becomes
\begin{equation}
\frac{1}{2}\left(m-n\right)=\frac{K}{4\pi},\label{eq: K solved}
\end{equation}
which implies that eq. (\ref{eq: differenza radici fisiche}) is satisfied
for $K=2\pi\left(m-n\right).$ Hence, we can rewrite (\ref{eq: differenza radici fisiche})
as
\begin{equation}
\Delta E_{n\rightarrow m}=E_{m}-E_{n}=2\pi\left(m-n\right)\left[T_{E}(\omega)\right]_{n\rightarrow m}.\label{eq: differenza radici fisiche finale}
\end{equation}
By using eq. (\ref{eq: Corda Probability}), the probability of emission
between the two levels $n$ and $m$ can be written in the intriguing
form
\begin{equation}
\Gamma_{n\rightarrow m}\sim\exp-\left\{ \frac{\Delta E_{n\rightarrow m}}{\left[T_{E}(\omega)\right]_{n\rightarrow m}}\right\} =\exp\left[-2\pi\left(m-n\right)\right].\label{eq: Corda Probability Intriguing}
\end{equation}
Thus, we have find the fundamental result that the probability of
emission between two arbitrary levels characterized by the two ``overtone''
quantum numbers $n$ and $m$ scales like $\exp\left[-2\pi\left(m-n\right)\right].$
In particular, for $m=n+1$ the probability of emission has its maximum
value $\sim\exp(-2\pi)$, i.e. the probability is maximum for two
adjacent levels, as one can intuitively expect. Notice that, if one
fixes $n$, the probabilities (\ref{eq: Corda Probability Intriguing})
can be normalized to the unity in the following way
\begin{equation}
\sum_{m=n}^{m_{max}}\Gamma_{n\rightarrow m}=\sum_{m=n}^{m_{max}}\alpha\exp\left[-2\pi\left(m-n\right)\right]=1,\label{eq: normalizzazione}
\end{equation}
where $\alpha$ is the prefactor of eq. (\ref{eq: Corda Probability Intriguing}),
$m_{max}$ is the maximum value for the ``overtone'' number $m$
and $m=n$ corresponds to the probability that the black hole does
not emit. Let us compute $m_{max}$. Following \cite{key-1}, as $(\omega_{0})_{m}=E_{m}\:$
is the total emitted energy for a black hole excited at a level $m$,
one needs
\begin{equation}
M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(m+\frac{1}{2})^{2}}\geq0\label{eq: need}
\end{equation}
in eq. (\ref{eq: radice fisica}). In fact, a black hole does not
emit more energy than its total mass. This implies that the countable
sequence of quasi-normal modes cannot be infinity \cite{key-1}. Eq.
(\ref{eq: need}) can be solved giving a maximum value for the ``overtone''
number $m$ \cite{key-1}
\begin{equation}
m\leq m_{max}=2\pi^{2}\left(\sqrt{16M^{4}-(\frac{\ln3}{\pi})^{2}}-1\right).\label{eq: n max}
\end{equation}
The maximum value (\ref{eq: n max}) corresponds to $(\omega_{0})_{m_{max}}=E_{m_{max}}=M.$
On the other hand, by using the Generalized Uncertainty Principle,
the approach in \cite{key-20} proposed that the total evaporation
of a black hole can be prevented exactly like the Uncertainty Principle
prevents the hydrogen atom from total collapse. In such an analysis,
the collapse is prevented by dynamic instead of by symmetry as \emph{Planck
scale} is approached \cite{key-20}. In that way, eq. (\ref{eq: need})
has to be slightly modified, becoming (notice that the Planck mass
is equal to $1$ in Planck units) \cite{key-25}
\begin{equation}
M^{2}-\frac{1}{4\pi}\sqrt{(\ln3)^{2}+4\pi^{2}(m+\frac{1}{2})^{2}}\geq1.\label{eq: need 1}
\end{equation}
The solution of eq. (\ref{eq: need 1}) gives a different value of
the maximum value for for the ``overtone'' number $m$ \cite{key-25}
\begin{equation}
m\leq m_{max}=2\pi^{2}\left(\sqrt{16(M^{2}-1)^{2}-(\frac{\ln3}{\pi})^{2}}-1\right).\label{eq: n max 1}
\end{equation}
Putting $k=m-n\:$ and $\exp\left[-2\pi\right]=X\:$ eq. (\ref{eq: normalizzazione})
becomes
\begin{equation}
\sum_{k=0}^{k_{max}}\Gamma_{0\rightarrow k}=\alpha\sum_{k=0}^{k_{max}}X^{k}=1.\label{eq: normalizzazione 2}
\end{equation}
The sum in eq. (\ref{eq: normalizzazione 2}) is well known. It is
the $kth$ partial sum of the geometric series and can be solved as
\cite{key-21}
\begin{equation}
\sum_{k=0}^{k_{max}}X^{k}=\alpha\frac{1-X^{\left(k_{max}+1\right)}}{1-X}.\label{eq: serie geometrica}
\end{equation}
Thus, one gets
\begin{equation}
\alpha\frac{1-X^{\left(k_{max}+1\right)}}{1-X}=1,\label{eq: serie geometrica risolta}
\end{equation}
which permits to solve for $\alpha$
\begin{equation}
\alpha\equiv\alpha_{n}=\frac{1-X}{1-X^{\left(k_{max}+1\right)}}=\frac{1-\exp-\left[2\pi\right]}{1-\exp\left[-2\pi\left(m_{max}-n+1\right)\right]}.\label{eq: mio alpha number}
\end{equation}
Hence, we find that the constant of proportionality $\alpha$ depends
on the black hole's quantum level $n.$ Notice that for $m_{max}\gg n$
one finds that such a dependence can be neglected
\begin{equation}
\alpha\simeq1-\exp-\left[2\pi\right]\simeq1-1.87*10^{-3}\sim1.\label{eq: alpha 2}
\end{equation}
This is not surprising as for $\Delta E_{n\rightarrow m}\ll M_{n},$
i.e. when the emitted energy is much minor than mass of the black
hole and the condition $m_{max}\gg m,\: n$ is guaranteed, the thermal
approximation is excellent as the back reaction due to the energy
conservation can be neglected. The dependence of the constant of proportionality
$\alpha$ on the black hole's quantum level $n$ becomes very important
when $m,\: n\sim m_{max}$, i.e. near the final stages of the black
hole's evaporation. In that case, in which we label the constant of
proportionality $\alpha_{n}$, the thermal approximation breaks down
as the condition $\Delta E_{n\rightarrow m}\ll M_{n}$ is no more
guaranteed and one needs to use the correct formula (\ref{eq: mio alpha number}).
An intermediate case can be considered too, i.e. when it is $m_{max}\sim m\gg n$.
In that case, eq. (\ref{eq: alpha 2}) can be used even if the condition
$\Delta E_{n\rightarrow m}\ll M_{n}$ is not guaranteed and the thermal
approximation breaks down. On the other hand, eq. (\ref{eq: Corda Probability Intriguing})
shows that transitions in which $m\gg n$ are highly improbable.
Inserting the result (\ref{eq: mio alpha number}) in eq. (\ref{eq: Corda Probability Intriguing})
we fix the probability of emission between the two levels $n$ and
$m$ as
\begin{equation}
\begin{array}{c}
\Gamma_{n\rightarrow m}=\alpha_{n}\exp-\left\{ \frac{\Delta E_{n\rightarrow m}}{\left[T_{E}(\omega)\right]_{n\rightarrow m}}\right\} =\alpha_{n}\exp\left[-2\pi\left(m-n\right)\right]=\\
\\
=\left\{ \frac{1-\exp-\left[2\pi\right]}{1-\exp\left[-2\pi\left(m_{max}-n+1\right)\right]}\right\} \exp\left[-2\pi\left(m-n\right)\right].
\end{array}\label{eq: Corda Probability Intriguing finalized}
\end{equation}
We recall that the quasi-normal frequencies (\ref{eq: quasinormal modes corrected})
are the eigenvalues of the equation \cite{key-1,key-2,key-17,key-25}
\noindent
\begin{equation}
\left(-\frac{\partial^{2}}{\partial x^{2}}+V(x)-\omega^{2}\right)\phi,\label{eq: diff.}
\end{equation}
where
\begin{equation}
V(x)\equiv 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}
is the (time independent) \emph{effective Regge-Wheeler potential}
\cite{key-1,key-2} and the relation between the Regge-Wheeler ``tortoise''
coordinate $x$ and the radial coordinate $r$ is \cite{key-1,key-2}
\noindent
\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: tortoise}
\end{equation}
Eq. (\ref{eq: diff.}) is interpreted like a \emph{Schroedinger equation}
\cite{key-1,key-2,key-17}. In fact, from a mathematical point of
view, the quasi-normal frequencies are discrete quasi-normal states,
analogous to the quasi-stationary states of quantum mechanics which
frequency is allowed to be complex \cite{key-17}. From the quantum
mechanical point of view, one can physically interpret Hawking radiation
like energies of quantum jumps among the unperturbed levels (\ref{eq: radice fisica})
\cite{key-1,key-2}. Hence, we have a perturbation of the type in
eq. (\ref{eq: delta dirac}) which acts on the quasi-normal modes
\cite{key-1,key-2}. Such a perturbation can be described by an operator
\cite{key-22}
\emph{
\begin{equation}
U(t)=\begin{array}{c}
W(t)\;\;\; for\;0\leq t\leq\tau\\
\\
0\;\;\; for\; t<0\; and\; t>\tau.
\end{array}\label{eq: perturbazione}
\end{equation}
}We need $W(t)$ to be conform to eq. (\ref{eq: delta dirac}). Thus,
in an appropriate orthonormal basis \cite{key-22}, the matrix elements
of $W(t)$ can be written as
\begin{equation}
W_{ij}(t)\equiv A_{ij}\delta(t),\label{eq: a delta}
\end{equation}
where the $A_{ij}$ are real. Thus, the complete (time dependent)
Hamiltonian is described by the operator \cite{key-22}
\begin{equation}
H(x,t)\equiv V(x)+U(t),\label{eq: Hamiltoniana completa}
\end{equation}
which permits to write the correspondent \emph{time dependent Schroedinger
equation }for the system \cite{key-22}
\begin{equation}
i\frac{d|\psi(x,t)>}{dt}=\left[V(x)+U(t)\right]|\psi(x,t)>=H(x,t)|\psi(x,t)>.\label{eq: Schroedinger equation}
\end{equation}
The\emph{ }state\emph{ }which satisfies eq. (\ref{eq: Schroedinger equation})
is \cite{key-22}
\begin{equation}
|\psi(x,t)>=\sum_{m}a_{m}(t)\exp\left(-i\omega_{m}t\right)|\varphi_{m}(x)>,\label{eq: Schroedinger wave-function}
\end{equation}
where the $\varphi_{m}(x)$ are the eigenfunctions of eq. (\ref{eq: diff.})
and the $\omega_{m}$ are the correspondent eigenvalues. In order
to solve the complete quantum mechanical problem described by the
operator (\ref{eq: Hamiltoniana completa}) one needs to know the
probability amplitudes $a_{m}(t)$ due to the application of the perturbation
descripted by the time dependent operator (\ref{eq: perturbazione})
\cite{key-22}, which represents the perturbation associated to the
emission of an Hawking quantum. For $t<0,$ i.e. before the perturbation
operator (\ref{eq: perturbazione}) starts to work, the system is
in a stationary state $|\varphi_{n}t(x)>,$ at the quantum level $n,$
with energy $E_{n}=|\omega_{n}|$ given by eq. (\ref{eq: radice fisica}).
Therefore, in eq. (\ref{eq: Schroedinger wave-function}) only the
term
\begin{equation}
|\psi_{n}(x,t)>=\exp\left(-i\omega_{n}t\right)|\varphi_{n}(x)>,\label{eq: Schroedinger wave-function in.}
\end{equation}
is not null for $t<0.$ This implies $a_{m}(t)=\delta_{mn}\:\:$for
$\: t<0.$ When the perturbation operator (\ref{eq: perturbazione})
stops to work, i.e. after that a quantum has been emitted, for $t>\tau$
the probability amplitudes $a_{m}(t)$ return to be time independent,
having the value $a_{n\rightarrow m}(\tau)$ \cite{key-22}. In other
words, for $t>\tau\:$ the system is described by the \emph{Schroedinger
wave-function}
\begin{equation}
|\psi_{final}(x,t)>=\sum_{m=n}^{m_{max}}a_{n\rightarrow m}(\tau)\exp\left(-i\omega_{m}t\right)|\varphi_{m}(x)>.\label{eq: Schroedinger wave-function fin.}
\end{equation}
Thus, the probability to find the system in an eigenstate having energy
$E_{m}=|\omega_{m}|$ is given by \cite{key-22}
\begin{equation}
\Gamma_{n\rightarrow m}(\tau)=|a_{n\rightarrow m}(\tau)|^{2}.\label{eq: ampiezza e probability}
\end{equation}
In order to solve the problem, one proceeds following \cite{key-22}.
By using a standard analysis, one can obtain the following differential
equation from eq. (\ref{eq: Schroedinger wave-function fin.}) \cite{key-22}
\begin{equation}
i\frac{d}{dt}a_{n\rightarrow m}(t)=\sum_{l=m}^{m_{max}}W_{ml}a_{n\rightarrow l}(t)\exp\left[i\left(\Delta E_{l\rightarrow m}\right)t\right].\label{eq: systema differenziale}
\end{equation}
To first order in $U(t)$, by using the Dayson series, one gets the
solution \cite{key-22}
\begin{equation}
a_{n\rightarrow m}=-i\int_{0}^{t}\left\{ W_{mn}(t')\exp\left[i\left(\Delta E_{n\rightarrow m}\right)t'\right]\right\} dt'.\label{eq: solution}
\end{equation}
By inserting (\ref{eq: a delta}) in (\ref{eq: solution}) we obtain
\begin{equation}
a_{n\rightarrow m}=-iA_{mn}\int_{0}^{t}\left\{ \delta(t')\exp\left[i\left(\Delta E_{n\rightarrow m}\right)t'\right]\right\} dt'=-\frac{i}{2}A_{mn}\label{eq: solution 2}
\end{equation}
Combining this equation with eqs. (\ref{eq: Corda Probability Intriguing finalized})
and (\ref{eq: ampiezza e probability}) one gets
\begin{equation}
\begin{array}{c}
\alpha_{n}\exp\left[-2\pi\left(m-n\right)\right]=\frac{1}{4}A_{mn}^{2}\\
\\
A_{mn}=2\sqrt{\alpha_{n}}\exp\left[-\pi\left(m-n\right)\right]\\
\\
a_{n\rightarrow m}=-i\sqrt{\alpha_{n}}\exp\left[-\pi\left(m-n\right)\right].
\end{array}\label{eq: uguale}
\end{equation}
We stress that, as $\sqrt{\alpha_{n}}\sim1,$ for $m=n+1$, i.e. when
$A_{mn}$ is maximum for an emission, one gets $A_{mn}\sim10^{-2}$.
This implies that the error in our solution is $\sim10^{-4},$ i.e.
our approximation is very good. Clearly, for $m>n+1$ the approximation
is better. Thus, one can write down the final form of the Schroedinger
wave-function of the system
\begin{equation}
|\psi_{final}(x,t)>=\sum_{m=n}^{m_{max}}-i\sqrt{\alpha_{n}}\exp\left[-\pi\left(m-n\right)\left(-i\omega_{m}t\right)\right]|\varphi_{m}(x)>.\label{eq: Schroedinger wave-function finalissima}
\end{equation}
Notice that the \emph{Schroedinger wave-function} of the system (\ref{eq: Schroedinger wave-function finalissima})
represents a \emph{pure final state instead of a mixed final state.}
Thus, the states are written in terms of a \emph{unitary} evolution
matrix instead of a density matrix\emph{ }and\emph{ }this implies
the fundamental conclusion that\emph{ }information is not loss in
black hole evaporation, in agreement with the results of string theory
\cite{key-7}, with the use of Feynman sum over histories \cite{key-6},
and with the using of the Kerr-Schild formalism in Kerr-Newman black
holes \cite{key-8}. Notice that we found this fundamental result,
which is in full agreement with quantum mechanics, by using a pure
semi-classical analysis, i.e. without extending the assumptions of
our theoretical approach like in the cited cases of string theory
\cite{key-7} and of the use of Feynman sum over histories \cite{key-6}.
The result is also in agreement with the assumption by 't Hooft that
Schroedinger equations can be used universally for all dynamics in
the universe \cite{key-24}.
In summary, in this letter we found a fundamental equation that directly
connects the probability of emission of an Hawking quantum with the
two emission levels which are involved in the transition. Such an
equation permitted us to interpret the correspondence between Hawking
radiation and black hole quasi-normal modes in terms of a time dependent
Schroedinger system. In that system, the energies of the quasi-normal
modes, which are also the total energies emitted by the black hole
in correspondence of the various quantum levels, represent the eigenvalues
of the unperturbed Hamiltonian of the system and the Hawking quanta
represent the energy transitions among the eigenvalues, which correspond
to a perturbed Hamiltonian $\propto\delta(t)$. In this way, we explicitly
wrote down a time dependent Schroedinger equation\emph{ }for the system
composed by Hawking radiation and black hole quasi-normal modes. The
states of the correspondent Schroedinger wave-function\emph{ }are
written in terms of a \emph{unitary} evolution matrix instead of a
density matrix\emph{,} and, in turn, result to be pure states instead
of mixed ones. In other words, the black hole obeys a time dependent
Schroedinger equation which allows pure states to evolve into pure
states. This implies the non-trivial consequence that information
comes out in black hole's evaporation. As a consequence, the underlying
quantum gravity theory should be also unitary. We stress that the
result of this paper are in full agreement with the results of string
theory \cite{key-7}, with the use of Feynman sum over histories \cite{key-6},
and with the using of the Kerr-Schild formalism in Kerr-Newman black
holes \cite{key-8}. On the other hand, an important difference is
that here we found this fundamental result, which is in full agreement
with quantum mechanics, by using a pure semi-classical analysis, i.e.
without extending the assumptions of our theoretical approach like
in the cited cases of string theory \cite{key-7} and of the use of
Feynman sum over histories \cite{key-6}.
We also emphasize that the results of this lettr are correct only
for $n\gg1,$ i.e. only for highly excited black holes. This is the
reason because we assumed an emission from the ground state to a state
with large $n\:$ in our discussion. On the other hand, a state with
large $n\:$ is always reached at late times, maybe not through a
sole emission from the ground state, but, indeed, through various
subsequent emissions of Hawking quanta.
\subsubsection*{Acknowledgements}
It is a pleasure to thank Hossein Hendi, Erasmo Recami, Lawrence Crowell,
Herman Mosquera Cuesta, Ram Vishwakarma and my students Reza Katebi
and Nathan Schmidt for useful discussions on black hole physics.
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\end{document}
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