\lang{The low-laying frequencies of characteristic quasi-normal modes (QNM) of Schwarzschild-de Sitter (SdS) black holes have been calculated for fields of different spin using the sixth-order WKB approximation and the approximation by the P\"{o}shl-Teller potential. The well-known asymptotic formula for large $l$ is generalized here on a case of the Schwarzschild-de Sitter black hole. In the limit of the near extreme $\L$ term the results given by both methods are in a very good agreement, and in this limit fields of different spin decay with the same rate.} {��������� ����� ������� ���������������� ��������������� ��� (quasi-normal modes) ������ �� �����������-�� ѳ���� (Schwarzschild-de Sitter) ��� ���� ������ ����, �������������� ���������� ������� ������� WKB � ���������� ����������� ������-�������. ����������� ����� ����������� ������� ��� ������� $l$ �� ������� ����� ��� �����������-�� ѳ����. ���� $\L$-���� ����� �� �������������� ��������, ���������� ���� ������, ����� ������������ �� �����, � ���� ����� ����� �������� � ��������� ��������.}
gr-qc0307012.tex
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%GREEK LETTERS
 \def\a{\alpha}\def\b{\beta}\def\g{\gamma}\def\d{\delta}\def\e{\epsilon}
 \def\E{\varepsilon} \def\k{\kappa}\def\l{\lambda}\def\L{\Lambda}
 \def\s{\sigma}\def\S{\Sigma} \def\Th{\Theta}\def\th{\theta}\def\om{\omega}
 \def\Om{\Omega}\def\G{\Gamma} \def\und{\underline}\def\r{\rho}\def\n{\nu}
 \def\m{\mu}\def\t{\tilde} \def\o{\overline} \def\z{\zeta}
 \def\P{\Psi}\def\Ph{\Phi} \def\ph{\phi}

%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\imo{i}

\def\be{\begin{equation}}
\def\ee{\end{equation}}
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\begin{document}
\lang{}{\selectlanguage{ukrainian}\cyr}
\thispagestyle{empty}
\centerline{\bf \Large \lang{Quasi-normal modes of Schwarzschild-de Sitter
black holes}{�������������� ���� ������ �� �����������-��
ѳ����}}
\centerline{\bf \lang{A. Zhidenko}{�. �������}}

\centerline{\lang{Department of Physics, Dniepropetrovsk National University}{����������������� ������������ �����������, �������� ���������}}
\centerline{\lang{St. Naukova 13, Dniepropetrovsk 49050, Ukraine}{���. �������, 13, ��������������� 49050, ������}}
\centerline{Z\_A\_V@ff.dsu.dp.ua}
\vskip3mm
\bigskip \nopagebreak

\begin{abstract}
\lang{The low-laying frequencies of characteristic quasi-normal modes
(QNM) of Schwarzschild-de Sitter (SdS) black holes have been
calculated for fields of different spin using the sixth-order WKB
approximation and the approximation by the P\"{o}shl-Teller
potential. The well-known asymptotic formula for large $l$ is
generalized here on a case of the Schwarzschild-de Sitter black
hole. In the limit of the near extreme $\L$ term the results given
by both methods are in a very good agreement, and in this limit
fields of different spin decay with the same rate.} {���������
����� ������� ���������������� ��������������� ��� (quasi-normal
modes) ������ �� �����������-�� ѳ���� (Schwarzschild-de Sitter)
��� ���� ������ ����, �������������� ���������� ������� �������
WKB � ���������� ����������� ������-�������. ����������� �����
����������� ������� ��� ������� $l$ �� ������� ����� ���
�����������-�� ѳ����. ���� $\L$-���� ����� �� ��������������
��������, ���������� ���� ������, ����� ������������ �� �����, �
���� ����� ����� �������� � ��������� ��������.}
\end{abstract}

\newpage\setcounter{page}1
\section{\lang{Introduction}{�����}}
\lang{Within general relativity it is well-known that there are three
stages to the dynamical evolution of field perturbation on a black
hole background: the initial outburst from the source of
perturbation, the damping (quasi-normal) oscillations and
asymptotic tails at very late time. The evolution significantly
depends on the asymptotic behavior of the background: for
different values of cosmological constant $\L$ there are three
types of behavior depending on the sign of $\L$. Thus, the
remarkable fact is that while for asymptotically flat backgrounds
we have inverse power law tails, for asymptotically Anti-de Sitter
background it is exponential low that governs the decay of a field
at all times. The same exponential decay at very late times was
observed by Brady \textit{et al} \cite{Brady-Chambers} for
asymptotically de Sitter background.} {� �������� ����
��������� ����� �����, �� ���� ��� ���䳿 �������� ��������
�������� ���� �� �� ����� ���: ��������� �� ������� ��������,
(��������������) ���������, �� ��������, �� ������������ ������ ��
����� ������. �������� ������� �������� �� ������������
�������� ���: ������� �� ����� ����������� ����� $\L$ ����
��� ���� ��������. ������ ����� � ����, �� ��� ������������
�������� ��� ������������ ������ � �����������, � ��� ��'����
����������� ����� ���� ��������������� �������� �� ���
������. ���� ���� ��������������� �������� �� ���� ����� ������
�������� ����� �� ��. \cite{Brady-Chambers} ��� ��������� $\L$.}

\lang{Apart from original interest coming from possibility to observe
quasi-normal ringing with the help of gravitational wave detectors
(see, e.g., \cite{QNMrev} for review), the quasi-normal modes
bring now a lot of interest in different contexts: in Anti-de
Sitter(de Sitter)/Conformal Field Theory correspondence
\cite{Horowitz-Hubeny,CFT-Correspond,Moss-Norman},
when considering thermodynamic properties of black holes in loop quantum gravity
\cite{LQG}, in the context of possible connection with critical
collapse \cite{Horowitz-Hubeny,BHCC}.}{��� ���������
������������� ���������������� ������ � ��������� ���������
������������ ����� (���. ��������� �����
\cite{QNMrev}), ����� ����� ���� ������� �� ���������������
��� � ����� ������: ����� ��������� ���������� �� ���������
����-�� ѳ���� (�� ѳ����) �� ���������� ����� ���� (Conformal
Field Theory) \cite{Horowitz-Hubeny,CFT-Correspond,Moss-Norman}, �
�������� �������� �������������� ������������ ������ �� �
��������� �������� ���������, � ����� ����� ������� ���������
��'���� � ��������� �������� \cite{Horowitz-Hubeny,BHCC}.}

\lang{The gravitational QNMs of Schwarzschild-de Sitter black hole were
considered in \cite{Otsuki,Mellor-Moss,Moss-Norman}. Recently,
asymptotic values of QNMs were studied for nearly extremal
Schwarzschild-de Sitter black hole and it was shown that in the
near extremal limit the effective potential of the wave equation
reduces to the P\"{o}shl-Teller potential, and it is expected
therefore the P\"{o}shl-Teller method provides the best accuracy
near the extremal value of $\L$
\cite{Cardoso-Lemos2}.}{����������� �������������� ���� ����� ���
� ������� �����������-�� ѳ���� ������������ �
\cite{Otsuki,Mellor-Moss,Moss-Norman}. ��������� ��� ��������
������������ ������� ��������������� ��� ��� ������������
����������� ����� ���� ��������, �� ���������� ���������
���������� ������� ��������� �� ��������� ������-�������, ����
���������, �� ����� ������-������� �� �������� �������� ���
�������������� �������� $\L$ \cite{Cardoso-Lemos2}.}

\lang{QNMs corresponding to decay of fields of different spin are
extensively studied for Schwarzschild and Schwarzschild-anti de
Sitter backgrounds. At the same time there is lack of such study
of the Schwarzschild-de Sitter case. This motivated us to
calculate QNMs of scalar, electromagnetic, gravitational and Dirac
fields for Schwarzschild-de Sitter black hole. We also compare the
results found here with those obtained recently in
\cite{Cho,Cardoso-Lemos2}.}{�������������� ���� ���� ������ ����
���� �������� ������� �� �� ������� ������������ ��
�����������-����-�� ѳ����, ��� �� ������� �� ѳ���� ����������
��������������. �� ������ ���������� ���������� ���������������
��� ����������, �����������������, ������������� �� ���� ĳ����
��� ����� ��� �����������-�� ѳ����, � ����� ��������� �
��������� ���������� ������������ \cite{Cho,Cardoso-Lemos2}.}

\lang{In section \ref{eq} we give the perturbation equations to describe
fields of different spin in SdS background. Section \ref{calc}
presents comparison of QNM frequencies obtained through sixth
order WKB formula and approximation by the P\"{o}shl-Teller
potential.}{� ����� \ref{eq} ���������� ������� ��� �����
�������� ���� ������ ���� �� �� ������� �����������-�� ѳ����.
����� \ref{calc} ����������� ��������� ��������������� ������,
��������� � ������� WKB ������� ������� � ����������� �����������
������ �������.}

\lang{There was shown that the frequencies of the massless Dirac field
perturbations \cite{Cho} are the same for both chiralities in SdS
background as well. Near the extremal value of $\L$ the sixth
order WKB formula gives the values which are very close to those
obtained within the P\"{o}shl-Teller potential approximation. This
agrees with recent work of Cardoso and Lemos
\cite{Cardoso-Lemos2}.}{��������, �� ������� ����������� ���� ĳ���� \cite{Cho}
� ����� ���������� ��� ���� ������������ (chirality) � �������
�����������-�� ѳ����. ���� �������������� �������� $\L$ ������
������� WKB �� �������� ���� ������ �� ���, �� �������� �
���������� ����������� ������-�������, �� ����������� � �������
������� � ������ \cite{Cardoso-Lemos2}.}


\section{\lang{Perturbation equations and methods}{г������ � ������ ����� �������}}\label{eq}
\lang{It is well known that the Schwarzschild-de Sitter black hole
of mass $M$ is described by the metric}{����� �����, �� �����
��� �����������-�� ѳ���� ���� $M$ ��������� ��������}:
\be\label{sds}
ds^2 = f(r)dt^2 - \frac{dr^2}{f(r)} - r^2 d\s^2; \qquad f(r) = 1 -
\frac{2M}{r} - \L\frac{r^2}{3}, \quad d\s^2 = d\th^2 + \sin^2\th
d\ph^2.
\ee
\lang{A perturbation equation on this background can be reduced to the
Schr\"{o}dinger wave-like equations}{г������ �������� �� ������
�� ����� ������ �� ������� ���� ���������}:
\be\label{wleq}
\left(\frac{d^2}{dr^{*2}} + \om^2 - V(r^*)\right)\P(r^*) = 0,
\ee
\lang{where we assume exponential dependence of}{�� �� ���������� ��������������� ����������} $\P$ \lang{on time}{�� ����} ($\sim e^{-\imo\om
t}$), $\om = \om_{Re} - i\om_{Im}$, $r^*$ \lang{is the "tortoise"
coordinate}{"����������"{} ("tortoise") ����������}:
\be\label{tortoise}
dr^* = \frac{dr}{f(r)}.
\ee
\lang{Under the positive real part QNMs, by definition, satisfy the following boundary conditions}
{�������������� ���� � �������� ������ �������� �� ����������
������������� �������� �����}:
\be\label{bounds}
\P(r^*) \sim C_\pm \exp(\pm\imo\om r^*), \qquad r\longrightarrow
\pm\infty,
\ee
\lang{which correspond to purely out-going waves at infinity and in-going
waves at the horizon}{�� ������� ����� ������, �� �������� ��
������������ �������� � ������� �� �������� ����}.

\lang{The corresponding perturbation equations can be rewritten in form \p{wleq}
with potentials}{³������� ������� ����� ���������� � ����
\p{wleq} � ������������}
\bea\label{sp}
V_s &=& f(r)\left(\frac{l(l+1)}{r^2} + \frac{1}{r}f'(r)\right);
\\\label{emp}
V_{el} &=& f(r)\frac{l(l+1)}{r^2};
\\\label{gpo}
V_{odd} &=& f(r)\left(\frac{l(l+1)}{r^2} - \frac{6M}{r^3}\right);
\\\label{gpe}
V_{even} &=& \frac{2f(r)}{r^3}\frac{9M^3 + 3c^2Mr^2 + c^2(1 +
c)r^3 + 3M^2(3cr - \L r^3)}{(3M + cr)^2}, \quad c =
\frac{l(l+1)}{2} - 1;
\\\label{mdp}
V_d &=& \frac{\k f^{1/2}(r)}{r^2}(\k f^{1/2}(r) \pm (\frac{3M}{r}
- 1));
\eea
\lang{for massless scalar, electromagnetic, gravitational (odd end even
parities) and Dirac fields respectively (see, e.g.,
\cite{Potentials} and references therein), where $l$ is the
multipole number, $\k = l+1$ for (+) sign and $\k = l$ for (-)
sign in \p{mdp}. All the potentials vanish at the horizon and tend
to a constant at infinity, having a peak near the horizon. It
allows to use the P\"{o}shl-Teller potential approximation as well
as WKB method.}{�������� ��� ���������� ����������,
�����������������, ������������� (����������� �� ���������) �
ĳ���������� ���� (���., ���������,
\cite{Potentials} � ��������� � ���), �� $l$ ������������ �����,
$\k = l+1$ ��� ����� (+) � $\k = l$ ��� ����� (-) � \p{mdp}. ��
���������� �������� �� ���� �� �������� ���� � �� ����� ��
������������� ��������, ����� �������� ��� ��������� ����. ֳ
���������� ���������� ��������������� � ����� ����������
����������� ������-�������, � ����� WKB.}


\lang{The WKB approach is based on the analogy with the problem of waves
scattering near the peak of the potential barrier $V(r^*)$ in
quantum mechanics, where $\om^2$ plays a role of energy. The
approach was used by Schutz and Will
\cite{SchutzWill}, developed to the third order by Iyer and Will
\cite{IyerWill}, and recently developed by Konoplya to the sixth
order beyond the eikonal approximation \cite{Konoplya}. The result
has the form}{ϳ���� WKB �������� � �������� ��
����������������� ������ ��� �������� ����� �� ��������������
���'�� $V(r^*)$, �� ���� ����㳿 ��� ������� ������� $\om^2$.
������� WKB ����������������� �� ����� �� ³��� \cite{SchutzWill},
���� ��������� ����� � ³��� �� �������� ������� \cite{IyerWill},
� ��������� �. �������� ���� �������� ���������� ���� �����������
�������� �� ������� ������� \cite{Konoplya}. ��������� ��
������}:
\be\label{WKB}
\imo\frac{\om^2 - V_0}{\sqrt{-2V_0^{\prime\prime}}} - \L_2 - \L_3 - \L_4 -
\L_5 - \L_6 = n + \frac{1}{2},
\ee
\lang{where $V_0$ is the height and $V_0^{\prime\prime}$ is the second
derivative with respect to the tortoise coordinate of the
potential at the maximum. $\L_2$ and $\L_3$ can be found in
\cite{IyerWill}, $\L_4$, $\L_5$ and $\L_6$ are presented in
\cite{Konoplya}; the corrections depend on the value of the potential
and higher derivatives of it at the maximum.}{�� $V_0$ --
��������, � $V_0^{\prime\prime}$ -- ����� ������� �� �����������
��������� ���������� � ��������. $\L_2$ �� $\L_3$ ����� ������ �
\cite{IyerWill}, $\L_4$, $\L_5$ � $\L_6$ �������� �
\cite{Konoplya}. ֳ �������� �������� �� �������� ���������� ��
���� ����� �������� � ��������.}

\lang{Generally, WKB method was effectively used for finding
quasi-normal frequencies in a lot of papers \cite{WKB}. Comparing
the values calculated by the different order WKB formula one can
judge about the convergence to the some unknown accurate result as
the WKB order increasing. We expect that the possible error is
less then the difference between the sixth and the fifth order WKB
values\footnote{The accuracy of values are to take as a half of
the number last order (e. g. $0.125$ means $0.125\pm 0.003$), but
it is much better for most of them.}. Note, that standard machine
precision in MATHEMATICA gives large errors due to numerical
truncations. To exclude such "noises" we had to hold 40 digits in
all intermediate values.}{������, ����� WKB ���������
���������������� ��� ����������� ��������������� ������ � ��������
������� \cite{WKB}. ��������� ��������, ��������� �� ���������
����� ������� WKB, ����� ������� �������� ������ ����
��������� �� ����� ������� �������� � ���������� �������
������. ���������, �� ������� ������� � ������ �� ������ ��
������ � �'���� �������� WKB\footnote{�������� ��������� �����
������� �������� ���������� ������� (��������� $0.125$ ������
$0.125\pm 0.003$), ��� � ������� ������� ���� ������ ����.}.
��� ���������, �� ���������� ������� �������� ��������
MATHEMATICA �� ������ ������� ����� ��������� ����� ������
�������, ���� ��������� ���������� 40 ����� � ��� ��������
���������.}

\lang{For comparison we will use the semi-analytical method for
calculations proposed by B. Mashhoon \cite{FerrariMashhoon} who
used P$\ddot\mathrm o$shl-Teller approximate potential}{���
��������� ����������� ����� ����������, �������������� �.
������� \cite{FerrariMashhoon}, ���� ��������� ��������������
��������� ������-�������}
\be\label{PT}
V_{PT} = \frac{V_0}{\cosh^2(r^*/b)}.
\ee
\lang{It contains two free parameters ($V_0$ and $b$) which are
used to fit the height and the second derivative of the potential
$V(r^*)$ at the maximum \cite{FerrariMashhoon}.}{��� ��������� ��
��� ������ ��������� ($V_0$ � $b$), �� ������������ �� ���������
������ � ����� ������� ���������� $V(r^*)$ � ��������
\cite{FerrariMashhoon}.}

\lang{The quasi-normal modes of the P\"{o}shl-Teller potential can be
evaluated analytically}{�������������� ���� ��� ����������
������-������� ����������� ���������}:
\be\label{PTqnm}
\om = \frac{1}{b}\left(\sqrt{V_0b^2 - \frac 1 4} -(n + \frac{1}{2})\imo\right).
\ee

\section{\lang{QNMs of SdS black holes}{�������������� ���� ������ �� �����������-�� ѳ����}}\label{calc}
\lang{Both methods give best result for low overtone number but it is known that the
higher modes give larger values of the imaginary part of
frequencies \cite{QNMrev}. Therefore these modes decay faster then
low-lying ones that are more interesting for us. So we calculated
some first quasi-normal mode frequencies for massless scalar,
electromagnetic and gravitational and Dirac fields using both of
the methods. The results are presented below (all values are
measured in the black hole mass (M) units).}{������ ������ �����
����� ���������� ��� ������ ���������, ����� �����, �� ���� ����
����� ����� ����� ������� \cite{QNMrev}. ���� �� ���� ��������
������ � � ���� �������� ��� ������ ��������. � ����� ��������
����� ������ ��������������� ������ ��� ���������� ����������,
�����������������, ������������� �� ĳ���������� ����,
��������� ����� �������� (�� ����� ������ � �������� ���� (M)
����� ���).}
\def\MSF{\lang{QNMs of massless scalar field}{������� ���������� ����}}
\bigskip\newline
\begin{minipage}[c]{9cm}
\centerline{\MSF: $l=1$, $n=0$}
\begin{longtable}{|r|r|r|}
\hline
$\L$ & $\om$ (WKB) & $\om$ (P-T)\\
\hline
   $0$&$0.2929-0.0978\imo$&$0.299-0.101\imo$\\
$0.02$&$0.2603-0.0911\imo$&$0.263-0.093\imo$\\
$0.04$&$0.2247-0.0821\imo$&$0.226-0.083\imo$\\
$0.06$&$0.1854-0.0701\imo$&$0.187-0.071\imo$\\
$0.08$&$0.1404-0.0542\imo$&$0.141-0.055\imo$\\
$0.09$&$0.11392-0.04397\imo$&$0.1147-0.0443\imo$\\
$0.10$&$0.08156-0.03121\imo$&$0.0819-0.0315\imo$\\
$0.11$&$0.02549-0.00965\imo$&$0.02550-0.00967\imo$\\
\hline
\end{longtable}
\end{minipage}
\begin{minipage}[c]{9cm}
\centerline{\MSF: $l=2$, $n=0$}
\begin{longtable}{|r|r|r|}
\hline
$\L$ & $\om$ (WKB) & $\om$ (P-T)\\
\hline
   $0$&$0.48364-0.09677\imo$&$0.487-0.098\imo$\\
$0.02$&$0.43461-0.08858\imo$&$0.437-0.089\imo$\\
$0.04$&$0.38078-0.07876\imo$&$0.382-0.079\imo$\\
$0.06$&$0.32002-0.06685\imo$&$0.321-0.067\imo$\\
$0.08$&$0.24747-0.05197\imo$&$0.248-0.052\imo$\\
$0.09$&$0.20296-0.04256\imo$&$0.2033-0.0427\imo$\\
$0.10$&$0.14661-0.03069\imo$&$0.1468-0.0308\imo$\\
$0.10$&$0.04617-0.00963\imo$&$0.04617-0.00963\imo$\\
\hline
\end{longtable}
\end{minipage}
\bigskip\newline
\begin{minipage}[c]{9cm}
\centerline{\MSF: $l=2$, $n=1$}
\begin{longtable}{|r|r|r|}
\hline
$\L$ & $\om$ (WKB) & $\om$ (P-T)\\
\hline
   $0$&$0.46385-0.29563\imo$&$0.487-0.294\imo$\\
$0.02$&$0.42084-0.26862\imo$&$0.437-0.268\imo$\\
$0.04$&$0.37165-0.23796\imo$&$0.382-0.238\imo$\\
$0.06$&$0.31422-0.20163\imo$&$0.321-0.202\imo$\\
$0.08$&$0.24426-0.15627\imo$&$0.248-0.156\imo$\\
$0.09$&$0.20098-0.12794\imo$&$0.2033-0.1282\imo$\\
$0.10$&$0.14576-0.09212\imo$&$0.1468-0.0923\imo$\\
$0.11$&$0.04614-0.02889\imo$&$0.04617-0.02890\imo$\\
\hline
\end{longtable}
\end{minipage}
\def\EMF{\lang{QNMs of electromagnetic field}{������� ����������������� ����}}
\begin{minipage}[c]{9cm}
\centerline{\EMF: $l=1$, $n=0$}
\begin{longtable}{|r|r|r|}
\hline
$\L$ & $\om$ (WKB) & $\om$ (P-T)\\
\hline
   $0$&$0.2482-0.0926\imo$&$0.255-0.096\imo$\\
$0.02$&$0.2259-0.0842\imo$&$0.231-0.087\imo$\\
$0.04$&$0.2006-0.0748\imo$&$0.204-0.077\imo$\\
$0.06$&$0.1709-0.0639\imo$&$0.173-0.065\imo$\\
$0.08$&$0.1339-0.0502\imo$&$0.135-0.051\imo$\\
$0.09$&$0.11053-0.04156\imo$&$0.1110-0.0419\imo$\\
$0.10$&$0.08035-0.03028\imo$&$0.0805-0.0304\imo$\\
$0.11$&$0.02545-0.00962\imo$&$0.02546-0.00962\imo$\\
\hline
\end{longtable}
\end{minipage}
\bigskip\newline
\begin{minipage}[c]{9cm}
\centerline{\EMF: $l=2$, $n=0$}
\begin{longtable}{|r|r|r|}
\hline
$\L$ & $\om$ (WKB) & $\om$ (P-T)\\
\hline
   $0$&$0.45759-0.09501\imo$&$0.461-0.096\imo$\\
$0.02$&$0.41502-0.08615\imo$&$0.418-0.087\imo$\\
$0.04$&$0.36723-0.07624\imo$&$0.369-0.077\imo$\\
$0.06$&$0.31182-0.06478\imo$&$0.313-0.065\imo$\\
$0.08$&$0.24365-0.05067\imo$&$0.244-0.051\imo$\\
$0.09$&$0.20085-0.04180\imo$&$0.2012-0.0419\imo$\\
$0.10$&$0.14582-0.03037\imo$&$0.1459-0.0304\imo$\\
$0.11$&$0.04614-0.00962\imo$&$0.04615-0.00962\imo$\\
\hline
\end{longtable}
\end{minipage}
\begin{minipage}[c]{9cm}
\centerline{\EMF: $l=2$, $n=1$}
\begin{longtable}{|r|r|r|}
\hline
$\L$ & $\om$ (WKB) & $\om$ (P-T)\\
\hline
   $0$&$0.43653-0.29073\imo$&$0.461-0.289\imo$\\
$0.02$&$0.39900-0.26202\imo$&$0.418-0.261\imo$\\
$0.04$&$0.35602-0.23065\imo$&$0.369-0.231\imo$\\
$0.06$&$0.30498-0.19516\imo$&$0.313-0.196\imo$\\
$0.08$&$0.24046-0.15223\imo$&$0.244-0.153\imo$\\
$0.09$&$0.19907-0.12549\imo$&$0.2012-0.1258\imo$\\
$0.10$&$0.14515-0.09114\imo$&$0.1459-0.0913\imo$\\
$0.11$&$0.04612-0.02886\imo$&$0.04615-0.02887\imo$\\
\hline
\end{longtable}
\end{minipage}
\def\GF{\lang{QNMs of gravitational field}{������� ������������� ����}}
\bigskip\newline
\begin{minipage}[c]{18cm}
\centerline{\GF: $l=2$, $n=0$}
\begin{longtable}{|l|r|r|r|r|}
\hline
$\L$ & $\om$ (WKB) (odd) & $\om$ (P-T) (odd) & $\om$ (WKB) (even) & $\om$ (P-T) (even)\\
\hline
   $0$&$0.3736-0.0889\imo$&$0.378-0.091\imo$&$0.3737-0.0889\imo$&$0.378-0.091\imo$\\
$0.02$&$0.3384-0.0817\imo$&$0.342-0.083\imo$&$0.3384-0.0817\imo$&$0.342-0.083\imo$\\
$0.04$&$0.2989-0.0733\imo$&$0.301-0.074\imo$&$0.2989-0.0733\imo$&$0.301-0.074\imo$\\
$0.06$&$0.2533-0.0630\imo$&$0.254-0.064\imo$&$0.2533-0.0630\imo$&$0.254-0.064\imo$\\
$0.08$&$0.1975-0.0499\imo$&$0.198-0.050\imo$&$0.1975-0.0499\imo$&$0.198-0.050\imo$\\
$0.09$&$0.16261-0.04136\imo$&$0.1629-0.0415\imo$&$0.16261-0.04137\imo$&$0.1629-0.0415\imo$\\
$0.10$&$0.11792-0.03021\imo$&$0.1180-0.0303\imo$&$0.11792-0.03021\imo$&$0.1180-0.0303\imo$\\
$0.11$&$0.03727-0.00962\imo$&$0.03727-0.00962\imo$&$0.03727-0.00962\imo$&$0.03727-0.00962\imo$\\
\hline
\end{longtable}
\end{minipage}
\bigskip\newline
\begin{minipage}[c]{18cm}
\centerline{\GF: $l=2$, $n=1$}
\begin{longtable}{|r|r|r|r|r|}
\hline
$\L$ & $\om$ (WKB) (odd) & $\om$ (P-T) (odd) & $\om$ (WKB) (even) & $\om$ (P-T) (even)\\
\hline
   $0$&$0.3463-0.2735\imo$&$0.378-0.272\imo$&$0.3467-0.2739\imo$&$0.378-0.272\imo$\\
$0.02$&$0.3185-0.2488\imo$&$0.342-0.249\imo$&$0.3188-0.2491\imo$&$0.342-0.249\imo$\\
$0.04$&$0.2858-0.2215\imo$&$0.301-0.223\imo$&$0.2858-0.2217\imo$&$0.301-0.223\imo$\\
$0.06$&$0.2457-0.1896\imo$&$0.254-0.191\imo$&$0.2457-0.1898\imo$&$0.254-0.191\imo$\\
$0.08$&$0.1941-0.1497\imo$&$0.198-0.150\imo$&$0.1941-0.1498\imo$&$0.198-0.150\imo$\\
$0.09$&$0.16079-0.12412\imo$&$0.1629-0.1246\imo$&$0.16079-0.12415\imo$&$0.1629-0.1246\imo$\\
$0.10$&$0.11724-0.09063\imo$&$0.1180-0.0908\imo$&$0.11724-0.09064\imo$&$0.1180-0.0908\imo$\\
$0.11$&$0.03725-0.02885\imo$&$0.03727-0.02885\imo$&$0.03725-0.02885\imo$&$0.03727-0.02885\imo$\\
\hline
\end{longtable}
\end{minipage}
\bigskip\newline
\begin{minipage}[c]{18cm}
\centerline{\GF: $l=3$, $n=0$}
\begin{longtable}{|r|r|r|r|r|}
\hline
$\L$ & $\om$ (WKB) (odd) & $\om$ (P-T) (odd) & $\om$ (WKB) (even) & $\om$ (P-T) (even)\\
\hline
   $0$&$0.599443-0.092703\imo$&$0.602-0.093\imo$&$0.599443-0.092703\imo$&$0.602-0.093\imo$\\
$0.02$&$0.543115-0.084495\imo$&$0.545-0.085\imo$&$0.543115-0.084496\imo$&$0.545-0.085\imo$\\
$0.04$&$0.480058-0.075146\imo$&$0.481-0.076\imo$&$0.480058-0.075146\imo$&$0.481-0.076\imo$\\
$0.06$&$0.407175-0.064140\imo$&$0.408-0.064\imo$&$0.407175-0.064140\imo$&$0.408-0.064\imo$\\
$0.08$&$0.317805-0.050382\imo$&$0.318-0.051\imo$&$0.317804-0.050382\imo$&$0.318-0.051\imo$\\
$0.09$&$0.261841-0.041644\imo$&$0.2621-0.0417\imo$&$0.261843-0.041643\imo$&$0.2621-0.0417\imo$\\
$0.10$&$0.189994-0.030314\imo$&$0.1901-0.0303\imo$&$0.189994-0.030314\imo$&$0.1901-0.0303\imo$\\
$0.11$&$0.060091-0.009619\imo$&$0.06009-0.00962\imo$&$0.060091-0.009619\imo$&$0.06009-0.00962\imo$\\
\hline
\end{longtable}
\end{minipage}
\bigskip\newline
\begin{minipage}[c]{18cm}
\centerline{\GF: $l=3$, $n=1$}
\begin{longtable}{|r|r|r|r|r|}
\hline
$\L$ & $\om$ (WKB) (odd) & $\om$ (P-T) (odd) & $\om$ (WKB) (even) & $\om$ (P-T) (even)\\
\hline
   $0$&$0.58264-0.28129\imo$&$0.602-0.280\imo$&$0.58264-0.28129\imo$&$0.602-0.280\imo$\\
$0.02$&$0.53074-0.25536\imo$&$0.545-0.255\imo$&$0.53074-0.25536\imo$&$0.545-0.255\imo$\\
$0.04$&$0.47166-0.22639\imo$&$0.481-0.227\imo$&$0.47166-0.22639\imo$&$0.481-0.227\imo$\\
$0.06$&$0.40217-0.19281\imo$&$0.408-0.193\imo$&$0.40217-0.19281\imo$&$0.408-0.193\imo$\\
$0.08$&$0.31550-0.15125\imo$&$0.318-0.152\imo$&$0.31549-0.15125\imo$&$0.318-0.152\imo$\\
$0.09$&$0.26056-0.12498\imo$&$0.2621-0.1251\imo$&$0.26057-0.12497\imo$&$0.2621-0.1251\imo$\\
$0.10$&$0.18952-0.09095\imo$&$0.1901-0.0910\imo$&$0.18952-0.09095\imo$&$0.1901-0.0910\imo$\\
$0.11$&$0.06008-0.02886\imo$&$0.06009-0.02886\imo$&$0.06008-0.02886\imo$&$0.06009-0.02886\imo$\\
\hline
\end{longtable}
\end{minipage}
\def\MDF{\lang{QNMs of massless Dirac field}{������� ����������� ���� ĳ����}}
\bigskip\newline
\begin{minipage}[c]{18cm}
\centerline{\MDF: $\k=1$, $n=0$}
\begin{longtable}{|r|r|r|r|r|}
\hline
$\L$ & $\om$ (WKB) (-)& $\om$ (P-T) (-)& $\om$ (WKB) (+) & $\om$ (P-T) (+)\\
\hline
   $0$&$0.183-0.097\imo$&$0.183-0.098\imo$&$0.183-0.095\imo$&$0.189-0.105\imo$\\
$0.02$&$0.167-0.087\imo$&$0.166-0.088\imo$&$0.167-0.085\imo$&$0.171-0.094\imo$\\
$0.04$&$0.149-0.077\imo$&$0.147-0.077\imo$&$0.149-0.075\imo$&$0.151-0.082\imo$\\
$0.06$&$0.128-0.069\imo$&$0.127-0.065\imo$&$0.128-0.064\imo$&$0.128-0.069\imo$\\
$0.08$&$0.1005-0.0507\imo$&$0.0985-0.0505\imo$&$0.1006-0.0501\imo$&$0.1001-0.0529\imo$\\
$0.09$&$0.0832-0.0418\imo$&$0.0813-0.0415\imo$&$0.0833-0.0414\imo$&$0.0824-0.0432\imo$\\
$0.10$&$0.0606-0.0304\imo$&$0.0592-0.0301\imo$&$0.0606-0.0303\imo$&$0.0597-0.0311\imo$\\
$0.11$&$0.0192-0.0096\imo$&$0.0188-0.0096\imo$&$0.0192-0.0096\imo$&$0.0189-0.0097\imo$\\
\hline
\end{longtable}
\end{minipage}
\bigskip\newline
\begin{minipage}[c]{18cm}
\centerline{\MDF: $\k=2$, $n=0$}
\begin{longtable}{|r|r|r|r|r|}
\hline
$\L$ & $\om$ (WKB) (-)& $\om$ (P-T) (-)& $\om$ (WKB) (+) & $\om$ (P-T) (+)\\
\hline
   $0$&$0.38004-0.09642\imo$&$0.382-0.097\imo$&$0.38007-0.09637\imo$&$0.386-0.099\imo$\\
$0.02$&$0.34491-0.08715\imo$&$0.346-0.088\imo$&$0.34493-0.08712\imo$&$0.349-0.089\imo$\\
$0.04$&$0.30542-0.07690\imo$&$0.306-0.077\imo$&$0.30543-0.07688\imo$&$0.308-0.079\imo$\\
$0.06$&$0.25953-0.06516\imo$&$0.260-0.065\imo$&$0.25954-0.06515\imo$&$0.261-0.066\imo$\\
$0.08$&$0.20296-0.05085\imo$&$0.203-0.051\imo$&$0.20296-0.05084\imo$&$0.204-0.052\imo$\\
$0.09$&$0.16738-0.04190\imo$&$0.167-0.0418\imo$&$0.16738-0.04190\imo$&$0.168-0.0423\imo$\\
$0.10$&$0.12157-0.03041\imo$&$0.121-0.0304\imo$&$0.12157-0.03041\imo$&$0.122-0.0305\imo$\\
$0.11$&$0.03849-0.00962\imo$&$0.0384-0.0096\imo$&$0.03849-0.00962\imo$&$0.0384-0.0096\imo$\\
\hline
\end{longtable}
\end{minipage}
\bigskip

\lang{The relative error of the WKB method depends significantly on a
field under consideration. Comparison of the sixth and fifth WKB
order let us to conclude that the worst WKB convergence gives the
Dirac field, where the expected error reaches 1\% in the sixth WKB
order. The results calculated in \cite{Cho} by using the third WKB
order differ from the sixth order values up to 4\%. Thus, the
possible relative error of the third order WKB results may exceed
4\%.}{³������ ������� ������ WKB ������� �������� �� ����, ��
������������. ��������� ������� � �'����� ������� ��������
������� ��������, �� ������� ������� WKB �������� ���
ĳ���������� ����, �� ������� ���������� � 1\% ��� �������
�������. ���������� ���������� � \cite{Cho} �� ��������� ��������
������� WKB ����������� �� ���������� ������� ������� �� 4\%,
���� ������� ������� ������� �������� ������� WKB ����
������������ 4\%.}

\lang{We can also evaluate the relative error order for gravitational modes by
comparison QNMs of odd and even perturbations. Because the
potentials can be expressed in the form}{³������ ������� ���
������������ ���, ����� ������� ����� ��������� �������, ��
���������� ���������� �� �������� ���������. ������� ��������
���������� ������ ���� ������������ � ������}
\be\label{goec}
V_{odd}= W_g^2 + \frac{dW_g}{dr^*}+\b; \qquad V_{even} = W_g^2 -
\frac{dW_g}{dr^*}+\b;
\ee
\lang{where}{��}
$$W_g = \frac{2M}{r^2}-\frac{3+2c}{3r}+\frac{3c^2+2c^3-9\L M^2}{3c(3M+cr)}-\frac{1}{3M}(c^2+c-\frac{3\L M^2}{c}); \qquad \b = -\frac{c^2(c+1)^2}{9M^2};$$
\lang{any solution for even parity can be found from the solution for
odd parity}{������ ����'���� ��� ������ ������� ����� ������ �
���������� ����'���� ��� �������� �������}
\cite{Chandrasekhar}:
$$\P_{even}(r^*)=q\left(W_g(r^*) - \frac{d}{dr^*}\right)\P_{odd}(r^*); \qquad q = const.$$
\lang{Obviously, if a solution $\P_{odd}$ describing quasi-normal
oscillations satisfies the boundary conditions \p{bounds}, the
$\P_{even}$ solution satisfies the same ones describing
quasi-normal oscillations too.}{��������, �� ���� ����'����
$\P_{odd}$, �� ����� �������������� ��������� �����������
�������� ����� \p{bounds}, ����'���� $\P_{even}$ ����� �����������
\p{bounds}.}

\lang{The same fact \cite{Potentials} takes place for massless Dirac
QNMs of opposite chiralities}{����� ����� ���� \cite{Potentials}
�� ���� ��� ��������������� ��� ���������� ����������
����������� ĳ�������� ����}:
\be\label{docc}
V_d = W_d^2 \pm \frac{dW_d}{dr^*}; \qquad W_d(r) =
\frac{\k\sqrt{f(r)}}{r}.
\ee

\lang{Therefore, the QNM spectrum is the same for odd and even
gravitational perturbations and for opposite chirality of Dirac
perturbations in SdS background as well as in asymptotically flat
universe. We see that the difference between the numerical results
does not exceed our expectations.}{����� ��, �� �� �����������-��
ѳ����, ��� ����, �� � �� ����������� �������� ��, ������
��������������� ������ � ��������� ��� ���������� � ��������
������� ������������� ����, � ����� ������� ĳ���������� ����
���������� ����������. ����� ������, �� ������ �� ����������
���������� �� �������� ��������� �������.}

\lang{We can see that the imaginary parts of QNMs, calculated using both
sixth-order WKB and P\"{o}shl-Teller approaches, depend on
multipole and overtone number and black hole mass only near the
extremal value of $\L$. This agrees with \cite{Cardoso-Lemos2},
where the QNMs of near extremal SdS space-time were found}{�����
������, �� ����� ������� ��������������� ������, ���������� ��
��������� ������� ������� WKB � ������ ������-�������, ���
�������������� �������� $\L$ �������� ����� �� ������ �������� �
����������, � ����� �� ���� ����� ���. �� ����������� �
\cite{Cardoso-Lemos2}, �� �������� �������������� ������� ���
����� ������������� ������ ��}:
\be\label{ese}
\om b = -\left(n+\frac 1 2\right)\imo + \sqrt{l(l+1)-\frac 1
4}
\ee
\lang{for scalar and electromagnetic perturbations}{��� ������� ���������� �� ����������������� ���� �},
\be\label{eg}
\om b = -\left(n+\frac 1 2\right)\imo + \sqrt{(l+2)(l-1)-\frac 1
4}
\ee
\lang{for gravitational perturbations. Here}{��� �������������. ���}
$$b = \frac{54M^3}{(r_c-3M)(r_c + 6M)}, \qquad r_c\rightarrow 3M.$$
\lang{Using the same technique, one can find that}{������������ �����
����� ��������, ����� ������, ��}
\be\label{eds}
\om b = -\left(n+\frac 1 2\right)\imo + \sqrt{\k^2-\frac 1
4}
\ee
\lang{for massless Dirac field perturbations. It means in the extremal
limit fields of different spin decay with the same rate. Note,
that formulae \p{ese}, \p{eg}, \p{eds} give right results for any
overtone number $n$ and may be useful for studying the
$n\rightarrow\infty$ limit.}{��� ������� ����������� ĳ����������
����. �� ������, �� � ������������� ������� ���� ������ ����
�������� � ��������� ��������. ���������, �� ������� \p{ese},
\p{eg}, \p{eds} � ����������� ��� ����-����� ������ �������� $n$ � ������
��������� ��������� ��� �������� ������� $n\rightarrow\infty$.}

\lang{The well-known approximate formula for large multipole number $l$ (or $\k$ in
\p{mdp})}{³���� ��������� ������� ��� ������� �������� ������ ���������� $l$ (��� $\k$
� \p{mdp})}
\cite{FerrariMashhoon,Press}:
\bea\label{aflls}
\om &=& \frac{1}{3\sqrt 3 M}\left(l + \frac 1 2 - \left(n + \frac 1 2\right)\imo\right) + O\left(\frac 1 l\right);
\\\nonumber\om &=& \frac{1}{3\sqrt 3 M}\left(\k - \left(n + \frac 1 2\right)\imo\right) + O\left(\frac 1 \k\right);
\eea
\lang{can be generalized for the case of non-zero $\L$. Making use of
the first-order WKB method or the formula of the approximation by
the P\"{o}shl-Teller potential one can find for the non-extremal
SdS black hole}{���� ���� ����������� �� ������� ���������� $\L$.
������������ ������ �������� ������� WKB ��� �����������
����������� ������-�������, ����� ������, �� ��� ��������������
����� ��� �����������-�� ѳ����}:
\bea\label{afllads}
\om &=& \frac{\sqrt{1 - 9M^2\L}}{3\sqrt 3 M}\left(l + \frac 1 2 - \left(n + \frac 1 2\right)\imo\right) + O\left(\frac 1 l\right);
\\\nonumber\om &=& \frac{\sqrt{1 - 9M^2\L}}{3\sqrt 3 M}\left(\k - \left(n + \frac 1 2\right)\imo\right) + O\left(\frac 1 \k\right).
\eea

\section{\lang{Conclusion}{����������}}
\lang{The QNMs of Schwarzschild-de Sitter black holes for fields of different spin have been
calculated. The modes are determined by the black hole mass $M$
and the cosmological constant $\L$ only. The frequencies all have
a negative imaginary part, which means that the black hole is
stable against these perturbations. The presence of the
cosmological constant leads to decrease of the real oscillation
frequency and to a slower decay. The interesting problem that was
outside our consideration is the search of asymptotic behavior of
SdS QNMs at large imaginary part ($n \rightarrow \infty$) for
which other approaches should be explored
\cite{Nollert}.}
{���� ��������� �������������� ���� ����� ��� �����������-��
ѳ����. ֳ ���� ������������ ����� ����� ����� ��� $M$ �
������������ ������ $\L$. �� ������� ����� ��'���� �����
�������, �� ������ ����������� ����� ��� ������� �����������
�������. ��������� ����������� ����� ���������� �� ���������
����� ������� �������� � ���� ��������� ��������. ֳ�����
���������, ��� ���������� ���� ���������, � ����� �����������
�������� ��������������� ��� � ������� �����������-�� ѳ���� ���
������ ����� ������� ($n \rightarrow \infty$), ��� ����'����
��� ��� ��������������� ���� ������ \cite{Nollert}.}


\section*{\lang{Acknowledgements}{������}}
\lang{I would like to thank R. Konoplya for proposing this problem to me
and reading the manuscript.}{���� ���������� ������ ������ ��
������������� ������ � �������� ���� �����.}

\begin{thebibliography}{99}
\def\PRD #1 {{\sl Phys. Rev.} {\bf D#1} }
\def\PRDatpress {{\sl Phys. Rev.} {\bf D} at press }
\def\PRev #1 {{\sl Phys. Rev.} {\bf  #1} }
\def\PRL #1 {{\sl Phys. Rev. Lett.} {\bf #1} }
\def\PLB #1 {{\sl Phys. Lett.} {\bf B#1} }
\def\JPA #1 {{\sl J. Physics} {\bf A#1} }
\def\AJ #1 {{\sl Astrophys. J.} {\bf #1} }
\def\CQG #1 {{\sl Class. Quant. Grav.} {\bf #1} }
\def\GRG #1 {{\sl Gen. Relat. Grav.} {\bf #1} }
\def\LRR #1 {{\sl Living. Reviews. Relativ.} {\bf #1} }

\bibitem{Brady-Chambers} P. Brady, C. Chambers, W. Krivan and P. Laguna, \PRD 55 7538 (1997)
\\ P. Brady, C. Chambers, W. Laarakkers and E. Poison, \PRD 60 064003 (1999)

\bibitem{QNMrev} K. Kokkotas and B. Schmidt, \LRR 2 2 (1999);\\
H. P. Nollert, \CQG 16 R159 (1999)

\bibitem{Horowitz-Hubeny} G. T. Horowitz and V. Hubeny, \PRD 62 024027 (2000)
\bibitem{CFT-Correspond} V. Cardoso and J. P. S. Lemos \PRD 63 124015 (2001)
\\ J. S. F. Chan and R. B. Mann, \PRD 55 7546-7562 (1997)
\\ J. S. F. Chan and R. B. Mann, \PRD 59 064025 (1999)
\\ D. Birmingham, I. Sachs, and S. N. Solodukhin, \PRL 88 151301 (2002)
\\ D. Birmingham, I. Sachs, and S. N. Solodukhin, \PRDatpress hep-th/0212308 (2002)
\\ R. A. Konoplya, \PRD 66 044009 (2002)
\\ R. A. Konoplya, \PRD 66 084007 (2002)
\\ A. O. Starinets, \PRD 66 124013 (2002)
\\ V. Cardoso, R. A. Konoplya and J. P. S. Lemos, \PRDatpress gr-qc/0305037
\\ E. Abdalla, K. H. C. Castello-Branco and A. Lima-Santos, hep-th/0208065
\\ R. Aros, C. Martinez, R. Troncoso, and J. Zanelli, \PRD 67 044014 (2003)
\\ S. Fernando, hep-th/0306214

\bibitem{Moss-Norman} I. G. Moss and J. P. Norman, \CQG 19 2323 (2002)

\bibitem{LQG} O. Dreyer, \PRL 90 081301 (2003)
\\ G. Kunstatter, \PRL 90 161301 (2003)
\\ L. Motl, gr-qc/0212096

\bibitem{BHCC} R.A.Konoplya, \PLB 550 117 (2002)
\\ W. T. Kim and J. J. Oh, \PLB 514 155 (2001)

\bibitem{Otsuki} H. Otsuki and T. Futamase, {\sl Progress of Theoretical
Physics} {\bf{85}} 771 (1991)
\bibitem{Mellor-Moss} F. Mellor and I. Moss, \PRD 41 403 (1990)

\bibitem{Cho} H. T. Cho, \PRD 68 024003 (gr-qc/0303078)
\bibitem{Cardoso-Lemos2} V. Cardoso and J. P. S. Lemos, \PRD 67 084020
(gr-qc/0301078)

\bibitem{Potentials} T. Regge and J. A. Wheeler \PRev 108 1063-1069 (1957)
\\U. Khanal \PRD 28 1291 (1983)

\bibitem{SchutzWill} B. F. Schutz and C. M. Will \AJ 291 L33-L36 (1985)
\bibitem{IyerWill} S. Iyer and C. M. Will, \PRD 35 3621 (1987)

\bibitem{Konoplya} R. A. Konoplya, \PRD 68 024018 (2003) (gr-qc/0303052)

\bibitem{WKB} R. A. Konoplya, \GRG 34 329 (2002)
\\ O. B. Zaslavskii, \PRD 43 605 (1991)
\\ L. E. Simone and C. M. Will, \CQG 9 963 (1992)
\\ S. Iyer, \PRD 35 3632 (1987)
\\ K. Kokkotas, and B. F. Schutz, \PRD 37 3378 (1988)
\\ E. Berti \& K. Kokkotas, \PRD 67 064020 (2003)
\\ E. Berti \& K. Kokkotas, hep-th/0303029
\\ E. Berti, V. Cardoso, K. Kokkotas \& Onozawa, hep-th/0307013

\bibitem{Chandrasekhar} S. Chandrasekhar, in \emph{The Mathematical
Theory of Black Holes}, Oxford University Press, New York (1983)

\bibitem{FerrariMashhoon} V. Ferrari, B. Mashhoon \PRD 30 295-304 (1984)
\bibitem{Press} W. H. Press, \AJ 170 L105 (1971)

\bibitem{Nollert} H.-P. Nollert \PRD 47 5253 (1993)

\bibitem{Suneeta} V. Suneeta \PRDatpress gr-qc/0303114
\end{thebibliography}
\end{document}

gr-qc0307012.extracted_bib
\begin{thebibliography}{99}
\def\PRD #1 {{\sl Phys. Rev.} {\bf D#1} }
\def\PRDatpress {{\sl Phys. Rev.} {\bf D} at press }
\def\PRev #1 {{\sl Phys. Rev.} {\bf  #1} }
\def\PRL #1 {{\sl Phys. Rev. Lett.} {\bf #1} }
\def\PLB #1 {{\sl Phys. Lett.} {\bf B#1} }
\def\JPA #1 {{\sl J. Physics} {\bf A#1} }
\def\AJ #1 {{\sl Astrophys. J.} {\bf #1} }
\def\CQG #1 {{\sl Class. Quant. Grav.} {\bf #1} }
\def\GRG #1 {{\sl Gen. Relat. Grav.} {\bf #1} }
\def\LRR #1 {{\sl Living. Reviews. Relativ.} {\bf #1} }

\bibitem{Brady-Chambers} P. Brady, C. Chambers, W. Krivan and P. Laguna, \PRD 55 7538 (1997)
\\ P. Brady, C. Chambers, W. Laarakkers and E. Poison, \PRD 60 064003 (1999)

\bibitem{QNMrev} K. Kokkotas and B. Schmidt, \LRR 2 2 (1999);\\
H. P. Nollert, \CQG 16 R159 (1999)

\bibitem{Horowitz-Hubeny} G. T. Horowitz and V. Hubeny, \PRD 62 024027 (2000)
\bibitem{CFT-Correspond} V. Cardoso and J. P. S. Lemos \PRD 63 124015 (2001)
\\ J. S. F. Chan and R. B. Mann, \PRD 55 7546-7562 (1997)
\\ J. S. F. Chan and R. B. Mann, \PRD 59 064025 (1999)
\\ D. Birmingham, I. Sachs, and S. N. Solodukhin, \PRL 88 151301 (2002)
\\ D. Birmingham, I. Sachs, and S. N. Solodukhin, \PRDatpress hep-th/0212308 (2002)
\\ R. A. Konoplya, \PRD 66 044009 (2002)
\\ R. A. Konoplya, \PRD 66 084007 (2002)
\\ A. O. Starinets, \PRD 66 124013 (2002)
\\ V. Cardoso, R. A. Konoplya and J. P. S. Lemos, \PRDatpress gr-qc/0305037
\\ E. Abdalla, K. H. C. Castello-Branco and A. Lima-Santos, hep-th/0208065
\\ R. Aros, C. Martinez, R. Troncoso, and J. Zanelli, \PRD 67 044014 (2003)
\\ S. Fernando, hep-th/0306214

\bibitem{Moss-Norman} I. G. Moss and J. P. Norman, \CQG 19 2323 (2002)

\bibitem{LQG} O. Dreyer, \PRL 90 081301 (2003)
\\ G. Kunstatter, \PRL 90 161301 (2003)
\\ L. Motl, gr-qc/0212096

\bibitem{BHCC} R.A.Konoplya, \PLB 550 117 (2002)
\\ W. T. Kim and J. J. Oh, \PLB 514 155 (2001)

\bibitem{Otsuki} H. Otsuki and T. Futamase, {\sl Progress of Theoretical
Physics} {\bf{85}} 771 (1991)
\bibitem{Mellor-Moss} F. Mellor and I. Moss, \PRD 41 403 (1990)

\bibitem{Cho} H. T. Cho, \PRD 68 024003 (gr-qc/0303078)
\bibitem{Cardoso-Lemos2} V. Cardoso and J. P. S. Lemos, \PRD 67 084020
(gr-qc/0301078)

\bibitem{Potentials} T. Regge and J. A. Wheeler \PRev 108 1063-1069 (1957)
\\U. Khanal \PRD 28 1291 (1983)

\bibitem{SchutzWill} B. F. Schutz and C. M. Will \AJ 291 L33-L36 (1985)
\bibitem{IyerWill} S. Iyer and C. M. Will, \PRD 35 3621 (1987)

\bibitem{Konoplya} R. A. Konoplya, \PRD 68 024018 (2003) (gr-qc/0303052)

\bibitem{WKB} R. A. Konoplya, \GRG 34 329 (2002)
\\ O. B. Zaslavskii, \PRD 43 605 (1991)
\\ L. E. Simone and C. M. Will, \CQG 9 963 (1992)
\\ S. Iyer, \PRD 35 3632 (1987)
\\ K. Kokkotas, and B. F. Schutz, \PRD 37 3378 (1988)
\\ E. Berti \& K. Kokkotas, \PRD 67 064020 (2003)
\\ E. Berti \& K. Kokkotas, hep-th/0303029
\\ E. Berti, V. Cardoso, K. Kokkotas \& Onozawa, hep-th/0307013

\bibitem{Chandrasekhar} S. Chandrasekhar, in \emph{The Mathematical
Theory of Black Holes}, Oxford University Press, New York (1983)

\bibitem{FerrariMashhoon} V. Ferrari, B. Mashhoon \PRD 30 295-304 (1984)
\bibitem{Press} W. H. Press, \AJ 170 L105 (1971)

\bibitem{Nollert} H.-P. Nollert \PRD 47 5253 (1993)

\bibitem{Suneeta} V. Suneeta \PRDatpress gr-qc/0303114
\end{thebibliography}