Therefore the spectrum contains all the information about
stability. To discuss stability for systems with quasi-normal
modes, let us consider a case like equation (5) with the assumption that
V
is of compact support but not necessarily positive.
Data of compact support define solutions which grow at most exponentially in time
where
a
is independent of the data. As outlined in appendix
9, eigenvalues necessarily have
and the eigenfunctions determine solutions growing exponentially
in time. If no eigenvalues exist, the solution can not grow
exponentially. Polynomial growth is still possible and related to
the properties of the Laplace transform of the Green function at
s
=0. As the potential has compact support, the functions
are analytic for all
s
. Hence, the Green function can at most have a pole at
s
=0. A pole of order two and higher implies polynomial growth in
time. If the potential is positive, energy conservation shows
that the field can grow at most linearly in time and therefore we
can have at most a pole of order 2 at
s
=0.
If we define stability as boundedness in time for all solutions with data of compact support, properties of quasi-normal modes can not decide the stability issue. However, the appearance of a normal mode proves instability. If the support of the potential is not compact everything becomes more complicated. In particular, it is a non trivial problem to obtain the behavior of the Green function at s =0.
In the case of the Schwarzschild black hole, stability is demonstrated by Kay and Wald [117] who showed the boundedness of all solutions with data of compact support.
The issue is more subtle for Kerr. There is a conserved
energy, but because of the ergoregion its integrand is not
positive definite, hence the conserved energy could be finite
while the field still might grow exponentially in parts of the
spacetime. Papers by Press and Teukolsky [165], Hartle and Wilkins [109], and Stewart [193] try to exclude the existence of an exponentially growing normal
mode. Their work makes the stability very plausible but is not as
conclusive as the Wald-Kay result. However this is a delicate
issue as we see if, for example, we multiply the Regge-Wheeler
potential by a factor
: For any
we obtain an infinite number of QNMs, for
, however there is no QNM! Whiting [208] has proven that there are no exponentially growing modes, and
in his proof he showed that the growth of the modes is at most
linear. Recent numerical evolution calculations [132
,
133
] for slowly and fast rotating Kerr black holes pick up all the
expected features (QNM ringing, tails) and show no sign of
exponential growth. It should be noted that the massive scalar
perturbations of Kerr are known to be unstable [72,
214,
77]. These unstable modes are known to be very slowly growing (with
growth times similar to the age of universe).
Let us finally turn to the ``completeness of QNMs''. A general
mathematical theorem (spectral theorem) implies that for systems
like strings or membranes the general solution can be expanded
into a converging sum of normal modes. A similar result can not
be expected for QNMs, the reasons are given in section
2
. There is, however, the possibility that an infinite sum of the
form (15) will be a representation of a solution for late times. This
property has been shown by Beyer [42] for the Pöschl-Teller potential which has a similar form as the
potential on Schwarzschild (23
). The main difference is its exponential decay at both ends.
In [158] Nollert and Price propose a definition of completeness and show
its adequateness for a particular model problem. There are also
systematic studies [63] about the relation between the structure of the QNM's of the
Klein-Gordon equation and the form of the potential. In these
studies there is a discussion on both the requirements for QNMs
to form a complete set and the definition of completeness.
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Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt http://www.livingreviews.org/lrr-1999-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |