To be more specific, by evolving a perturbation on a black hole background one can get a QNM signal as the one shown in figure 4, but in this signal the Fourier transform will show that there are present at most two frequencies (the slowest damped ones); then by doing ``matched filtering'' of this signal we will get the right frequencies and damping times, but then all the extra modes of the spectrum are missing! See for example the work by Bachelot and Motet-Bachelot [35, 34], and Krivan et al [132, 133]. This is true also for stars; in recent evolutions of axial perturbations of stellar backgrounds Andersson and Kokkotas [18] saw only a few of the w -modes (the ones that damped slowest), while in similar calculations for even parity stellar perturbations Allen et al. [5] have seen only the f -mode, 2-3 p -modes and two of the w -modes.
Of course with more detailed studies for various sets of initial data one might be successful to get a few more modes, but more important than deriving extra modes is understanding the physical situation which generates the appropriate initial data.
................................... Finally, the evolutions of the time dependent perturbation equations can be extremely useful (and probably will be the only way) for the calculation of the QNM frequencies and waveforms for the perturbations of the Kerr-Newman black hole and for slowly and fast rotating relativistic stars.
The approach used by Nollert and Schmidt [156,
159] is more elaborate and based on a better estimate of the values
of the quasi-eigenfunctions on both boundaries (
); this leads to a more accurate estimate of frequencies and one
also finds frequencies which damp extremely fast.
Andersson [8] suggested an alternative integration scheme. The key idea is to
separate ingoing and outgoing wave solutions by numerically
calculating their analytic continuations to a place in the
complex
r
-coordinate plane where they have comparable amplitudes. This
method is extremely accurate.
where
are the two roots (turning points) of
. A more general form can be found [82,
41] which is valid for complex potentials. This form can be
extended to the complex
r
plane where the contour encircles the two turning points which
are connected by a branch cut. In this way one can calculate the
eigenfrequencies even in the case of complex potentials, as it is
the case for Kerr black holes.
This method has been used for the calculation of the
eigenfrequencies of the Schwarzschild [113], Reissner-Nordström [129], Kerr [185,
123] and Kerr-Newman [124
] black holes (restricted case). In general with this approach
one can calculate quite accurately the low-lying (relatively
small imaginary part) QNM modes, but it fails to give accurate
results for higher-order modes.
This WKB approach was improved considerably when the phase integral formalism of Fröman and Fröman [97] was introduced. In a series of papers [96, 25, 15, 32] the method was developed and a great number of even extremely fast damped QNMs of the Schwarzschild black hole have been calculated with a remarkable accuracy. The application of the method for the calculation of QNM frequencies of the Reissner-Nordström black hole [16] has considerably improved earlier results [129] for the QNMs which damp very fast. Close to the logic of this WKB approach were the attempts of Blome, Mashhoon and Ferrari [46, 86, 85] to calculate the QNM modes using an inversion of the black hole potential. Their method was not very accurate but stimulated future work using semianalytic methods for estimating QNMs.
His approach was analogous to the determination of the
eigenvalues of the H
ion developed in [33]. A series representation of the solution
is assumed to represent also
for the value of the quasi-normal mode frequency. For normal
modes the method may work because
is certainly bounded at infinity. In the case of quasi-normal
modes this is not so clear because
grows exponentially. Nevertheless, the method works very well
numerically and was improved by Nollert [157] such that he was able to calculate very high mode numbers (up
to 100,000!). In this way he obtained the asymptotic
distributions of modes described in (31
). An alternative way of using the recurrence relations was
suggested in [145].
Nollert [156] explains in his PhD thesis why the method works. As initiated
by Heisenberg et al. [111] he considers potentials depending analytically on a parameter
such that for
the potential has bound states - normal modes - and for for
just quasi-normal modes. This is, for example, the case if we
multiply the Regge-Wheeler potential (23
) by
. Assuming that the modes depend continuously on
, one can try to relate normal modes to quasi-normal modes and
their methods of calculation.
In the case of QNMs of the Kerr black hole, one has to deal in
practice with two coupled equations, one which governs the radial
part (40) and another which governs the angular dependence of the
perturbation (39
). For both of them one can construct recurrence relations for
the coefficients of the series expansion of their solutions, and
through them calculate the QNM frequencies.
For the case of the QNMs of the Reissner-Nordström black hole,
the asymptotic form of the solutions is similar to that shown in
equation (28) but the coefficients
are determined via a four term recurrence relation. This means
that the nice properties of convergence of the three term
recurrence relations have been lost and one should treat the
problem with great caution. Nevertheless, Leaver [137] has overcome this problem and showed how to calculate the QNMs
for this case.
As a final comment on this excellent method we should point out that it has a disadvantage compared to the WKB based methods in that it is a purely numerical method and it cannot provide much intuition about the properties of the QNM spectrum.
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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 |