Stability and completeness of quasi-normal modes

3.3 Stability and completeness of quasi-normal modes

From the normal modes one can learn a lot about stability. Take as an example linear stellar oscillation within the framework of Newton’s theory of gravity. As outlined at the beginning of Section 2, we have a sequence of normal modes $ωn$ with $2 ω$ real. The general solution is a convergent linear combination of the corresponding eigenfunctions. Hence all eigenfunctions are bounded if $ω$ is purely imaginary, i.e. $ω2 < 0$ .

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 A, eigenvalues necessarily have $sn > 0$ 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 $f (s,x) ±$ 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 $𝜖$ : For any $𝜖 > 0$ we obtain an infinite number of QNMs, for $𝜖 = 0$ , 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.