Black holes

6.1 Black holes

We would like to point out here that the attempts to calculate the QNM frequencies date back to the beginning of 1970s. More specifically in their study of the black hole oscillations excited by an infalling particle Davis et al. [73 ] found that the peak of the spectrum is (for a Schwarzschild black hole) at around $M ω = 0.32$ (geometrical units). This number is very close to the one calculated by much more accurate methods later. In what will follow we will describe the various methods used to calculate the QNM frequencies.

6.1.1 Evolving the time dependent wave equation

This approach was actually the first one used to study the QNM excitation by Vishveshwara [207 ] but it has only been recently revived thanks to increased power of computers. In general, one does not need to Fourier decompose the perturbed Einstein equations but instead evolve them for given sets of initial data and at the end to Fourier analyze the resulting waveform. This procedure has certain advantages since one does not need to be so careful in considering the appropriate boundary conditions on the horizon, at infinity, on the surface or at the center of the star. This does not mean that these boundaries are not important for the evolution schemes. The difference is that in the time independent case one formulates a boundary value problem, and the eigenvalues (quasi-normal frequencies) depend critically on the correct conditions on the various boundaries. A major disadvantage of the evolution schemes is that one cannot get the complete spectrum of the QNMs neither for a star nor for a black hole. The reason is that although any perturbation is the sum of the harmonics involved, in practice only a few of them will be observed and in the best case one can succeed in getting a few extra modes by “playing” around with the initial data.

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 [34 , 35 ], 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. [6 ] 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.

6.1.2 Integration of the time independent wave equation

This technique was used by Chandrasekhar and Detweiler [58 ] and is based on the definition of QNMs given in Section 2. They assumed that a QNM is a solution corresponding to incoming waves on the horizon and outgoing at infinity. Then by taking a series expansion of the Zerilli wave equation (21 , 23 , 24 ) at both limits (horizon and infinity) of the form given by (27 ) they found initial values for the numerical integration of the equation. Their integration goes from both limits towards a common point which was set close to the peak of the potential i.e. around $r = 3M$ . The values of $ω$ for which the Wronskian of the two numerically taken solutions vanishes are the quasi-eigensolutions of the problem. In this way they managed to calculate the first 2 – 3 QNM frequencies of the Schwarzschild black hole for various harmonic indices. The accuracy of the method improves for increasing $ℓ$ . Later, Gunter [108 ] and Kokkotas with Schutz [129 ] used the same approach to calculate the QNMs of the Reissner–Nordström black hole.

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.

6.1.3 WKB methods

This technique, originally in the form suggested by Schutz and Will [180 ], based on elementary quantum mechanical arguments, was later developed into a powerful technique with which accurate results have been derived. The idea is that one can reduce the QNM problem into the standard WKB treatment of scattering of waves on the peak of the potential barrier. The simplest way to find the QNM frequencies is to use the well known Bohr–Sommerfeld (BS) rule. Using this rule it is possible to reproduce not only the Schutz–Will formula (30 ) but also to give a way to extend the accuracy of that formula by taking higher order terms [114 , 105 ]. The classical form of the BS rule for equations like (21 ) is

where $rA,rB$ are the two roots (turning points) of $2 ω − V (r) = 0$ . 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 semi-analytic methods for estimating QNMs.

6.1.4 The method of continued fractions

In 1985, Leaver [135 ] presented a very accurate method for calculating the QNM frequencies. His method can be applied to the calculation of the QNMs of Schwarzschild, Kerr and, with some modifications, Reissner–Nordström black holes [137 ]. This method is very accurate also for the high-order modes.

His approach was analogous to the determination of the eigenvalues of the H $+$ ion developed in [33 ]. A series representation of the solution $f −$ is assumed to represent also $f+$ for the value of the quasi-normal mode frequency. For normal modes the method may work because $f+$ is certainly bounded at infinity. In the case of quasi-normal modes this is not so clear because $f +$ 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 $λ$ such that for $λ = 1$ the potential has bound states – normal modes – and for for $λ = − 1$ just quasi-normal modes. This is, for example, the case if we multiply the Regge–Wheeler potential (23 ) by $− λ$ . Assuming that the modes depend continuously on $λ$ , 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 $an$ 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.