3.2 Kerr Black Holes3 Quasi-Normal Modes of Black 3 Quasi-Normal Modes of Black

3.1 Schwarzschild Black Holes 

The study of perturbations of Schwarzschild black holes assumes a small perturbation tex2html_wrap_inline2816 on a static spherically symmetric background metric

  equation161

with the perturbed metric having the form

  equation169

which leads to a variation of the Einstein equations i.e.

equation175

By assuming a decomposition into tensor spherical harmonics for each tex2html_wrap_inline2816 of the form

  equation180

the perturbation problem is reduced to a single wave equation, for the function tex2html_wrap_inline2820 (which is a combination of the various components of tex2html_wrap_inline2816). It should be pointed out that equation (20Popup Equation) is an expansion for scalar quantities only. From the 10 independent components of the tex2html_wrap_inline2816 only tex2html_wrap_inline2826, tex2html_wrap_inline2828, and tex2html_wrap_inline2830 transform as scalars under rotations. The tex2html_wrap_inline2832, tex2html_wrap_inline2834, tex2html_wrap_inline2836, and tex2html_wrap_inline2838 transform as components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components tex2html_wrap_inline2840, tex2html_wrap_inline2842, and tex2html_wrap_inline2844 transform as components of a tex2html_wrap_inline2846 tensor and can be expanded in a series of tensor spherical harmonics (see [202Jump To The Next Citation Point In The Article, 212Jump To The Next Citation Point In The Article, 152Jump To The Next Citation Point In The Article] for details). There are two classes of vector spherical harmonics (polar and axial) which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics. The difference between the two families is their parity. Under the parity operator tex2html_wrap_inline2848 a spherical harmonic with index tex2html_wrap_inline2850 transforms as tex2html_wrap_inline2852, the polar class of perturbations transform under parity in the same way, as tex2html_wrap_inline2852, and the axial perturbations as tex2html_wrap_inline2856 Popup Footnote . Finally, since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.

The radial component of a perturbation outside the event horizon satisfies the following wave equation,

  equation212

where tex2html_wrap_inline2874 is the ``tortoise'' radial coordinate defined by

equation219

and M is the mass of the black hole.

For ``axial'' perturbations

  equation221

is the effective potential or (as it is known in the literature) Regge-Wheeler potential [173], which is a single potential barrier with a peak around r =3 M, which is the location of the unstable photon orbit. The form (23Popup Equation) is true even if we consider scalar or electromagnetic test fields as perturbations. The parameter tex2html_wrap_inline2880 takes the values 1 for scalar perturbations, 0 for electromagnetic perturbations, and -3 for gravitational perturbations and can be expressed as tex2html_wrap_inline2884, where s =0, 1, 2 is the spin of the perturbing field.

For ``polar'' perturbations the effective potential was derived by Zerilli [212] and has the form

  equation230

where

equation236

Chandrasekhar [54] has shown that one can transform the equation (21Popup Equation) for ``axial'' modes to the corresponding one for ``polar'' modes via a transformation involving differential operations. It can also be shown that both forms are connected to the Bardeen-Press [38] perturbation equation derived via the Newman-Penrose formalism. The potential tex2html_wrap_inline2888 decays exponentially near the horizon, tex2html_wrap_inline2890, and as tex2html_wrap_inline2892 for tex2html_wrap_inline2894 .

From the form of equation (21Popup Equation) it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section  2 .

Kay and Wald [117Jump To The Next Citation Point In The Article] have shown that solutions with data of compact support are bounded. Hence we know that the time independent Green function tex2html_wrap_inline2896 is analytic for Re (s)>0. The essential difficulty is now to obtain the solutions tex2html_wrap_inline2782 (cf. equation (10Popup Equation)) of the equation

  equation246

(prime denotes differentiation with respect to tex2html_wrap_inline2874) which satisfy for real, positive s :

  equation252

To determine the quasi-normal modes we need the analytic continuations of these functions.

As the horizon (tex2html_wrap_inline2890) is a regular singular point of (26Popup Equation), a representation of tex2html_wrap_inline2908 as a converging series exists. For tex2html_wrap_inline2910 it reads:

  equation261

The series converges for all complex s and | r -1|<1 [162]. (The analytic extension of tex2html_wrap_inline2676 is investigated in [115Jump To The Next Citation Point In The Article].) The result is that tex2html_wrap_inline2676 has an extension to the complex s plane with poles only at negative real integers. The representation of tex2html_wrap_inline2678 is more complicated: Because infinity is a singular point no power series expansion like (28Popup Equation) exists. A representation coming from the iteration of the defining integral equation is given by Jensen and Candelas [115Jump To The Next Citation Point In The Article], see also [159Jump To The Next Citation Point In The Article]. It turns out that the continuation of tex2html_wrap_inline2678 has a branch cut tex2html_wrap_inline2926 due to the decay tex2html_wrap_inline2928 for large r  [115].

The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot [35Jump To The Next Citation Point In The Article]. Here the existence of an infinite number of quasi-normal modes is demonstrated. Truncating the potential (23Popup Equation) to make it of compact support leads to the estimate (16Popup Equation).

The decay of solutions in time is not exponential because of the weak decay of the potential for large r . At late times, the quasi-normal oscillations are swamped by the radiative tail [166Jump To The Next Citation Point In The Article, 167Jump To The Next Citation Point In The Article]. This tail radiation is of interest in its own right since it originates on the background spacetime. The first authoritative study of nearly spherical collapse, exhibiting radiative tails, was performed by Price [166, 167].

Studying the behavior of a massless scalar field propagating on a fixed Schwarzschild background, he showed that the field dies off with the power-law tail,

equation277

at late times, where P =1 if the field is initially static, and P =2 otherwise. This behavior has been seen in various calculations, for example the gravitational collapse simulations by Cunningham, Price and Moncrief [66, 67, 68]. Today it is apparent in any simulation involving evolutions of various fields on a black hole background including Schwarzschild, Reissner-Nordström [106Jump To The Next Citation Point In The Article], and Kerr [132Jump To The Next Citation Point In The Article, 133Jump To The Next Citation Point In The Article]. It has also been observed in simulations of axial oscillations of neutron stars [18Jump To The Next Citation Point In The Article], and should also be present for polar oscillations. Leaver [136Jump To The Next Citation Point In The Article] has studied in detail these tails and associated this power low tail with the branch-cut integral along the negative imaginary tex2html_wrap_inline2624 axis in the complex tex2html_wrap_inline2624 plane. His suggestion that there will be radiative tails observable at tex2html_wrap_inline2942 and tex2html_wrap_inline2944 has been verified by Gundlach, Price, and Pullin [106]. Similar results were arrived at recently by Ching et al. [62] in a more extensive study of the late time behavior. In a nonlinear study Gundlach, Price, and Pullin [107] have shown that tails develop even when the collapsing field fails to produce a black hole. Finally, for a study of tails in the presence of a cosmological constant refer to [49], while for a recent study, using analytic methods, of the late-time tails of linear scalar fields outside Schwarzschild and Kerr black holes refer to [36, 37].

Using the properties of the waves at the horizon and infinity given in equation (27Popup Equation) one can search for the quasi-normal mode frequencies since practically the whole problem has been reduced to a boundary value problem with tex2html_wrap_inline2564 being the complex eigenvalue. The procedure and techniques used to solve the problem will be discussed later in section  6, but it is worth mentioning here a simple approach to calculate the QNM frequencies proposed by Schutz and Will [180Jump To The Next Citation Point In The Article]. The approach is based on the standard WKB treatment of wave scattering on the peak of the potential barrier, and it can be easily shown that the complex frequency can be estimated from the relation

  equation295

where tex2html_wrap_inline2948 is the peak of the potential barrier. For tex2html_wrap_inline2556 and n =0 (the fundamental mode) the complex frequency is tex2html_wrap_inline2954, which for a tex2html_wrap_inline2956 black hole corresponds to a frequency of 1.2 kHz and damping time of 0.55 ms. A few more QNM frequencies for tex2html_wrap_inline2550 and 4 are listed in table  1 .

  

table303

Table 1: The first four QNM frequencies (tex2html_wrap_inline2548) of the Schwarzschild black hole for tex2html_wrap_inline2550, and 4 [135Jump To The Next Citation Point In The Article]. The frequencies are given in geometrical units and for conversion into kHz one should multiply by tex2html_wrap_inline2554 .

Figure  2 shows some of the modes of the Schwarzschild black hole. The number of modes for each harmonic index tex2html_wrap_inline2850 is infinite, as was mathematically proven by Bachelot and Motet-Bachelot [35Jump To The Next Citation Point In The Article]. This was also implied in an earlier work by Ferrari and Mashhoon [85Jump To The Next Citation Point In The Article], and it has been seen in the numerical calculations in [25Jump To The Next Citation Point In The Article, 157Jump To The Next Citation Point In The Article]. It can be also seen that the imaginary part of the frequency grows very quickly. This means that the higher modes do not contribute significantly in the emitted gravitational wave signal, and this is also true for the higher tex2html_wrap_inline2850 modes (octapole etc.).

  

Click on thumbnail to view image

Figure 2: The spectrum of QNM for a Schwarzschild black-hole, for tex2html_wrap_inline2556 (diamonds) and tex2html_wrap_inline2558 (crosses) [25Jump To The Next Citation Point In The Article]. The 9th mode for tex2html_wrap_inline2556 and the 41st for tex2html_wrap_inline2558 are ``special'', i.e. the real part of the frequency is zero (tex2html_wrap_inline2564).

As is apparent in figure  2 that there is a special purely imaginary QNM frequency. The existence of ``algebraically special'' solutions for perturbations of Schwarzschild, Reissner-Nordström and Kerr black holes were first pointed out by Chandrasekhar [57]. It is still questionable whether these frequencies should be considered as QNMs [137Jump To The Next Citation Point In The Article] and there is a suggestion that the potential might become transparent for these frequencies [11]. For a more detailed discussion refer to [144].

As a final comment we should mention that as the order of the modes increases the real part of the frequency remains constant, while the imaginary part increases proportionally to the order of the mode. Nollert [157Jump To The Next Citation Point In The Article] derived the following approximate formula for the asymptotic behavior of QNMs of a Schwarzschild black hole,

  equation327

where tex2html_wrap_inline2992, 0.7545 and 2.81 for tex2html_wrap_inline2998 2, 3 and 6, correspondingly, and tex2html_wrap_inline3000 . The above relation was later verified in [10] and [143].

For large values of tex2html_wrap_inline2850 the distribution of QNMs is given by [164, 86Jump To The Next Citation Point In The Article, 85Jump To The Next Citation Point In The Article, 113Jump To The Next Citation Point In The Article]

equation338

For a mathematical proof refer to [39].

The perturbations of Reissner-Nordström black holes, due to the spherical symmetry of the solution, follow the footsteps of the analysis that we have presented in this section. Most of the work was done during the seventies by Zerilli [213], Moncrief [153, 154] and later by Chandrasekhar and Xanthopoulos [55Jump To The Next Citation Point In The Article, 209]. For an extensive discussion refer to [56]. We have again wave equations of the form (21Popup Equation), one for each parity with potentials which are like (23Popup Equation) and (24Popup Equation) plus extra terms which relate to the charge of the black hole. An interesting feature of the charged black holes is that any perturbation of the gravitational (electromagnetic) field will also induce electromagnetic (gravitational) perturbations. In other words, any perturbation of the Reissner-Nordström spacetime will produce both electromagnetic and gravitational radiation. Again it has been shown that the solutions for the odd parity oscillations can be deduced from the solutions for even parity oscillations and vice versa [55]. The QNM frequencies of the Reissner-Nordström black hole have been calculated by Gunter [108Jump To The Next Citation Point In The Article], Kokkotas, and Schutz [129Jump To The Next Citation Point In The Article], Leaver [137Jump To The Next Citation Point In The Article], Andersson [9], and lately for the nearly extreme case by Andersson and Onozawa [26].



3.2 Kerr Black Holes3 Quasi-Normal Modes of Black 3 Quasi-Normal Modes of Black

image 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
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