5.1 Studies of black hole QNM excitation

In the study of QNMs of black holes the attention, in most cases, was focused on estimating the spectrum and its properties. But there is limited work in the direction of understanding what details of the perturbation determine the strength of the QNM ringing.

In 1977 Detweiler [76] discussed the resonant oscillations of a rotating black hole, and after identifying the QNMs as “resonance peaks” in the emitted spectrum he showed that the modes formally correspond to poles of a Green function to the inhomogeneous Teukolsky equation [197]. This idea has been extended in a more mathematically rigorous way by Leaver [136Jump To The Next Citation Point]. Leaver extracts the QNM contribution to the emitted radiation as a sum over residues. This sum arises when the inversion contour of the Laplace transform, which was used to separate the dependence on the spatial variables from the time dependence, is deformed analytically in the complex frequency plane. In this way the contribution from the QNM can be accounted for. Sun and Price [194195] discussed in detail the way that QNM are excited by given Cauchy data based to some extent on numerical results obtained by Leaver [136]. Lately, Andersson [12] used the phase-integral method to determine some characteristics of the QNM excitation.

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Figure 4: The response of a Schwarzschild black hole as a Gaussian wave packet impinges upon it. The QNM signal dominates the signal after t ≈ 70M while at later times (after t ≈ 300M) the signal is dominated by a power-law fall-off with time.

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