3.5 Quasi-normal modes of a black hole

In 1970, Vishveshwara [381Jump To The Next Citation Point] discussed a gedanken experiment, similar in philosophy to Rutherford’s (real) experiment with the atom. In Vishveshwara’s experiment, he scattered gravitational radiation off a black hole to explore its properties. With the aid of such a gedanken experiment, he demonstrated for the first time that gravitational waves scattered off a black hole will have a characteristic waveform, when the incident wave has frequencies beyond a certain value, depending on the size of the black hole. It was soon realized that perturbed black holes have quasi-normal modes (QNMs) of vibration and in the process emit gravitational radiation, whose amplitude, frequency and damping time are characteristic of its mass and angular momentum [298Jump To The Next Citation Point221]. We will discuss in Section 6.4 how observations of QNMs could be used in testing strong field predictions of general relativity.

We can easily estimate the amplitude of gravitational waves emitted when a black hole forms at a distance r from Earth as a result of the coalescence of compact objects in a binary. The effective amplitude is given by Equation (20View Equation), which involves the energy E put into gravitational waves and the frequency f at which the waves come off. By dimensional arguments E is proportional to the total mass M of the resulting black hole. The efficiency at which the energy is converted into radiation depends on the symmetric mass ratio ν of the merging objects. One does not know the fraction of the total mass emitted nor the exact dependence on ν. Flanagan and Hughes [164Jump To The Next Citation Point] argue that E ∼ 0.03(4ν)2M. The frequency f is inversely proportional to M; indeed, for Schwarzschild black holes f = (2πM )−1. Thus, the formula for the effective amplitude takes the form

4ανM heff ∼ ------, (35 ) πr
where α is a number that depends on the (dimensionless) angular momentum a of the black hole and has a value between 0.7 (for a = 0, Schwarzschild black hole) and 0.4 (for a = 1, maximally spinning Kerr black hole). For stellar mass black holes at a distance of 200 Mpc the amplitude is:
( ) ( ) ( )−1 heff ≃ 10−21 -ν-- --M--- ----r---- . (36 ) 0.25 20M ⊙ 200 Mpc
For SMBHs, even at cosmological distances, the amplitude of quasinormal mode signals is pretty large:
( ) ( ) ( )−1 −17 -ν-- ----M------ ---r---- heff ≃ 3 × 10 0.25 2 × 106M ⊙ 6.5 Gpc . (37 )
In the first case we have a pair of 10M ⊙ black holes inspiraling and merging to form a single black hole. In this case the waves come off at a frequency of around 500 Hz [cf. Equation (13View Equation)]. The initial ground-based network of detectors might be able to pick these waves up by matched filtering, especially when an inspiral event precedes the ringdown signal. A 100M ⊙ black hole plunging into a 6 10 M ⊙ black hole at a distance of 6.5 Gpc (z ≃ 1) gives out radiation at a frequency of about 15 mHz. Note that in the latter case the radiation is redshifted from 30 mHz to 15 mHz. Such an event produces an amplitude just large enough to be detected by LISA. At the same distance, a pair of 106M ⊙ SMBHs spiral in and merge to produce a fantastic amplitude of − 17 3 × 10, way above the LISA background noise. In this case, the signals would be given off at about 7.5 mHz and will be loud and clear to LISA. It will not only be possible to detect these events, but also to accurately measure the masses and spins of the objects before and after merger and thereby test the black hole no-hair theorem and confirm whether the result of the merger is indeed a black hole or some other exotic object (e.g., a boson star or a naked singularity).
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