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Black holes in asymptotically flat space
The study of quasi-normal modes in asymptotically flat space-times has attracted
a lot of attention recently [15,16,17,18,19] because the asymptotic form of quasi-normal
frequencies was shown to be related to the Barbero-Immirzi parameter [20] of Loop Quantum Gravity (see [21] and references therein).
The asymptotic form of high overtones is
Motl and Neitzke [15] showed that (3) may be extended to
arbitrary spin
j and analytically derived the generalized expression
where
Extending the above calculation to rotating (Kerr) black holes appears to be far from straightforward. One needs to solve a set of coupled differential
equations, because the angular and radial equations do not neatly separate as
in the Schwarzschild case [28].
Based on Bohr's correspondence principle, Hod has argued [29] that
the real part of the quasi-normal frequencies of gravitational waves
ought to be given by the asymptotic expression (cf. eq. (3))
To shed light on this issue, Dr. Siopsis and Dr. Musiri obtained an expression for the asymptotic form of quasi-normal modes of a Kerr black hole [30] by solving the Teukolsky wave equation [28] and applying a monodromy argument similar to the Schwarzschild case [15]. The equation reduced to Whittaker's equation and the final expression was in agreement with Hod's formula (7). However, this result is only valid for asymptotic values of which are bounded from above by , where is the angular momentum per unit mass of the Kerr black hole. Thus, its validity is limited to small values of the parameter . This includes the Schwarzschild limit in which the range of frequencies extends to infinity. It appears that Bohr's correspondence principle is not applicable to the true asymptotic regime in general and the numerical result (8) may well be valid. Understanding it analytically is a challenge. |
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