Estimate of the radius responsible for quasinormal modes in the extreme Kerr limit and asymptotic behavior of the Sasaki-Nakamura transformation

Abstract

The Sasaki-Nakamura transformation gives a short-ranged potential and a convergent source term for the master equation of perturbations in the Kerr space-time. In this paper, we study the asymptotic behavior of the transformation, and present a new relaxed necessary and sufficient condition of the transformation to obtain the short-ranged potential in the assumption that the transformation converges in the far distance. Also, we discuss quasinormal mode frequencies which are determined by the information around the peak of the potential in the WKB analysis. Finally, in the extreme Kerr limit, $a/M\to 1$, where $M$ and $a$ denote the mass and spin parameter of a Kerr black hole, respectively, we find the peak location of the potential, $r_p/M\lesssim 1+1.8(1-a/M{)}^{1/2}$ by using the new transformation. The uncertainty of the location is as large as that expected from the equivalence principle.

arXiv comment12 pages, 2 figures, added some references, modifications to match the published version in PTEP

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Cited by in corpus (3)

• Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (II) Ringdown
• Synchronous frequencies of extremal Kerr black holes: resonances, scattering and stability
• Exceptional point and hysteresis in perturbations of Kerr black holes