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

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 for the transformation to obtain the short-ranged potential in the assumption that the transformation converges in the far distance. Also, we discuss the peak location of the potential which is responsible for quasinormal mode frequencies inWKBanalysis. Finally, in the extreme Kerr limit, a/M → 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 ≲ 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.