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 in tWKB 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 .