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

TL;DR

This paper analyzes the asymptotic behavior of the Sasaki-Nakamura transformation in Kerr black holes, especially near the extremal limit, and estimates the potential peak location influencing quasinormal modes.

Contribution

It introduces a new relaxed condition for the transformation to produce a short-ranged potential and estimates the potential peak location in the extreme Kerr limit.

Findings

The new condition ensures the potential is short-ranged if the transformation converges.

The peak location of the potential in the extreme Kerr limit is estimated as r_p/M \,\lesssim\, 1 + 1.8 (1-a/M)^{1/2}.

The uncertainty in the peak location aligns with the expectations from the equivalence principle.

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.