Quasi normal mode of kerr black hole and kerr ads black hole at extremal limit

Abstract

This report is based on the quasi normal mode of Kerr black hole for massless spin-s field perturbation and also focused on the quasi normal mode of Kerr Ads black hole at extremal limit for scalar perturbation. At first we discuss Kerr black hole’s properties and the role of Killing -Yano tensor in generating conserved quantities which simplify the study of geodesic motion, wave propagation etc .We also analyze geometric structure of Kerr black hole at extremal limit and at near horizon limit . Furthermore, we establish a correspondence between high-frequency QNMs (eikonal regime) and unstable photon orbits, demonstrating how the real part of QNM frequencies relates to orbital energy and angular momentum, while the imaginary part encodes instability via Lyapunov exponents. Due to disturbance of black hole by some external perturbation, the quasi normal mode tells about the oscillation.They are defined by complex frequency which give information about properties of black hole .The Teukolsky equation describes the perturbation of black hole and for the Kerr black hole it is divided into two parts, one is radial and another one is angular in Boyer-Lindquist coordinates.Solution for both parts are made by series solution and the coefficients follow some recurrence relation.By using it, the QNF(Quasi normal frequency) are derived numerically for different field spin . This analysis offers a comprehensive framework to study black hole perturbation and also the results are compared with Leaver’s paper .We here also try to find quasi normal frequencies for others different spin field perturbation like s=-1 for electromagnetic field and s=0 for outgoing scalar field . After that,I try to find the quasi-normal modes of a Kerr-AdS black hole in the extremal limit under scalar perturbations, and demonstrate that it remains linearly stable in this extremal case.