The hidden conformal symmetry and quasinormal modes of the four dimensional Kerr black hole

Abstract

This dissertation has two areas of interest with regard to the four dimensional Kerr black hole; the rst being its conformal nature in its near region and second it characteristic frequencies. With it now known that the scalar solution space of the four dimensional Kerr black hole has a two dimensional conformal symmetry in its near region, it was the rst focus of this dissertation to see if this conformal symmetry is unique to the near region scalar solution space or if it is also present in the spin-half solution space. The second focus of this dissertation was to explore techniques which can be used to calculate these quasinormal mode (characteristic) frequencies, such as the WKB(J) approximation which has been improved from third order to sixth order recently and applied to the perturbations of a Schwarzschild black hole. The additional correction terms show a signi cant increase of accuracy when comparing to numerical methods. This dissertation shall use the sixth order WKB(J) method to calculate the quasinormal mode frequencies for both the scalar and spin-half perturbations of a four dimensional Kerr black hole. An additional method used was the asymptotic iteration method, a relatively new technique being used to calculate the quasinormal mode frequencies of black holes that have been perturbed. Prior to this dissertation it had only been used on a variety of Schwarzschild black holes and their possible perturbations. For this dissertation the asymptotic iteration method has been used to calculate the quasinormal frequencies for both the scalar and spin-half perturbations of the four dimensional Kerr black hole. The quasinormal mode frequencies calculated using both the sixth order WKB(J) method and the asymptotic iteration method were compared to previously published values and each other. For the most part, they both compare favourably with the numerical values, with di erences that are near negligible. The di erences did become more apparent when the mode number (or angular momentum per unit mass increased), but less so when the angular number increased. The only factor that separates these two methods signi cantly, was that the computational time for the sixth order WKB(J) method is less than than that of the asymptotic iteration method.