An Initial Understanding On Spin-Weighted Spheroidal Wave Equation

The Kerr black hole determines the final state of large rotating stars in general relativity, therefore, its stability is of astrophysical importance. Teukolsky equation describes the linear stability of Kerr black hole, a stationary rotating black hole, suffers from the linear perturbation of a field. Teukolsky equation can be separated into two Sturm-Liouville problem, they are called respectively the radical and the angular, where the latter is also called the spin-weighted spheroidal equation, it is whereθ∈(0,π), (0.0.5) and its solutions in the end points of interval (0,π) are finite. In this thesis we will solve the eigenvalues and eigenfunctions of the spin-weighted spheroidal wave equation in case of small c by the way of supersymmetric quantum mechanics, it contains two parts. The first one is to solve the spin-weighted spheroidal wave equation with spin is zero in case of small c, which is called spheroidal wave equation, its eigenvalues and eigenfunctions are given in series in c2, and also giving an immediate relation between the eigenvalues of excited state and the ground eigen-value. The second part is to solve the spin-weighted spheroidal wave equation with spin is not zero in case of small c, similarly, its eigenvalues and eigenfunctions are also given in series in c.There are three chapters in this thesis as following:In Chapter 1:there ia a brief introduction on the background, the meaning of the content of this thesis, the related fundamental conceptions and principles. It contains the linear perturbation of Kerr black hole and the Teukolsky equation, and the way of supersymmetric quantum mechanics.In Chapter 2:solving the spheroidal wave equation in case of small c.In Chapter 3:solving the spin-weighted spheroidal wave equation with spin is not zero in case of small c...