This thesis presents an expository account of the quasinormal modes of black holes in General Relativity, and the hyperboloidal approach to black hole perturbation theory, along with an extension of the hyperboloidal framework to the calculation of the quasinormal modes of the Kerr-Newman black hole. Perturbation equations are derived for the Schwarzschild, Kerr, and Kerr-Newman spacetimes, using axial metric perturbations for the former, and perturbations in the Newman-Penrose formalism for the latter two.

For the Schwarzschild spacetime, quasinormal modes are calculated successfully using a direct numerical integration scheme. A novel hyperboloidal foliation of the Schwarzschild spacetime is then implemented, and quasinormal modes are calculated in this hyperboloidal framework using spectral methods. The two approaches are found to be in good agreement with each other, along with sources [41, 60] from the literature.

For the Kerr spacetime, the direct numerical integration scheme is demonstrated not to generalise with sufficient accuracy. Following [66], the hyperboloidal framework is extended to the Kerr spacetime and the quasinormal modes are calculated using spectral methods. For comparison, we also calculate the quasinormal modes of the Kerr black hole using Leaver’s method [74]. The two methods are found to be in good agreement with each other.

For the Kerr-Newman spacetime, the hyperboloidal framework is generalised using the I-fixing gauge of [51]. The perturbation equations of the Kerr-Newman black hole are recast in the corresponding compactified hyperboloidal coordinates, and asymptotic analysis in the Schwarzschild limit is found to justify our extension of the hyperboloidal framework.