ABSTRACT
This chapter discusses the dynamics of small metric perturbations of a Schwarzschild black hole. It follows the approach pioneered by Regge and Wheeler in which the metric perturbations are decomposed in a complete basis of spherical harmonics according to their parity. A powerful approach to study the behaviour of physical systems near equilibrium is to analyse their normal modes, i.e. the characteristic modes of oscillation in the absence of an external force. This approach is also effective to study the behaviour of black holes near equilibrium: if a black hole is perturbed by an external event, for instance a mass falling toward the horizon, after a transient it starts oscillating at some characteristic frequencies. In principle the quasi-normal modes of the axial and polar perturbations of a Schwarzschild black hole should be different.
