A characteristic approach to the quasi-normal mode problem

In this paper we discuss a new approach to the quasi-normal mode problem in general relativity. By combining a characteristic formulation of the perturbation equations with the integration of a suitable phase-function for a complex-valued radial coordinate, we reformulate the standard outgoing-wave boundary condition as a zero Dirichlet condition. This has a number of important advantages over previous strategies. The characteristic formulation permits coordinate compactification, which means that we can impose the boundary condition at future null infinity. The phase function avoids oscillatory behaviour in the solution, and the use of a complex radial variable allows a clean distinction between out- and ingoing waves. We demonstrate that the method is easy to implement, and that it leads to high precision numerical results. Finally, we argue that the method should generalize to the important problem of rapidly rotating neutron star spacetimes.