The non-linear perturbation of a black hole by gravitational waves. II. Quasinormal modes and the compactification problem

Abstract: Recently, Friedrich's Generalized Conformal Field Equations (GCFE) have been implemented numerically and global quantities such as the Bondi energy and the Bondi-Sachs mass loss have been successfully calculated directly on null-infinity. Although being an attractive option for studying global quantities by way of local differential geometrical methods, how viable are the GCFE for study of quantities arising in the physical space-time? In particular, how long can the evolution track phenomena that need a constant proper physical timestep to be accurately resolved? We address this question by studying the curvature oscillations induced on the Schwarzschild space-time by a non-linear gravitational perturbation. For small enough amplitudes, these are the well approximated by the linear quasinormal modes, where each mode rings at a frequency determined solely by the Schwarzschild mass. We find that the GCFE can indeed resolve these oscillations, which quickly approach the linear regime, but only for a short time before the compactification becomes ``too fast'' to handle numerically.