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

TL;DR

This paper investigates the ability of Friedrich's Generalized Conformal Field Equations to numerically resolve curvature oscillations in perturbed Schwarzschild space-time, focusing on the limitations imposed by compactification effects.

Contribution

It demonstrates that GCFE can resolve non-linear gravitational perturbations and their transition to linear quasinormal modes, but only for a limited duration due to numerical compactification issues.

Findings

GCFE successfully captures curvature oscillations in perturbed Schwarzschild space-time.

Oscillations quickly approach the linear quasinormal mode regime.

Numerical compactification limits the duration of accurate evolution tracking.

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.