The ringdown of a black hole surrounded by a thin shell of matter

Recent studies have shown that far-field perturbations to the curvature potential of a black hole spacetime may destabilize its quasinormal mode (QNM) spectrum while only mildly affecting time-domain ringdown signals. In this work, we study the QNM spectrum and ringdown behavior of a Schwarzschild black hole with a far-field perturbation to its physical environment -- a thin matter shell with finite surface tension. After accounting for the dynamics of the interaction between GWs and the shell, we find that the fundamental mode can migrate perturbatively or be destabilized by the appearance of new modes with no analogue in the vacuum case, much like studies of ``bumps" in the curvature potential. However, unlike these previous works, we find that the coupling between metric perturbations and oscillations of the shell also sources weakly-damped QNMs which are exclusive to the polar sector. We then study whether the analysis tools of least-squares QNM fits and the full and rational ringdown filters can clearly identify the signatures of the shell in representative ringdown waveforms. We conclude that ringdown at sufficiently early times is insensitive to the shell; weakly-damped QNMs (in the polar sector) and echoes, which may enable the analysis methods considered here to infer the presence of a shell, only appear at late times and are generally weak.