Spectroscopy of magnetized black holes and topological stars

We study the linear response of four dimensional magnetized black holes and regular topological stars arising from dimensional compactification of Einstein-Maxwell theory in five dimensions. We consider both radial and nonradial perturbations and study the stability of these solutions, both in the frequency and in the time domain. Due to the presence of magnetic fluxes in the background, axial (i.e., odd-parity) gravitational perturbations are coupled to polar (i.e., even-parity) electromagnetic perturbations (Type-I sector) whereas polar gravitational and scalar perturbations are coupled to axial electromagnetic ones (Type-II sector). We provide a comprehensive analytical and numerical study of the radial perturbations and of the Type I sector, finding no evidence of linear instabilities (besides the already known Gregory-Laflamme instability of black strings occurring only in a certain range of the parameters), even despite the fact that the effective potential for radial perturbations of topological stars is negative and divergent near the inner boundary. Ultracompact topological stars exhibit long-lived trapped modes that give rise to echoes in the time-domain response. Their prompt response is very similar to that of the corresponding black hole with comparable charge-to-mass ratio. This provides a concrete realization of ultracompact objects arising from a well-defined theory. The numerical analysis of the Type-II sector will appear in a companion paper.