We present a new covariant, gauge-invariant formalism describing linear metric perturbation fields on any spherically symmetric background in general relativity. The advantage of this formalism relies in the fact that it does not require a decomposition of the perturbations into spherical tensor harmonics. Furthermore, it does not assume the background to be vacuum, nor does it require its staticity. In the particular case of vacuum perturbations, we derive two master equations describing the propagation of arbitrary linear gravitational waves on a Schwarzschild black hole. When decomposed into spherical harmonics, they reduce to covariant generalizations of the well-known Regge-Wheeler and Zerilli equations. Next, we discuss the general case where the metric perturbations are coupled to matter fields and derive a new constrained wave system describing the propagation of three gauge-invariant scalars from which the complete metric perturbations can be reconstructed. We apply our formalism to the Einstein-Euler system, dividing the fluid perturbations into two parts. The first part, which decouples from the metric perturbations, obeys simple advection equations along the background flow and describes the propagation of the entropy and the vorticity. The second part describes a perturbed potential flow, and together with the metric perturbations it forms a closed wave system.

We analyze the steady radial accretion of matter into a nonrotating black hole. Neglecting the selfgravity of the accreting matter, we consider a rather general class of static, spherically symmetric and asymptotically flat background spacetimes with a regular horizon. In addition to the Schwarzschild metric, this class contains certain deformation of it which could arise in alternative gravity theories or from solutions of the classical Einstein equations in the presence of external matter fields. Modeling the ambient matter surrounding the black hole by a relativistic perfect fluid, we reformulate the accretion problem as a dynamical system, and under rather general assumptions on the fluid equation of state, we determine the local and global qualitative behavior of its phase flow. Based on our analysis and generalizing previous work by Michel, we prove that for any given positive particle density number at infinity, there exists a unique radial, steady-state accretion flow which is regular at the horizon. We determine the physical parameters of the flow, including its accretion and compression rates, and discuss their dependency on the background metric.

We discuss the quasi-normal oscillations of black holes which are sourced by a nonlinear electrodynamic field. While previous studies have focused on the computation of quasi-normal frequencies for the wave or higher spin equation on a fixed background geometry described by such black holes, here we compute for the first time the quasi-normal frequencies for the coupled electromagneticgravitational linear perturbations.To this purpose, we consider a parametrized family of Lagrangians for the electromagnetic field which contains the Maxwell Lagrangian as a special case. In the Maxwell case, the unique spherically symmetric black hole solutions are described by the Reissner-Nordström family and in this case it is well-known that the quasi-normal spectra in the even-and odd-parity sectors are identical to each other. However, when moving away from the Maxwell case, we obtain deformed ReissnerNordström black holes, and we show that in this case there is a parity splitting in the quasi-normal mode spectra. A partial explanation for this phenomena is provided by considering the eikonal (high-frequency) limit.