Yu-Qian Zhao, Paolo Pani

We develop a perturbative framework for a black hole embedded in a generic, possibly anisotropic, matter environment under spherical symmetry. Our approach extends previous analyses restricted to vanishing radial pressure or to perturbative matter configurations. Within this framework, we derive an analytical generalization of the Einstein cluster that incorporates a polytropic radial pressure, and we investigate the properties of this solution. We show that both the geodesic structure and the axial quasinormal-mode spectrum remain predominantly governed by an overall gravitational redshift effect, while the radial pressure systematically enhances the environmental corrections. In contrast, the tidal Love numbers are substantially more sensitive, and can exhibit order-unity deviations, including vanishing and negative strictly static magnetic Love numbers for sufficiently large anisotropy. We present the full linearized equations, which can be applied to various extensions, including ringdown analysis and extreme-mass-ratio inspirals.