Geometric properties of slowly rotating black holes embedded in matter environments

Astrophysical black holes are embedded in surrounding dark and baryonic matter that can measurably perturb the spacetime. We construct a self-consistent spacetime describing a slowly rotating black hole embedded in an external matter distribution, modeling the surrounding dark matter halo as an anisotropic fluid. Working within the slow-rotation approximation, we capture leading-order spin and frame-dragging effects while retaining analytic transparency. We show that the presence and rotation of the halo induce distinct deviations from the vacuum black hole geometry, modifying inertial frame dragging, equatorial circular geodesics, the light ring, the innermost stable circular orbit, and radial and vertical epicyclic frequencies. These effects produce systematic shifts in orbital constants of motion and the locations of epicyclic resonances. In particular, the epicyclic frequency ratios develop nonmonotonic behavior, such as local minima. We further demonstrate that these features depend on the angular velocity of the surrounding fluid, reflecting the interplay between environmental gravity and frame dragging. Our results demonstrate that environmental and rotational effects can leave observable imprints on precision strong-field probes, particularly extreme mass-ratio inspirals, where small corrections accumulate over many orbital cycles. This work provides a minimal and extensible framework for incorporating realistic astrophysical environments into strong-field tests of gravity with future space-based gravitational-wave detectors.