Geometric properties of slowly rotating black holes embedded in matter environments

Extreme mass-ratio inspirals (EMRIs) provide a precise probe of strong-field gravity, where small deviations from vacuum Kerr geometry can accumulate over many orbital cycles. In realistic astrophysical settings, black holes are embedded in surrounding dark and baryonic matter whose presence and motion can perturb the spacetime. A systematic semi-analytic framework incorporating the rotation of the environment itself into a black-hole geometry, and propagating its effects consistently into conserved quantities and epicyclic observables, has not been explored exhaustively, limiting consistent assessments of environmental effects on precision observables. In this work, we construct a slowly rotating black hole spacetime embedded in an anisotropic matter distribution and explicitly include the angular velocity of the surrounding medium within a controlled slow-rotation expansion. We demonstrate that the environment's velocity field induces corrections to the metric coefficients that propagate into modifications of conserved quantities governing geodesics. We also derive semi-analytic shifts in the innermost stable circular orbit, light-ring location, and radial and vertical epicyclic frequencies, showing that environmental rotation produces systematic and in certain regimes qualitatively distinct behavior relative to static configurations. Consequently, we explicitly show that the environment's nature and motion shift the positions of the epicyclic resonances. The formalism applies to generic anisotropic matter profiles and is not restricted to the specific halo model adopted for numerical illustration. These results establish a direct and quantitatively controlled link between environmental rotation and strong-field orbital observables, enabling consistent incorporation of rotating matter environments into precision EMRI modeling.