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In this work, we study the impact of the environment around a black hole indetail. We introduce non-vanishing radial pressure in a manner analogous tocompact stars. We examine both isotropic and anisotropic fluid configurationswith and without radial pressure respectively. Our focus extends beyond justdark matter density to the vital role of the energy condition and sound speedin the spacetime of a black hole immersed in matter. In cases of anisotropicpressure with vanishing radial pressure, all profiles violate the dominantenergy condition near the BH, and the tangential sound speed exceeds lightspeed for all dark matter profiles. In our second approach, without assumingvanishing radial pressure, we observe similar violations and superluminal soundspeeds. To rectify this, we introduce a hard cutoff for the sound speed,ensuring it remains subluminal. As a consequence, the energy condition is alsosatisfied. However, this results in increased density and pressure near the BH.This raises questions about the sound speed and its impact on the densitystructure, as well as questions about the validity of the model itself. Withthe matter distribution, we also compute the metric for differentconfigurations. It reveals sensitivity to the profile structure. The metriccomponents point towards the horizon structure.

Black holes immersed in dark matter: energy condition and sound speed