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Abstract

In this work, we study the impact of the environment around a black hole in
detail. We introduce non-vanishing radial pressure in a manner analogous to
compact stars. We examine both isotropic and anisotropic fluid configurations
with and without radial pressure respectively. Our focus extends beyond just
dark matter density to the vital role of the energy condition and sound speed
in the spacetime of a black hole immersed in matter. In cases of anisotropic
pressure with vanishing radial pressure, all profiles violate the dominant
energy condition near the BH, and the tangential sound speed exceeds light
speed for all dark matter profiles. In our second approach, without assuming
vanishing radial pressure, we observe similar violations and superluminal sound
speeds. To rectify this, we introduce a hard cutoff for the sound speed,
ensuring it remains subluminal. As a consequence, the energy condition is also
satisfied. However, this results in increased density and pressure near the BH.
This raises questions about the sound speed and its impact on the density
structure, as well as questions about the validity of the model itself. With
the matter distribution, we also compute the metric for different
configurations. It reveals sensitivity to the profile structure. The metric
components point towards the horizon structure.