In this work, we study the impact of the environment around a black hole in detail. We introduce nonvanishing 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 toward the horizon structure.

中文翻译：

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沉浸在暗物质中的黑洞：能量条件和声速

在这项工作中，我们详细研究了黑洞周围环境的影响。我们以类似于致密恒星的方式引入非零径向压力。我们分别检查有和没有径向压力的各向同性和各向异性流体构型。我们的关注点不仅限于暗物质密度，还包括浸没在物质中的黑洞时空中的能量条件和声速的重要作用。在各向异性压力和径向压力消失的情况下，所有剖面都违反了 BH 附近的主导能量条件，并且所有暗物质剖面的切向声速都超过了光速。在我们的第二种方法中，在不假设径向压力消失的情况下，我们观察到类似的违规和超光速声速。为了纠正这个问题，我们引入了声速的硬截止，确保其保持亚光速。因此，能量条件也得到满足。然而，这会导致 BH 附近的密度和压力增加。这引发了有关声速及其对密度结构的影响的问题，以及有关模型本身有效性的问题。通过物质分布，我们还计算不同配置的度量。它揭示了对轮廓结构的敏感性。度量组件指向水平结构。