A black hole inside dark matter and the rotation curves of galaxies

In this paper, we find a four-dimensional metric for a large black hole immersed in dark matter. Specifically, we look for and find a static spherically symmetric black hole solution to the Einstein equations which gives, in the Newtonian limit, the rotation curves of galaxies, including the flat region and the baryonic Tully-Fisher relation, and which has a regular horizon. We obtain as well the energy-momentum tensor of the dark matter sourcing this spacetime and it turns, in special, to satisfy the four energy conditions (dominant, weak, null and strong) everywhere outside the horizon. This black-hole-dark-matter system represents a successful simplified model for galaxies, opens a new area for exploring the relativistic regime of dark matter, and shows that the theory of General Relativity together with dark matter can account for the rotation curves of galaxies.