Black hole space-time in dark matter halo

For the first time, we obtain the analytical form of a stationary black hole space-time metric in dark matter halos. First, using the relation between the rotational velocity (in the equatorial plane) and the spherically symmetric space-time metric coefficients, we obtain the space-time metric for pure dark matter. By considering the dark matter halo in spherically symmetric space-time as part of the energy-momentum tensors in the Einstein field equation, we then obtain the spherically symmetric black hole solutions with a dark matter halo. Finally, utilizing the Newman-Janis method, we further generalize to rotational black holes. As examples, we obtain the space-time metric of black holes surrounded by Cold Dark Matter and Scalar Field Dark Matter halos, respectively. Our main results regarding the interaction between black holes and dark matter halos are as follows: (i) for both dark matter models, the density profile always produces a cusp phenomenon at small scales; (ii) the dark matter halo increases the black hole horizon but shrinks the ergosphere, while the magnitude is small; (iii) dark matter does not change the singularity of black holes. These results are useful to study the interaction of a stationary black hole and dark matter halo system. Particularly, the cusp produced at the 0 similar to 1 kpc scale would be observable in the Milky Way. Perspectives on future work regarding the applications of our results in astrophysics are also briefly discussed.