Black Hole Space-time In Dark Matter Halo
  For the first time, we obtain the analytical form of black hole space-time
metric in dark matter halo for the stationary situation. Using the relation
between the rotation velocity (in the equatorial plane) and the spherical
symmetric space-time metric coefficient, we obtain the space-time metric for
pure dark matter. By considering the dark matter halo in spherical symmetric
space-time as part of the energy-momentum tensors in the Einstein field
equation, we then obtain the spherical symmetric black hole solutions in dark
matter halo. Utilizing Newman-Jains method, we further generalize spherical
symmetric black holes 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 hole and dark matter halo are as follows: (i) For
both dark matter models, the density profile always produces "cusp" phenomenon
in small scale in the relativity situation; (ii) Dark matter halo makes the
black hole horizon to increase but the ergosphere to decrease, 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 black hole and dark
matter halo in stationary situation. Particularly, the "cusp" produced in the
$0\sim 1$ 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.
