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 ∼ 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.

We study, for the first time, the shadow of the supermassive black hole Sgr A * at the center of the Milky Way in dark matter halos. For the Cold Dark Matter and Scalar Field Dark Matter models considered in this work, the apparent shape of the shadow depends upon the black hole spin a and the dark matter parameter k. We find that both dark matter models influence the shadow in a similar way. The shadow is a perfect circle in the non-rotating case (a = 0) and a deformed one in the rotating case (a = 0). The size of the shadow increases with increasing k in both non-rotating and rotating cases, while the shadow gets more and more distorted with increasing a in the rotating case. We further investigate the black hole emission rate in both dark matter halos. We find that the emission rate decreases with increasing k and the peak of the emission shifts to lower frequency. Finally, by calculating the angular radius of the shadow, we estimate that the dark matter halo could influence the shadow of Sgr A * at a level of order of magnitude of 10 −3 µas and 10 −5 µas, for CDM and SFDM, respectively. Future astronomical instruments with high angular resolution would be able to observe this effect and shed light on the nature of Sgr A * . More interestingly, it may be possible to distinguish between CDM and SFDM models given the resolutions required differing by two orders of magnitude from each other.

Rastall gravity is the modify Einstein general relativity, in which the energymomentum conservation is assumed to be unsatisfied and generalized to T µν ;µ = λR ,ν . In this paper we investigate the Kerr-Newman-AdS black hole solutions surrounded by the scalar field matter of the Einstein-Maxwell equation in Rastall gravity using Newman-Jains method and complex computations. Through analyzing the root numbers of the horizon equation for two situations, we obtain the relations of α, M and κλ. For the quintessence dark energy (ω = −2/3), if 0 < κλ and κλ = 1, the maximum of α will decrease with the increase of κλ; if −2 < κλ < 0, the maximum of α will become large with the increase of | κλ |. For the scalar field dark matter (ω = −1/3), if 0 < κλ < 1 and κλ = 1/2, the maximum of α will decrease with the increase of κλ; if κλ < 0, the maximum of α will become large with the increase of | κλ |. We also study the influence of scalar field dark matter and quintessential dark energy on ergosphere size. It is found that scalar field matter has significant effect on ergosphere size, while quintessential dark energy has small influence for ergosphere size. The scalar field does not change the singularity of black hole. We also study the rotation velocity on the equatorial plane for quintessential dark energy (ω = −2/3) and scalar field dark matter (ω = −1/3), the parameters α and κλ make the shape of rotation velocity to be more different. These results could explain the diversity of rotation curve theoretically.