The interaction between dark matter and supermassive black holes in the galaxy center is one of the most interesting topics in astrophysics. This problem is mainly reflected in two aspects, the first is how the dark matter halo changes the black hole space-time, in turn affecting various physical properties of black holes. Second, the existence of a central supermassive black hole will change the spatial distribution of dark matter near the black hole, also known as the Spike phenomenon. This article has mainly done a series of studies on the previous issue. The research content is divided into two parts. The first is how the phenomenological dark matter model (perfect fluid dark matter) changes the black hole spacetime, and expands the black hole solution of Einstein field equation. We also use PFDM to explain the rotation curves diversity of LSB galaxies and power-law total mass-density profile for the inner regions of ETGs. The second is develop general method to study the black hole solution of Einstein field equation when dark matter halo exists. For Cold dark matter model and Scalar field dark matter model, we obtain the specific black hole solution. Next we will introduce the working ideas and main results.Firstly, we obtain the KN-ADS space-time of the Einstein-Maxwell field equation of the perfect fluid around a black hole by Newman-Janis algorithm and complex computations. We analyzed the nature of these black holes and found that for the equation of state is $-1/3$, the perfect fluid is dark matter. Using this black hole solution, we can study the interaction between black holes and perfect fluid dark matter. We also got another form of black hole space-time in perfect fluid dark matter background.Secondly, we study the KN-ADS black hole space-time surrounded by the perfect fluid matter in the Rastall gravity by the Newman-Janis method and Mathematica package. Next we discuss the effects of dark energy ($-2/3$) and dark matter ($-1/3$) on the properties of KN-ADS black holes. On the other hand, for the perfect fluid dark matter ($-1/3$), we could explain the rotation curve diversity in Low Surface Brightness (LSB) galaxies when the perfect fluid dark matter halo and baryon disk are both taken into account.Thirdly, we tried to explain the origin of the power distribution of the elliptical galaxy mass. We obtain the general expressions of galactic state equations and exponents. By analyzing the data of 119 elliptical galaxy gravitational lenses, we found that the average of the state equation is near $-1/3$. This shows that elliptical galaxies are dominated by dark matter. On the other hand, using the PFDM, we performed a good data fit for the rotation curve diversity of spiral galaxies.Finally, using the dark matter distribution model obtained by numerical simulation or empirical approximation, we constructed a space-time metric in the context of pure dark matter. Substituting the energy-momentum tensors corresponding to the dark matter halo into Einstein's gravitational field equation, we obtain the Schwarzschild black hole space-time in dark matter halo. By NJ method, we generalize Schwarzschild black holes to rotational black holes in dark matter halo. We have studied the properties of dark matter that affect black holes.
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