Rotating black holes surrounded by a dark matter halo in the galactic center of M87 and Sgr A

In this paper, we construct the solutions of modified Schwarzschild spacetime with three kinds of dark matter profiles, i.e. the Beta profiles, Moore profiles and Fermionic halos profiles. In order to make the models more realistic, by using the modified Newman–Janis algorithm (MNJA), we extend our solution to obtain three rotating black holes surrounded by dark matter halo and discuss the black hole properties such as the event horizon, ergoregion, geodesics of null particles and the black hole shadow. We consider two specific examples, i.e. using the present values for the parameters of the dark matter density profiles for the Sgr A∗ black hole and M87 central black hole, it shows that the effects of the dark matter on the size of shadows are almost negligible compared to the shadow of Kerr black hole. Besides, in specific condition having dark matter with very high density, we show that the shadow images of Beta model rotating black hole increase, but the shadow images of Moore model and Fermionic model rotating black holes decrease, furthermore, the Moore profile has the largest deviation, the Beta profile has the least deviation and the deviation of Fermionic halo model is in the middle. These phenomena may provides us with a possibility: the effect of dark matter on the apparent shadow shape could shed some light on mass density profiles of dark matter halo of a galaxy, by studying black holes in dark matter, it may provide an inspiration for us to indirectly detect dark matter.