Modeling the black holes surrounded by a dark matter halo in the galactic center of M87

Abstract

In this paper, the structure of a dark matter halo can be well described by the mass model of M87 and the Einasto profile for the cold dark matter model, i.e., ρeina(r)=ρeexp(-2α-1((r/re)α-1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _{\ ext {eina}} (r)=\\rho _\ ext {e} \\exp ( -2 \\alpha ^{-1} ((r/r_\ ext {e})^\\alpha -1 ) )$$\\end{document}(Wang et al. in Nature 585:39–42, 2020). Under these conditions, we construct a solution of a static spherically symmetric black hole in a dark matter halo. Then, using the Newman–Janis algorithm, we extend this static solution to the case of rotation, and obtain a solution for the Kerr-like black hole. We prove that this solution of the Kerr-like black hole is indeed a solution to the Einstein field equations. Finally, taking M87 as an example, we study and analyze some physical properties of this Kerr-like black hole, and then compare them with the Kerr black hole. Particularly, from the perspective of the black hole shadow and the fact that the Kerr-like black hole and the Kerr black hole is distinguishable, we give the upper limit of the shape parameter of the Einasto density profile, that is approximately α<0.22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha <0.22$$\\end{document}, which may provide a new method to further improve and perfect the density profile of dark matter model. These research results for the black hole in a dark matter halo may indirectly provide an effective method for detecting the existence of dark matter.