Modeling the black holes surrounded by a dark matter halo in the galactic center of M87
Dong Liu, Yi Yang, Zhaoyi Xu, Zheng‐Wen Long
Abstract
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., $$\rho _{\text {eina}} (r)=\rho _\text {e} \exp ( -2 \alpha ^{-1} ((r/r_\text {e})^\alpha -1 ) )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mtext>eina</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mtext>e</mml:mtext> </mml:msub> <mml:mo>exp</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>α</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mtext>e</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> (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 $$\alpha <0.22$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mn>0.22</mml:mn> </mml:mrow> </mml:math> , 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.