[Paper Review] Modeling the black holes surrounded by a dark matter halo in the galactic center of M87

Dong Liu, Yi Yang+2 more|arXiv (Cornell University)|Jul 25, 2023

Galaxies: Formation, Evolution, PhenomenaPhysics and Astronomy79 references3 citations

TL;DR

This paper proposes a Kerr-like black hole solution embedded in a dark matter halo described by the Einasto profile in M87’s galactic center. Using the Newman-Janis algorithm to extend a static solution, the authors derive a metric satisfying Einstein’s equations and show that the black hole shadow distinguishes the Einasto model from standard Kerr black holes, yielding an upper bound of α < 0.22 for the Einasto shape parameter, offering a new observational constraint on dark matter density profiles.

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., $ρ_{ ext{eina}} (r)=ρ_ ext{e} \exp ( -2 α^{-1} ((r/r_ ext{e})^α-1 ) )$ (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 $α&lt;0.22$, 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.

Motivation & Objective

• To model a rotating black hole in a dark matter halo consistent with the Einasto profile in M87’s galactic center.
• To extend a static spherically symmetric black hole solution to a rotating (Kerr-like) solution using the Newman-Janis algorithm.
• To verify that the resulting metric satisfies the Einstein field equations.
• To analyze physical properties of the black hole, particularly its shadow, and compare them with the standard Kerr black hole.
• To constrain the Einasto profile’s shape parameter α using shadow distinguishability, offering a new method to refine dark matter models.

Proposed method

• Construct a static, spherically symmetric black hole solution within a dark matter halo using the Einasto density profile: ρ_eina(r) = ρ_e exp(-2α⁻¹((r/re)^α - 1)).
• Apply the Newman-Janis algorithm to generate a rotating solution from the static metric, yielding a Kerr-like spacetime with modified mass function R(r) = M + r/2 - g(r)/2.
• Verify the resulting metric satisfies the Einstein field equations by computing non-zero components of the Einstein tensor and matching them to the energy-momentum tensor in orthonormal basis.
• Derive expressions for energy density ρ_ε and radial/azimuthal pressures p_r, p_θ, p_ϕ from the Einstein tensor components.
• Use the metric to compute the black hole shadow and analyze its dependence on the Einasto parameter α.
• Compare the shadow morphology with that of the standard Kerr black hole to determine the maximum α for which the two remain distinguishable.

Experimental results

Research questions

• RQ1 Can a Kerr-like black hole solution be consistently derived in a dark matter halo described by the Einasto profile within the framework of general relativity?
• RQ2 How does the presence of a dark matter halo affect the shadow of a black hole in M87 compared to a standard Kerr black hole?
• RQ3 What is the maximum value of the Einasto shape parameter α for which the black hole shadow in a dark matter halo remains distinguishable from the standard Kerr shadow?
• RQ4 Does the derived solution satisfy the full set of Einstein field equations with a physically consistent energy-momentum tensor?
• RQ5 Can observational constraints on the black hole shadow be used to refine or constrain the parameters of the dark matter density profile?

Key findings

• The derived Kerr-like black hole solution satisfies the Einstein field equations, as confirmed by explicit computation of the Einstein tensor and matching to the energy-momentum tensor.
• The energy density ρ_ε and pressure components p_r, p_θ, p_ϕ are derived in terms of R′(r) and R′′(r), confirming the solution’s physical consistency.
• The black hole shadow in the Einasto dark matter halo differs from the standard Kerr shadow, enabling observational distinction.
• The upper limit for the Einasto shape parameter is found to be α < 0.22, based on shadow distinguishability from the Kerr case.
• This constraint provides a new observational method to refine the Einasto dark matter profile and indirectly probe dark matter structure.
• The results suggest that black hole shadow observations can serve as a probe for dark matter density profiles, offering a novel indirect detection strategy.