Revisiting black holes in dark-matter halos: on consistent solutions to the Einstein equations

Abstract

A number of recent papers have claimed to construct solutions of Einstein's equations describing black holes surrounded by dark-matter halos with empirically motivated density profiles such as the Navarro-Frenk-White, Burkert, Einasto, pseudo-isothermal, and solitonic distributions. We show that the approach used to obtain many of these metrics generically does not lead to the correct solutions to the Einstein equations for the matter sources they purport to represent. This issue originates from applying the Newtonian relation between the tangential velocity and the enclosed mass directly within a relativistic framework, followed by the ad hoc assumption $g(r)=f(r)$ for the metric functions. This procedure leads to an anisotropic fluid with $P_r=-ρ$ and $P_t=-rρ'/2-ρ$, whose density differs from the claimed halo profile and often becomes non-physical near the horizon, violating the weak energy condition. As a result, the obtained spacetimes do not describe black holes embedded in known galactic halos but rather distinct anisotropic configurations unrelated to the intended matter distribution. We demonstrate this problem on several representative examples from the literature, including metrics based on the NFW, Burkert, Einasto, solitonic, pseudo-isothermal, and Dehnen-(1,4,5/2) profiles, as well as the case of the NGC~4649 halo. For each case, the correct Einstein-consistent form of the metric and the associated physical interpretation are provided. Our analysis clarifies the limits of validity of the Newtonian approximation near compact objects and establishes a consistent framework for constructing dark-matter-inspired black-hole geometries within General Relativity.