Exact Einsteinian Black Hole Solution in Dark Matter Environment

Abstract

Motivated by the growing recent interest in black hole solutions immersed in astrophysical dark matter environments, we construct an exact static, spherically symmetric black hole solution sourced by a King dark matter halo through the full Einstein field equations and investigate the physical consequences of the surrounding halo on the resulting spacetime geometry. The influence of the halo on optical phenomena is analyzed via null geodesics, where we show that the dark matter environment substantially modifies photon trajectories, displaces the circular photon orbits, and deforms the associated gravitational lensing structure. By evaluating the Lyapunov exponent of unstable null geodesics, we further determine the corresponding behavior of massless quasinormal modes in the eikonal regime, revealing explicit corrections to the oscillation and damping spectrum induced by the halo. We then explore the thermodynamic properties of the black hole--halo system by computing the conserved mass, Hawking temperature, entropy, heat capacity, and Gibbs free energy, allowing for a detailed assessment of both local and global thermal stability. Our analysis demonstrates that the dark matter halo increases the radius of the photon sphere and the apparent shadow, enlarges the domain of thermodynamic stability, and generates nontrivial phase structures absent in the vacuum Schwarzschild case. These results highlight that realistic dark matter environments can produce observable and thermodynamic deviations from isolated black hole geometries, potentially offering novel signatures of halo-induced gravitational effects.