Black hole in a generalized Chaplygin–Jacobi dark fluid: Shadow and light deflection angle

We investigate a generalized Chaplygin-like gas with an anisotropic equation of state, characterizing a dark fluid within which a static spherically symmetric black hole is assumed. By solving the Einstein equations for this black hole spacetime, we explicitly derive the metric function. The spacetime is parametrized by two critical parameters, $B$ and $\alpha$, which measure the deviation from the Schwarzschild black hole and the extent of the dark fluid’s anisotropy, respectively. We explore the behavior of light rays in the vicinity of the black hole by calculating its shadow and comparing our results with the Event Horizon Telescope observations. This comparison constrains the parameters to $0\le B\lesssim 0.03$ and $0<\alpha \lesssim 0.1$. Additionally, we calculate the deflection angles to determine the extent to which light is bent by the black hole. These calculations are further utilized to formulate possible Einstein rings, estimating the angular radius of the rings to be approximately $37.6\mathrm{\mu as}$. Throughout this work, we present analytical solutions wherever feasible, and employ reliable approximations where necessary to provide comprehensive insights into the spacetime characteristics and their observable effects.