Photon Spheres near Black Holes in a Model with an Anisotropic Fluid

This semi-review paper studies null geodesics which exist for black hole solutions in a gravitational 4D model with an anisotropic fluid. The equations of state for the fluid and the solutions depends on the integer parameter $$q=1,2,...$$ : $$p_{r}=-\rho c^{2}(2q-1)^{-1},\quad p_{t}=-p_{r}$$ , where $$\rho$$ is the mass density, $$c$$ is the speed of light, $$p_{r}$$ and $$p_{t}$$ are pressures in the radial and transverse directions, respectively. Circular null geodesics are explored, and a master equation for the radius $$r_{*}$$ of a photon sphere is found, as well as the proposition on the existence and uniqueness of a solution to the master equation, obeying $$r_{*}>r_{h}$$ , where $$r_{h}$$ is the horizon radius. Relations for the spectrum of quasinormal modes for a test massless scalar field in the eikonal approximation are overviewed and compared with the cyclic frequencies of circular null geo desics. Shadow angles are explored.