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 $$q=1,2,...$$ : pᵣ=-ρ c²(2q-1)⁻¹, pₜ=-pᵣ , where ρ is the mass density, c is the speed of light, pᵣ and pₜ 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*>rₕ , where rₕ 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.