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

Abstract

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.