Photon spheres near black holes in a model with anisotropic fluid

Abstract

The semi-review paper studies the null geodesics which appear for black hole solutions in the gravitational $4d$ model with anisotropic fluid. The equations of state for the fluid and solutions itselves depend upon 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 speed of light, $p_r$ and $p_t$ are pressures in radial and transverse to radial directions, respectively. The circular null geodesics are explored and the master equation for radius $r_*$ of photon sphere is outlined as well as the proposition on existence and uniqueness of the solution to master equation obeying $r_* > r_h$, where $r_h$ is horizon radius. The relations for spectrum of quasinormal modes for a test massless scalar field in the eikonal approximation are overviewed and compared with cyclic frequencies of circular null geodesics. The shadow angles are explored.