Ringdown waves from hairy black holes

Abstract

We study how quasinormal-mode frequencies may encode information about the effective matter source responsible for black-hole hair. Using the established eikonal correspondence between quasinormal modes and unstable null geodesics, we relate shifts in the ringdown spectrum to perturbations of the photon-orbit frequency and Lyapunov exponent. The black hole hair is treated as an anisotropic fluid perturbatively added to the vacuum black holes (Schwarzschild and Kerr black holes). In particular, we derive formulas which allow one to directly read off deviations from the Schwarzschild or Kerr QNM spectrum in terms of the corresponding equation-of-state parameters of the anisotropic fluid. In this setting, regardless of the energy conditions, our formulas provide a systematic method for computing quasi-normal mode frequencies for a broad class of hairy black holes.