Unstable circular null geodesics of static spherically symmetric black holes, Regge poles, and quasinormal frequencies

We consider a wide class of static spherically symmetric black holes of arbitrary dimension with a photon sphere (a hypersurface on which a massless particle can orbit the black hole on unstable circular null geodesics). This class includes various spacetimes of physical interest such as Schwarzschild, Schwarzschild-Tangherlini, and Reissner-Nordström black holes, the canonical acoustic black hole, or the Schwarzschild–de Sitter black hole. For this class of black holes, we provide general analytical expressions for the Regge poles of the $S$ matrix associated with a massless scalar field theory. This is achieved by using third-order WKB approximations to solve the associated radial wave equation. These results permit us to obtain analytically the nonlinear dispersion relation and the damping of the “surface waves” lying close to the photon sphere as well as, from Bohr-Sommerfeld–type resonance conditions, formulas beyond the leading-order terms for the complex frequencies corresponding to the weakly damped quasinormal modes.

Figure 1

The Watson integration contour.