Regge poles of the Schwarzschild black hole: A WKB approach

We provide simple and accurate analytical expressions for the Regge poles of the Schwarzschild black hole. This is achieved by using third-order WKB approximations to solve the radial wave equations for spins 0, 1, and 2. These results permit us to obtain analytically the dispersion relation and the damping of the surface waves lying close to the photon sphere of the Schwarzschild black hole and which generate the weakly damped quasinormal modes of its spectrum. Our results could be helpful in order to simplify considerably the description of wave scattering from the Schwarzschild black hole as well as the analysis of the gravitational radiation created in many black hole processes. Furthermore, the existence of nonlinear dispersion relations for the photons propagating close to the photon sphere could have also important consequences in the context of gravitational lensing.