Spectroscopy of the Schwarzschild Black Hole at Arbitrary Frequencies

Abstract

Linear field perturbations of a black hole are described by the Green function of the wave equation that they obey. After Fourier decomposing the Green function, its two natural contributions are given by poles (quasinormal modes) and a largely unexplored branch cut in the complex-frequency plane. We present new analytic methods for calculating the branch cut on a Schwarzschild black hole for arbitrary values of the frequency. The branch cut yields a power-law tail decay for late times in the response of a black hole to an initial perturbation. We determine explicitly the first three orders in the power-law and show that the branch cut also yields a new logarithmic behaviour $T^{-2\ell-5}\ln(T)$ for late times. Before the tail sets in, the quasinormal modes dominate the black hole response. For electromagnetic perturbations, the quasinormal mode frequencies approach the branch cut at large overtone index $n$. We determine these frequencies up to $n^{-5/2}$ and, formally, to arbitrary order. Highly-damped quasinormal modes are of particular interest in that they have been linked to quantum properties of black holes.