Spectroscopy of the Schwarzschild Black Hole at Arbitrary Frequencies

TL;DR

This paper develops new analytic methods to analyze the Green function of Schwarzschild black holes across all frequencies, revealing detailed late-time decay behaviors and the relationship between quasinormal modes and the branch cut.

Contribution

It introduces novel analytic techniques for calculating the branch cut contributions at arbitrary frequencies in Schwarzschild black holes, enhancing understanding of black hole perturbation responses.

Findings

Derived explicit power-law decay rates for late-time tails.

Discovered a new logarithmic decay behavior in the response.

Linked highly-damped quasinormal modes to quantum black hole properties.

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.