An analytical computation of asymptotic Schwarzschild quasinormal frequencies

TL;DR

The paper analyzes the asymptotic quasinormal frequencies of Schwarzschild black holes and their possible link to loop quantum gravity. It develops an analytic method using a Taylor-like expansion of the wavefunction and continued fractions to solve the master equation in the large-frequency limit, expressing the condition in terms of Gamma functions and sine factors. The key result is that for even-spin perturbations the asymptotic real part of the frequency is ln(3)/(4pi) (in appropriate units) and that for odd or half-integer spins the real part vanishes, with a simple imaginary spacing; this supports Dreyer's observation and connects to the Barbero-Immirzi parameter. The paper also surveys generalizations to charged or rotating black holes and higher dimensions, discusses potential thermodynamic interpretations, and outlines speculative links to spin networks and black hole chemistry.

Abstract

Recently it has been proposed that a strange logarithmic expression for the so-called Barbero-Immirzi parameter, which is one of the ingredients that are necessary for Loop Quantum Gravity (LQG) to predict the correct black hole entropy, is not another sign of the inconsistency of this approach to quantization of General Relativity, but is rather a meaningful number that can be independently justified in classical GR. The alternative justification involves the knowledge of the real part of the frequencies of black hole quasinormal states whose imaginary part blows up. In this paper we present an analytical derivation of the states with frequencies approaching a large imaginary number plus ln 3 / 8 pi M; this constant has been only known numerically so far. We discuss the structure of the quasinormal states for perturbations of various spin. Possible implications of these states for thermal physics of black holes and quantum gravity are mentioned and interpreted in a new way. A general conjecture about the asymptotic states is stated. Although our main result lends some credibility to LQG, we also review some of its claims in a critical fashion and speculate about its possible future relevance for Quantum Gravity.