WKB analysis of the Regge-Wheeler equation down in the frequency plane

TL;DR

This paper applies WKB analysis to the Regge-Wheeler equation for black-hole gravitational waves at large negative imaginary frequencies, deriving an analytic formula for highly damped quasinormal modes and confirming results with numerical data.

Contribution

It introduces a novel WKB-based method involving an expansion around the black-hole singularity to analytically compute highly damped quasinormal modes.

Findings

Analytic formula for highly damped Schwarzschild quasinormal modes

Good agreement with numerical results for wave outgoing to infinity

Connection formula involving expansion around the black-hole singularity

Abstract

The Regge-Wheeler equation for black-hole gravitational waves is analyzed for large negative imaginary frequencies, leading to a calculation of the cut strength for waves outgoing to infinity. In the--limited--region of overlap, the results agree well with numerical findings [Class. Quantum Grav._20_, L217 (2003)]. Requiring these waves to be outgoing into the horizon as well subsequently yields an analytic formula for the highly damped Schwarzschild quasinormal modes,_including_ the leading correction. Just as in the WKB quantization of, e.g., the harmonic oscillator, solutions in different regions of space have to be joined through a connection formula, valid near the boundary between them where WKB breaks down. For the oscillator, this boundary is given by the classical turning points; fascinatingly, the connection here involves an expansion around the black-hole singularity r=0.