The Branch Cut and Quasi-normal Modes at Large Imaginary Frequency in Schwarzschild Space-time

TL;DR

This paper analyzes the branch cut contribution to the Green function and calculates highly-damped quasinormal modes in Schwarzschild spacetime, revealing their connection and implications for quantum gravity and black hole perturbations.

Contribution

It introduces a novel analytic-continuation technique to compute the branch cut contribution at large imaginary frequencies and determines the first highly-damped electromagnetic quasinormal mode frequencies.

Findings

Branch cut contribution calculated for large imaginary frequencies.

First determination of highly-damped electromagnetic quasinormal modes.

Reveals a deep connection between quasinormal modes and the branch cut.

Abstract

The 'retarded' Green function for fields propagating on a Schwarzschild black hole spacetime possesses a branch cut on the complex frequency plane. Classically, the branch cut is important, for example, in order to fully determine the response of the black hole to a linear field perturbation. The branch cut is also useful for the calculation of the self-force on a point particle moving in the Schwarzschild background. In this paper we use techniques of analytic-continuation to the complex plane of the radial coordinate in order to calculate the branch cut contribution to the Green function in the limit of large imaginary frequency. It is expected that the contribution of this frequency regime to the perturbation response and to the self-force will be mostly for short time intervals. We also determine the highly-damped quasinormal mode frequencies for electromagnetic perturbations in Schwarzschild for the first time (previously only the leading imaginary part was known), which seem to have a 'deep connection' with the branch cut. We find that these frequencies behave like $\omega_{\ell,n}= -\dfrac{in}{2}-\dfrac{i[\ell(\ell+1)]^2}{2n}+\dfrac{\pi^{1/2}(1-i)[\ell(\ell+1)]^3}{2^{3/2}n^{3/2}}+O(n^{-2})$. The highly-damped quasinormal modes are particularly interesting for theories of quantum gravity in that they are believed to probe the small scale structure of the spacetime.