1112.2695 (Marc Casals et al.)

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

Marc Casals, Adrian C. Ottewill

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.

View original: http://arxiv.org/abs/1112.2695