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

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 ${\omega }_{\ell ,n}=-\frac{in}{2}-\frac{i[\ell (\ell +1){]}^2}{2n}+\frac{{\pi }^{1/2}(1-i)[\ell (\ell +1){]}^3}{2^{3/2}{n}^{3/2}}+O(n^{-2})$. 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.

26 pages, 14 figures. Modifications in version 2: inclusion of spin=1 as well as expressions for the QNM sum and some corrections, all in Sec.V. Minor corrections in the rest of the paper

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• Hawking radiation - quasi-normal modes correspondence and effective states for nonextremal Reissner-Nordström black holes
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• Perturbations of Extremal Kerr Spacetime: Analytic Framework and Late-time Tails
• Wavefront twisting by rotating black holes: orbital angular momentum generation and phase coherent detection
• Hawking radiation-quasinormal modes correspondence for large AdS black holes
• Spin-1 Quasi-normal Frequencies in Schwarzschild space-time for Large Overtone Number
• Numerical stability of the electromagnetic quasinormal and quasibound modes of Kerr black holes