Spectral decomposition of black-hole perturbations on hyperboloidal slices

TL;DR

This paper develops a spectral decomposition method for black-hole perturbations on hyperboloidal slices, explicitly calculating quasi-normal modes and their amplitudes using Laplace transforms and Taylor series analysis.

Contribution

It introduces a novel spectral decomposition approach for wave equations on hyperboloidal slices, explicitly computing all spectral components with high accuracy.

Findings

Explicit spectral decomposition including quasi-normal modes and branch cut contributions

Accurate calculation of mode amplitudes and contributions to black hole response

Detailed analysis of the role of infinity frequency modes in early-time response

Abstract

In this paper we present a spectral decomposition of solutions to relativistic wave equations described on horizon penetrating hyperboloidal slices within a given Schwarzschild-black-hole background. The wave equa- tion in question is Laplace-transformed which leads to a spatial differential equation with a complex parameter. For initial data which are analytic with respect to a compactified spatial coordinate, this equation is treated with the help of the Mathematica-package in terms of a sophisticated Taylor series analysis. Thereby, all ingredients of the desired spectral decomposition arise explicitly to arbitrarily prescribed accuracy, including quasi normal modes, quasi normal mode amplitudes as well as the jump of the Laplace-transform along the branch cut. Finally, all contributions are put together to obtain via the inverse Laplace transformation the spectral de- composition in question. The paper explains extensively this procedure and includes detailed discussions of relevant aspects, such as the definition of quasi normal modes and the question regarding the contribution of infinity frequencies modes to the early time response of the black hole.