Linearized Perturbations of a Black Hole: Continuum Spectrum

TL;DR

This paper analyzes the continuum spectrum of linearized perturbations of Schwarzschild black holes, revealing properties of the cut strength and its relation to quasinormal modes, with implications for gravitational wave signals.

Contribution

It characterizes the continuum spectrum and its interaction with quasinormal modes in black hole perturbations, providing new insights into their spectral properties and Green's function behavior.

Findings

The cut strength $q(\gamma)$ is proportional to $\gamma$ as $\gamma o 0$.

The cut strength vanishes at specific frequencies $\Gamma$ related to angular momentum $\ell$.

For $\ell=2$, a pair of QNMs are found near the cut, causing a large dipole in the Green's function.

Abstract

Linearized perturbations of a Schwarzschild black hole are described, for each angular momentum $\ell$, by the well-studied discrete quasinormal modes (QNMs), and in addition a continuum. The latter is characterized by a cut strength $q(\gamma>0)$ for frequencies $\omega = -i\gamma$. We show that: (a) $q(\gamma\downarrow0) \propto \gamma$, (b) $q(\Gamma) = 0$ at $\Gamma = (\ell+2)!/[6(\ell-2)!]$, and (c) $q(\gamma)$ oscillates with period $\sim 1$ ($2M\equiv1$). For $\ell=2$, a pair of QNMs are found beyond the cut on the unphysical sheet very close to $\Gamma$, leading to a large dipole in the Green's function_near_ $\Gamma$. For a source near the horizon and a distant observer, the continuum contribution relative to that of the QNMs is small.