Linearized Perturbations of a Black Hole: Continuum Spectrum

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 $q(γ>0)$ for frequencies $ω= -iγ$. We show that: (a) $q(γ\downarrow0) \propto γ$, (b) $q(Γ) = 0$ at $Γ= (\ell+2)!/[6(\ell-2)!]$, and (c) $q(γ)$ 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 $Γ$, leading to a large dipole in the Green's function_near_ $Γ$. For a source near the horizon and a distant observer, the continuum contribution relative to that of the QNMs is small.