Linearized Perturbations of a Black Hole: Continuum Spectrum

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.