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 $omega = -igamma$. We show that: (a) $q(gammadownarrow0) propto gamma$, (b) $q(Gamma) = 0$ at $Gamma = (ell+2)!/[6(ell-2)!]$, and (c) $q(gamma)$ oscillates with period $sim 1$ ($2Mequiv1$). 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 Greens function_near_ $Gamma$. For a source near the horizon and a distant observer, the continuum contribution relative to that of the QNMs is small.