Linearized Perturbations of a Black Hole: Continuum Spectrum

Linearized perturbations of a Schwarzschild black hole are described, for each angular momentum $\mathrm{\ell }$, 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) $q(\gamma \downarrow 0)\propto \gamma$, (b) $q(\mathrm{\Gamma })=0$ at $\Gamma = (\ell+2)!/[6(\ell-2)!]$,and(c)$q(\gamma)$oscillateswithperiod$\sim 1$($2M\equiv1$)\mathrm{.}For$\ell=2$, a pair of QNMs are found beyond the cut on the unphysical sheet very close to $\mathrm{\Gamma }$, leading to a large dipole in the Green’s function_near_ $\mathrm{\Gamma }$. For a source near the horizon and a distant observer, the continuum contribution relative to that of the QNMs is small.