Excitation of Schwarzschild black-hole quasinormal modes

The evolution of a scalar test field from initial data in a space-time that contains a Schwarzschild black hole is studied. The investigation involves a Green’s function representation of the solution, and general formulas determining the excitation of the quasinormal modes are discussed. We use the semianalytic phase-integral method to evaluate these formulas approximately. As an example that can be studied analytically we use Gaussian initial data. For intermediate times, when the quasinormal ringing dominates the radiation, the approximate results are shown to agree perfectly with the results of numerical evolutions of the same initial data. Our approximate analysis also reveals that the slowest damped mode for a certain radiating multipole l is maximally excited when the Gaussian has a half-width 12.24M(l+1/2${)}^{\mathrm{-}1}$ (M is the mass of the black hole in geometrized units). For broader pulses the fundamental mode is exponentially suppressed. Most of the presented results remain valid in the physically more interesting case of a gravitationally perturbed black hole.