Spectral decomposition of the perturbation response of the Schwarzschild geometry

The radiative Green’s function for the one-dimensional wave equation with the Regge-Wheeler and Zerilli potentials is formally constructed from recently developed analytic representations for generalized spheroidal wave functions, and decomposed into a convergent sum over quasinormal modes, an integral around a branch cut in the frequency domain, and a high-frequency remnant of the free-space propagator. This paper discusses the contribution to the time response made by the quasinormal modes and, at very late times, by the branch-cut integral. The initial-value problem is considered for source fields with both compact and extended radial dependences, and the problem of the formal divergence of the integrals of extended sources over quasinormal-mode wave functions is solved. The branch-cut integral produces a weak late-time radiative power-law decay tail that will characterize the astrophysically observed radiation spectrum for times subsequent to the exponential decay of the quasinormal ringing, when (ct-$r_{\mathrm{*}}$)≫2MG/$c^2$ and (ct-$r_{\mathrm{*}}$)/$r_{\mathrm{*}}$≪1. This radiative decay tail is shown to diminish to Price’s nonradiative tail in the final limit ct/$r_{\mathrm{*}}$≫1. The method is applied to a characteristic-value problem used to model the gravitational collapse of massive stars, and to the small-body radial in-fall problem. The analysis presented is generalizable, through the Newman-Penrose formalism and Teukolsky’s equations, to obtain the radiative Green’s function for perturbations to the Kerr geometry.