Spectral decomposition of the perturbation response of the Schwarzschild geometry

Details Spectral decomposition of the perturbation response of the Schwarzschild geometry Spectral decomposition of the perturbation response of the Schwarzschild geometry

Jul 1986

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1986phrvd..34..384l&link_type=abstract

Physical Review D (Particles and Fields), Volume 34, Issue 2, 15 July 1986, pp.384-408

Astronomy and Astrophysics

Astrophysics

173

Black Holes

Scientific paper

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*)>>2MG/c2 and (ct-r*)/r*<<1. This radiative decay tail is shown to diminish to Price's nonradiative tail in the final limit ct/r*>>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.

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