Mixing of spherical and spheroidal modes in perturbed Kerr black holes

Emanuele Berti; Antoine Klein

doi:10.1103/PhysRevD.90.064012

paper

Mixing of spherical and spheroidal modes in perturbed Kerr black holes

arXiv:1408.1860 · doi:10.1103/PhysRevD.90.064012

Abstract

The angular dependence of the gravitational radiation emitted in compact binary mergers and gravitational collapse is usually separated using spin-weighted spherical harmonics $_{s} Y_{ℓ m}$ of spin weight $s$ , that reduce to the ordinary spherical harmonics $Y_{ℓ m}$ when $s = 0$ . Teukolsky first showed that the perturbations of the Kerr black hole that may be produced as a result of these events are separable in terms of a different set of angular functions: the spin-weighted spheroidal harmonics $_{s} S_{ℓ mn}$ , where $n$ denotes the "overtone index" of the corresponding Kerr quasinormal mode frequency $ω_{ℓ mn}$ . In this paper we compute the complex-valued scalar products of the $_{s} S_{ℓ mn}$ 's with the $_{s} Y_{ℓ m}$ 's ("spherical-spheroidal mixing coefficients") and with themselves ("spheroidal-spheroidal mixing coefficients") as functions of the dimensionless Kerr parameter $j$ . Tables of these coefficients and analytical fits of their dependence on $j$ are available online for use in gravitational-wave source modeling and in other applications of black-hole perturbation theory.

8 pages, 6 figures, 1 table. Very minor changes to match version in press in PRD. Tables and fits of the mixing coefficients are available at http://www.phy.olemiss.edu/~berti/qnms.html

Mixing of spherical and spheroidal modes in perturbed Kerr black holes

Abstract

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