Towards a wave-extraction method for numerical relativity. V. Extracting the Weyl scalars in the quasi-Kinnersley tetrad from spatial data

Abstract

We extract the Weyl scalars $Ψ_0$ and $Ψ_4$ in the quasi-Kinnersley tetrad by finding initially the (gauge--, tetrad--, and background--independent) transverse quasi-Kinnersley frame. This step still leaves two undetermined degrees of freedom: the ratio $|Ψ_0|/|Ψ_4|$, and one of the phases (the product $|Ψ_0|\cdot |Ψ_4|$ and the {\em sum} of the phases are determined by the so-called BB radiation scalar). The residual symmetry ("spin/boost") can be removed by gauge fixing of spin coefficients in two steps: First, we break the boost symmetry by requiring that $ρ$ corresponds to a global constant mass parameter that equals the ADM mass (or, equivalently in perturbation theory, that $ρ$ or $μ$ equal their values in the no-radiation limits), thus determining the two moduli of the Weyl scalars $|Ψ_0|, |Ψ_4|$, while leaving their phases as yet undetermined. Second, we break the spin symmetry by requiring that the ratio $π/τ$ gives the expected polarization state for the gravitational waves, thus determining the phases. Our method of gauge fixing--specifically its second step--is appropriate for cases for which the Weyl curvature is purely electric. Applying this method to Misner and Brill--Lindquist data, we explicitly find the Weyl scalars $Ψ_0$ and $Ψ_4$ perturbatively in the quasi-Kinnersley tetrad.