Towards a wave-extraction method for numerical relativity. V. Estimating the gravitational-wave content of spatial hypersurfaces

We extract the Weyl scalars 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 |Ψ₀|/|Ψ₄|, and one of the phases (the product |Ψ₀|·|Ψ₄| and the sum of the phases are determined by the so-called Beetle-Burko 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 |Ψ₀|, |Ψ₄|, 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 Ψ₀ and Ψ₄ in the quasi-Kinnersley tetrad.