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

Abstract

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 |{psi}{sub 0}|/|{psi}{sub 4}|, and one of the phases (the product |{psi}{sub 0}|{center_dot}|{psi}{sub 4}| 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 {rho} corresponds to a global constant mass parameter that equals the ADM mass (or, equivalently in perturbation theory, that {rho} or {mu} equal their values in the no-radiation limits), thus determining the two moduli of the Weyl scalars |{psi}{sub 0}|, |{psi}{sub 4}|, while leaving their phases as yet undetermined. Second, we break the spin symmetry by requiring that the ratio {pi}/{tau} 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 {psi}{sub 0} and {psi}{sub 4} in the quasi-Kinnersley tetrad.