Wave extraction and asymptotic observables

Retain signature minus-plus-plus-plus. In a locally orthonormal asymptotic frame, choose outgoing and ingoing null legs from the time and radial directions with reciprocal square-root-two normalization. Choose the complex angular leg from the theta direction plus i times the phi direction. The null legs have mutual product minus one, and the complex angular leg has product one with its conjugate. Spin rotations and null boosts change tetrad components even when the underlying curvature is unchanged.

Use the Riemann convention with derivative order mu then nu in the upper-index curvature definition, and define the outgoing Newman-Penrose scalar with a minus Weyl contraction on two ingoing legs and two conjugate angular legs. Fix polarizations so the leading asymptotic relation is the second retarded-time derivative of h-plus minus i h-cross. A different curvature sign or angular-leg convention can flip or conjugate familiar formulas, so comparisons must translate conventions explicitly.

At future null infinity in an appropriate asymptotic frame, multiply strain and curvature scalar by areal radius before taking their leading limits. Let H denote the radius-weighted complex strain and Q the corresponding radius-weighted curvature. Q is the second derivative of H with respect to retarded time. H has length units, Q inverse-length units, and the ordinary strain is dimensionless. A nonzero-frequency sinusoid therefore acquires a negative inverse-frequency-squared factor when integrated twice.

Two integration constants remain, corresponding to a constant and a term linear in retarded time. Finite noisy time series introduce further difficulties: low-frequency contamination can create large secular drift even after those two constants are chosen. Frequency-domain prescriptions can reduce such artifacts but require a cutoff and assessment of physical low-frequency content. Removing every slow component without analysis can also remove genuine signal information, including relevant memory contributions.

Finite-radius extraction carries near-zone, tetrad and coordinate effects. Comparing spheres should use a consistent retarded-time prescription and a radius appropriate to the assumed expansion, often an areal rather than a raw coordinate radius. Extrapolating in inverse radius is a model for asymptotic behavior; its order, sphere selection and gauge dependence require checks. More points in a fit do not automatically remove an incorrect asymptotic ansatz.

Characteristic extraction can instead propagate data on null hypersurfaces toward future null infinity, introducing its own worldtube, initial-data and numerical errors. Either approach still needs resolution studies, mode decomposition conventions and treatment of startup radiation. The two-radius arithmetic below tests one stipulated leading correction, not a proof that a real waveform has only that correction. A converged finite-radius answer can remain systematically different from the desired asymptotic observable.