Transverse-traceless polarizations

Consider a locally plane wave traveling along positive z on a weakly perturbed flat background with the previous signature. Its retarded argument is t minus z over c. In TT gauge the time components vanish, the spatial tensor is transverse, and its spatial trace vanishes. The remaining x-y block contains two real functions. This construction applies to the radiative vacuum field, not to every constrained near-source component.

Normalize each polarization tensor to have squared tensor norm two. The cross tensor therefore includes both xy and yx entries, a detail that later produces a factor of two in energy formulas. The strain functions themselves are dimensionless. A transverse circular motion of coordinate labels is not their definition: they are components of a metric perturbation connected to measurable tidal curvature and light-propagation responses in an appropriate observer frame.

Rotate the transverse basis axes by an angle psi. The plus tensor is quadratic in those basis vectors, so its transformation uses cosine and sine of twice psi. The cross tensor transforms as its orthogonal partner. Projecting the same physical tensor onto the new basis mixes the strain components by the corresponding two-dimensional rotation. A forty-five-degree basis change consequently interchanges the roles of plus and cross with a sign.

This spin-two behavior differs from rotating an ordinary planar arrow by the same angle. The squared component sum is invariant, and the tensor contraction is twice that sum. A detector's polarization response coefficients must transform alongside the strains, preserving their scalar contraction. Rotating only the strain labels while holding response coefficients in their old basis would create an artificial change in the predicted signal.

At leading quadrupole order a nonprecessing circular binary can be written with plus strain proportional to one half of one plus cosine-squared inclination times a cosine phase, and cross strain proportional to cosine inclination times a sine phase. Let their common scale be A. A face-on source has equal quadrature components of amplitude A; an edge-on source has only plus amplitude A over two. This is a physical change of viewing direction, not a passive basis rotation.

These formulas assume a slowly evolving circular orbit, a dominant quadrupole harmonic and far-zone propagation. Higher harmonics, precession and eccentricity can alter the polarization history. Averaging over source inclination is distinct from averaging a detector's sky pattern, and neither should be silently inserted into an optimally oriented amplitude. A particular interferometer records its projection of the two strains, so a source polarization amplitude is not automatically the measured instrument strain.