Gravitational wave polarization

A gravitational wave in four-dimensional general relativity has two independent linear polarizations, conventionally called plus and cross. A rotation by angle $\vartheta$ in the transverse plane rotates their pair by $2\vartheta$.

A transverse-traceless spatial tensor is orthogonal to its Fourier mode wavevector in each index and has zero trace. The transverse-traceless projector extracts this part. Trace terms and symmetrized coordinate-displacement gradients have no such part, explaining the linear gauge invariance of tensor cosmological perturbations.

A gravitational wave of plus polarization has opposite transverse strain or geodesic deviation eigenvalues along two fixed perpendicular axes. For a sinusoidal periodic function profile, the stretching and compressing roles alternate every half-period.

The cross polarization is the plus pattern rotated through $45$ degrees in the transverse plane.

The transverse-traceless projector maps a symmetric tensor to the part orthogonal to a chosen wavevector and with zero trace. It is the identity on physical gravitational-wave polarizations.

A polarization completeness relation expresses a projector as a sum of outer products of an orthonormal polarization basis. For transverse-traceless tensors, ${\sum }_s{ϵ}_{ij}^s{ϵ}_{mn}^{s\ast }$ is the transverse-traceless projector up to the chosen normalization.