Transverse-traceless projector (${\Pi }_{ij,mn}^{\mathrm{TT}}$)

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.