Transverse-traceless tensor ($k_i{h}_{ij}=0,h_{ii}=0$)

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.