 Pure-gauge plane wave in linearized gravity

Under the convention $\delta h_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, choose ${\xi }^{\mu }=b^{\mu }\sin (k\cdot x)$ with constant $b^{\mu }$ and null $k^{\mu }$. The trace-reversed metric perturbation has amplitudeIt satisfies $k_{\mu }{ϵ}_{\mathrm{gauge}}^{\mu \nu }=0$ and $\square {\xi }^{\mu }=0$, so it is an allowed residual gauge symmetry of linearized gravity. Its linearized Riemann curvature operator is zero: the four wavevector factors in each second-derivative term cancel in the antisymmetric curvature combinations. Nonzero transverse-traceless curvature distinguishes the two physical gravitational wave polarizations from these coordinate modes.^[ref]