= 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 amplitude
$$
\epsilon_{\rm gauge}^{\mu\nu}=k^\mu b^\nu+k^\nu b^\mu-\eta^{\mu\nu}(k\cdot b).
$$
It satisfies $k_\mu\epsilon_{\rm gauge}^{\mu\nu}=0$ and $\Box\xi^\mu=0$, 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.https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S5.html[Section 5.3.1, equations 5.232–5.234: residual plane-wave gauge transformation and transverse-traceless reduction]{ref}
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