For a nonzero plane-wave trace-reversed metric perturbation,
(1)
The wave equation therefore requires a null wave vector. The Lorenz gauge in linearized gravity supplies transversality:
(2)
The amplitude is symmetric because it represents a metric perturbation; its real part is the physical wave. The two displayed conditions do not by themselves make it spatially transverse and traceless: the additional reduction to transverse-traceless gauge uses the remaining residual gauge symmetry of linearized gravity.

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