Linearized vacuum Einstein equations ($G_{\mu \nu }^{(1)}[h]=0$)

The source-free Linearized Einstein equations. For a metric perturbation about the Minkowski metric, imposing the Lorenz gauge in linearized gravity reduces them to $\square h_{\mu \nu }=0$ together with ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$, where $\bar{h}$ is the trace-reversed metric perturbation. A linearized coordinate gauge transformation preserves $G^{(1)}[h]=0$, while preserving the imposed gauge additionally requires its generator to satisfy the wave equation.