 Solution

Write $g_{\mu \nu }={\eta }_{\mu \nu }+ϵh_{\mu \nu }$. The linearized inverse metric is $g^{\mu \nu }={\eta }^{\mu \nu }-ϵh^{\mu \nu }$, and all quadratic Christoffel products are $O({ϵ}^2)$. The wave-coordinate condition linearizes toUsing the supplied Ricci formula then gives $G_{\mu \nu }^{(1)}=-\frac{1}{2}\square {\bar{h}}_{\mu \nu }$. Consequently the Linearized Einstein equations in Lorenz gauge in linearized gravity areThis Lorenz condition is the first-order form of the harmonic coordinate equations.