 Linearized Einstein equations

The Linearized Einstein equations retain first-order terms in a metric perturbation $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ about the Minkowski metric. Use metric signature $(-,+,+,+)$, coordinates $x^0=ct$, andIn Lorenz gauge in linearized gravity, ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$, the first-order Einstein tensor is $G_{\mu \nu }^{(1)}=-\square {\bar{h}}_{\mu \nu }/2$. HenceThe source must satisfy background stress-energy conservation, ${\partial }^{\mu }{T}_{\mu \nu }=0$, since taking the divergence of the left-hand side gives zero. In units $G=c=1$ this becomes $\square {\bar{h}}_{\mu \nu }=-16\pi T_{\mu \nu }$.

The equation is a sourced wave equation, not just a static Poisson equation. A retarded solution additionally needs a radiation boundary condition for linearized gravity; homogeneous gravitational waves can otherwise be added. The expansion is controlled only where the neglected higher-order metric perturbations remain small.