= Linearized Einstein equations
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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 $x^0=ct$, and
$$
\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h,\qquad
\Box=\eta^{\alpha\beta}\partial_\alpha\partial_\beta=-c^{-2}\partial_t^2+\nabla^2.
$$
In <Lorenz gauge in linearized gravity>, $\partial^\mu\bar h_{\mu\nu}=0$, the first-order <Einstein tensor> is $G^{(1)}_{\mu\nu}=-\Box\bar h_{\mu\nu}/2$. Hence
$$
\Box\bar h_{\mu\nu}=-\frac{16\pi G}{c^4}T_{\mu\nu}.
$$
The 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 $\Box\bar h_{\mu\nu}=-16\pi T_{\mu\nu}$.
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.
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