= Quadratic gravitational effective stress-energy tensor
{title2=$t_{\mu\nu}=-G^{(2)}_{\mu\nu}[h,h]/(8\pi)$}
= Quadratic gravitational stress-energy
{synonym}
Expand the <Einstein tensor> about the <Minkowski metric>. The quadratic term in the <metric perturbation> defines $t_{\mu\nu}=-G^{(2)}_{\mu\nu}[h,h]/(8\pi)$, a symmetric tensor relative to the background. For a first-order vacuum solution, the expanded <contracted Bianchi identity> gives $\partial^\mu t_{\mu\nu}=0$. Equivalently, a second-order correction $j$ obeys $G^{(1)}[j]=8\pi t$, whose flat divergence vanishes identically. This conservation is an on-shell statement. The local $t$ depends on perturbative gauge and the background split; conservation alone does not make it a gauge-independent local gravitational <stress-energy tensor>.
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