Definition: Background de Donder gauge

For a Lorentzian metric $\bar{q}$ with Levi-Civita connection $\bar{\nabla }$ and a smooth symmetric $u$,

and $\mathcal{F}[u;\bar{q}]:=\delta S_{\mathrm{gf}}[u;\bar{q}]/\delta u_{\mu \nu }$ its Euler derivative with respect to the fluctuation, a contravariant symmetric tensor density, defined through compactly supported variations of $u$. At $\bar{q}=\eta$, ${C^{\mu }}_{\eta }=C^{\mu }$ and $S_{\mathrm{gf}}[u;\eta ]={S^{(0)}}_{\mathrm{gf}}[u]$ of xe:gf. For $\bar{q}=\eta +\kappa h_{\mathrm{cl}}$ write

with ${\mathcal{F}}^{(1)}[w]u$ linear in $w$ and in $u$.