The quadratic part of the Einstein–Hilbert density

For ${\partial }_{\mu }{h}^{\mu \nu }=0$, $H=0$, (eq:G2) gives $G_{\mathrm{lin}}{}_{\mu \nu }[h]=-\frac{1}{2}\square h_{\mu \nu }$, so $S_{\mathrm{FP}}=-\frac{1}{8}\int h_{\mu \nu }\square h^{\mu \nu }=\frac{1}{8}\int {\partial }_{\lambda }{h}_{\mu \nu }{\partial }^{\lambda }{h}^{\mu \nu }$, whose spatial components read $\frac{1}{8}\int ({\partial }_t{h}_{ij}{\partial }_t{h}_{ij}-{\partial }_k{h}_{ij}{\partial }_k{h}_{ij})$, the same sign as $\frac{1}{2}\int {\dot{\varphi }}^2$. Independently, with the dictionary ${\kappa }^2=8\pi G_{\mathrm{N}}$, Lemma lem:gf below and a static source $T^{00}=\varrho >0$ give $\Delta (\kappa {\bar{h}}_{00})=2{\kappa }^2\varrho =16\pi G_{\mathrm{N}}\varrho$, i.e.\ ${\bar{h}}_{00}=4{\Phi }_{\mathrm{N}}$ with $\Delta {\Phi }_{\mathrm{N}}=4\pi G_{\mathrm{N}}\varrho$, ${\Phi }_{\mathrm{N}}<0$: attraction. Both checks fix the sign of Definition def:action; it agrees with the Landau–Lifshitz convention LL and is not copied from it.