Lemma: Newtonian sign comparison

Let $H=q-\eta$ denote the physical metric perturbation. For an auxiliary static conserved source $T_{00}=\rho \ge 0$, $T_{0i}=T_{ij}=0$, $\rho$ smooth compactly supported and nonzero, the linear unswitched equation is solved by

where

Thus ${\kappa }^2=32\pi G_N$ gives the attractive Newtonian sign.

Proof. The scalar metric variation computed below is negative, so the unswitched metric equation is $G_{\mu \nu }={\kappa }^2{T}_{\mu \nu }/4$. For the displayed perturbation, ${\bar{H}}_{00}=4{\Phi }_N$, ${\bar{H}}_{ij}=0$, and $D(H)=0$. Equation (eq:Glin) gives $G_{\mathrm{lin},00}(H)=2\Delta {\Phi }_N$ and zero spatial components. The Newton inverse of $\Delta$ gives (eq:Newton).