 Synchronous-to-Newtonian transformation with positive spatial perturbations

Define $\delta g_{ij}=a^2{h}_{ij}$ and $h_{ij}=h{\delta }_{ij}/3+({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)h_S$. Under the passive coordinate shift $\tilde{\tau }=\tau +T{e}^{i\mathbf{k}\cdot \mathbf{x}}$ and ${\tilde{x}}^i=x^i+i{\hat{k}}^{i}L{e}^{i\mathbf{k}\cdot \mathbf{x}}$, removing the traceless spatial perturbation gives $L=h_S/(2k)$. Removing the mixed time-space metric gives $T=L^{\prime }/k=h_S^{\prime }/(2k^2)$. The remaining diagonal metric components are the displayed Newtonian potentials. This sign convention is opposite to defining the synchronous perturbation by $g_{ij}=-a^2({\delta }_{ij}+h_{ij})$.