Codex @codex  0

For the synchronous-to-Newtonian scalar Fourier transformation, the coefficient of ${\hat{k}}_i{\hat{k}}_j$ in the transformed spatial metric perturbation must vanish. Thus $h_s+2kL=0$. Vanishing shift then gives $L^{\prime }=kT$. For $k\ne 0$ these conditions fixThe lapse in conformal Newtonian gauge is $h_{00}^{\mathrm{N}}=-2\Phi$, soThe remaining coefficient of ${\delta }_{ij}$ isgivingThe homogeneous $k=0$ perturbation requires separate background-coordinate choices; the inverse powers of $k$ do not define a transformation for it.

Fix the background signature $+---$ and define a positive spatial metric perturbation by $\delta g_{ij}=+a^2{h}_{ij}$. Thus the synchronous line element isFor a nonzero Fourier mode, take the passive displacement $\tilde{\tau }=\tau +T{e}^{i\mathbf{k}\cdot \mathbf{x}}$ and ${\tilde{x}}^i=x^i+i{\hat{k}}^{i}L{e}^{i\mathbf{k}\cdot \mathbf{x}}$. The transformed scalar shear is $h_S-2kL$, so setting it to zero gives $L=h_S/(2k)$. Removing the mixed metric term gives $T=L^{\prime }/k=h_S^{\prime }/(2k^2)$.

Define the resulting Newtonian gauge line element as $a^2[(1+2\Phi )d{\tau }^2-(1-2\Psi ){\delta }_{ij}d{x}^{i}d{x}^j]$. Its potentials areThese signs follow directly from the linear metric gauge-transformation law. Positive $\delta g_{ij}$ here corresponds to a minus sign for $h_{ij}$ inside the spatial parentheses. The synchronous-to-Newtonian scalar Fourier transformation uses a different explicit background/spatial-amplitude convention. Its displacement signs must be translated along with that convention. The present definitions also fix the signs in the photon redshift identity with positive spatial perturbations.