= Synchronous-to-Newtonian scalar Fourier transformation
{title2=$T=-h_s'/(2k^2),\quad L=-h_s/(2k)$}
For a nonzero <Fourier mode> in <synchronous gauge>, write $h_{ij}=h\delta_{ij}/3+h_s(\widehat k_i\widehat k_j-\delta_{ij}/3)$. Under the passive displacement $\xi^\mu=(T,i\widehat k_iL)e^{i\mathbf k\cdot\mathbf x}$, cancellation of spatial scalar shear requires $h_s+2kL=0$, and cancellation of the shift requires $kT-L'=0$. The resulting <cosmological Newtonian potential> is $\Phi=(h_s''+\mathcal Hh_s')/(2k^2)$ and the spatial potential is $\Psi=-(h-h_s)/6-\mathcal Hh_s'/(2k^2)$. Both remain unchanged under <residual synchronous-gauge freedom>, for which $T'+\mathcal HT=0$ and $L'=kT$.
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