Nonlinear scalar metric in a cosmological gradient expansion ($h_{ij}=a_{\mathrm{bg}}^2{e}^{2\zeta }{\delta }_{ij}$)

With vectors and independent tensors neglected, finite scalar cosmological perturbations of the local spatial scale can be encoded by $h_{ij}=a_{\mathrm{bg}}^2{e}^{2\zeta }{\delta }_{ij}$. Scalar lapse and shift potentials complete its 3+1 decomposition of spacetime. The exact shift-square term is $h^{ij}{N}_i{N}_j$, independently of the parametrization of the scalar shift.