= Nonlinear scalar metric in a cosmological gradient expansion
{title2=$h_{ij}=a_{\mathrm{bg}}^2e^{2\zeta}\delta_{ij}$}
With vectors and independent tensors neglected, finite <scalar cosmological perturbations> of the local spatial scale can be encoded by $h_{ij}=a_{\mathrm{bg}}^2e^{2\zeta}\delta_{ij}$. Scalar lapse and shift potentials complete its <3+1 decomposition of spacetime>. The exact shift-square term is $h^{ij}N_iN_j$, independently of the parametrization of the scalar shift.
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