= Vacuum constraints for gauge-invariant Minkowski perturbations
{title2=$\Delta\Psi=0,\quad\Delta V_i=0,\quad\Delta\Phi=0,\quad\Box h_{ij}^{\rm TT}=0$}
With the reversed convention in <curvature-sign convention in Killing derivative identities>, choose the representative $h_{00}=2\Phi$, $h_{0i}=V_i$, $h_{ij}=2\Psi\delta_{ij}+h_{ij}^{\rm TT}$. The linearized Einstein components are $G_{00}^{(1)}=-2\Delta\Psi$, $G_{0i}^{(1)}=-2\partial_i\dot\Psi-\Delta V_i/2$ and $G_{ij}^{(1)}=(\partial_i\partial_j-\delta_{ij}\Delta)(\Phi-\Psi)-2\delta_{ij}\ddot\Psi-(\partial_i\dot V_j+\partial_j\dot V_i)/2+\Box h_{ij}^{\rm TT}/2$. On all space with regular decaying fields, or for nonzero spatial Fourier modes, the vacuum constraints remove the scalar and vector parts, leaving two radiative <gravitational wave polarizations>. Singular Coulomb fields, spatial boundaries and homogeneous modes are not covered by that removal argument.
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