Gauge-invariant scalar-vector-tensor perturbations of Minkowski spacetime ($\Phi =A-\dot{B}+\ddot{E},\Psi =C-\Delta E/3,V_i=B_i-{\dot{E}}_i$)

For signature $(+---)$, decompose a metric perturbation into $h_{00}=2A$, $h_{0i}={\partial }_{i}B+B_i$ and $h_{ij}=2C{\delta }_{ij}+2({\partial }_i{\partial }_j-{\delta }_{ij}\Delta /3)E+{\partial }_i{E}_j+{\partial }_j{E}_i+h_{ij}^{\mathrm{TT}}$, where the spatial vectors are divergence-free and $h^{\mathrm{TT}}$ is a transverse-traceless tensor. Under a passive coordinate shift ${\xi }^0=\tau$, ${\xi }^i={\partial }_{i}L+L_i$, with ${\partial }_i{L}_i=0$, the changes are $\delta A=-\dot{\tau }$, $\delta B=\dot{L}-\tau$, $\delta C=\Delta L/3$, $\delta E=L$, $\delta B_i={\dot{L}}_i$, and $\delta E_i=L_i$. The displayed combinations and $h^{\mathrm{TT}}$ are therefore gauge invariant. The decomposition is unique after appropriate spatial decay or mode restrictions; homogeneous and boundary modes require separate conventions.

With the reversed convention in curvature-sign convention in Killing derivative identities, choose the representative $h_{00}=2\Phi$, $h_{0i}=V_i$, $h_{ij}=2\Psi {\delta }_{ij}+h_{ij}^{\mathrm{TT}}$. The linearized Einstein components are $G_{00}^{(1)}=-2\Delta \Psi$, $G_{0i}^{(1)}=-2{\partial }_i\dot{\Psi }-\Delta V_i/2$ and $G_{ij}^{(1)}=({\partial }_i{\partial }_j-{\delta }_{ij}\Delta )(\Phi -\Psi )-2{\delta }_{ij}\ddot{\Psi }-({\partial }_i{\dot{V}}_j+{\partial }_j{\dot{V}}_i)/2+\square h_{ij}^{\mathrm{TT}}/2$. 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.