= Gauge-invariant scalar-vector-tensor perturbations of Minkowski spacetime
{title2=$\Phi=A-\dot B+\ddot E,\quad\Psi=C-\Delta E/3,\quad V_i=B_i-\dot E_i$}
= Gauge-invariant Minkowski perturbation
{synonym}
For signature $(+---)$, decompose a <metric perturbation> into $h_{00}=2A$, $h_{0i}=\partial_iB+B_i$ and $h_{ij}=2C\delta_{ij}+2(\partial_i\partial_j-\delta_{ij}\Delta/3)E+\partial_iE_j+\partial_jE_i+h_{ij}^{\rm TT}$, where the spatial vectors are divergence-free and $h^{\rm TT}$ is a <transverse-traceless tensor>. Under a passive coordinate shift $\xi^0=\tau$, $\xi^i=\partial_iL+L_i$, with $\partial_iL_i=0$, 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 $h^{\rm TT}$ are therefore gauge invariant. The decomposition is unique after appropriate spatial decay or mode restrictions; homogeneous and boundary modes require separate conventions.
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