= Flat linearized metric perturbations are locally pure gauge
{title2=$R^{(1)}[h]=0\ \Longrightarrow\ h_{ab}=2\partial_{(a}\xi_{b)}$}
On a <contractible> Cartesian patch of a flat background, zero <linearized Riemann curvature operator> says that the <connection 1 forms> $\Gamma^{(1)a}{}_{bc}\,dx^c$ are closed. The <Poincaré lemma> writes them as <derivatives> of a matrix-valued potential. Symmetry of the two lower <affine connection> indices then supplies a second potential, giving $\Gamma^{(1)a}{}_{bc}=\partial_b\partial_c\xi^a$. Linearized <metric compatibility> implies that $h_{ab}-2\partial_{(a}\xi_{b)}$ is constant. Its symmetric constant part is absorbed by an affine <vector field>. Thus the <metric perturbation> is locally a coordinate change. Global <topology> and imposed <boundary conditions> can obstruct the corresponding global gauge choice.
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