= Linearized coordinate gauge transformation
{title2=$h_{\mu\nu}\mapsto h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$}
An infinitesimal coordinate change generated by $\xi$ changes a metric perturbation about <Minkowski spacetime> by $h_{\mu\nu}\mapsto h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$, in the passive sign convention. For a passive coordinate change $x^\mu\mapsto x^\mu+\xi^\mu$, the first-order change is minus the <Lie derivative of a tensor field> of the background metric, equivalently the pullback by the inverse flow. The <gauge invariance of the linearized Riemann tensor> follows because the added terms have commuting third derivatives, which cancel in the curvature antisymmetrizations. Gauge choices alter the metric components while preserving linearized tidal effects.
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