 Linearized coordinate gauge transformation

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.