Linear metric gauge-transformation law ($\widetilde{\delta g}=\delta g-{\mathcal{L}}_{\xi }\bar{g}$)

For the passive convention $\tilde{x}=x+\xi$, expanding the tensor coordinate-transformation law around a fixed background gives this Lie derivative formula. The background time shift and Jacobian terms must both be retained. On an FLRW metric, the scalar lapse, shift and spatial potentials transform under two scalar coordinate functions.