Linear metric gauge-transformation law
= Linear metric gauge-transformation law
{title2=$\widetilde{\delta g}=\delta g-\mathcal L_\xi\bar g$}
For the passive convention $\widetilde x=x+\xi$, 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.