= Scalar gauge transformations with a background lapse
{title2=$\delta\widetilde g=\delta g-\mathcal L_\xi\bar g$}
For $N=\bar N(1+\Psi)$, $g_{0i}=a^2\partial_iB$ and spatial scalar convention $h_{ij}=a^2[(1-2\Phi)\delta_{ij}-2\partial_i\partial_jE]$, a passive displacement $(\xi^0,\partial^i\lambda)$ gives $\widetilde\Psi=\Psi-\dot\xi^0-(\dot{\bar N}/\bar N)\xi^0$, $\widetilde B=B+\bar N^2\xi^0/a^2-\dot\lambda$, $\widetilde\Phi=\Phi+(\dot a/a)\xi^0$ and $\widetilde E=E+\lambda$. These follow from subtracting the background metric Lie derivative.
Back to article page
