Lie derivatives and types

${\mathcal{L}}_{\xi }$ acts on covariant symmetric two-tensors by $({\mathcal{L}}_{\xi }u{)}_{\mu \nu }={\xi }^{\lambda }{\partial }_{\lambda }{u}_{\mu \nu }+u_{\lambda \nu }{\partial }_{\mu }{\xi }^{\lambda }+u_{\mu \lambda }{\partial }_{\nu }{\xi }^{\lambda }$ and on contravariant symmetric tensor densities of weight one by ${\mathcal{L}}_{\xi }{T}^{\mu \nu }={\partial }_{\lambda }({\xi }^{\lambda }{T}^{\mu \nu })-T^{\lambda \nu }{\partial }_{\lambda }{\xi }^{\mu }-T^{\mu \lambda }{\partial }_{\lambda }{\xi }^{\nu }$. For the flow $F_s$ of $\xi$, $\frac{d}{ds}{F}_s^{\ast }T=F_s^{\ast }{\mathcal{L}}_{\xi }T$ for every tensor density $T$. Metrics and fluctuations are covariant tensors; Euler derivatives with respect to them, in particular $\mathcal{E}$, $G_{\mathrm{lin}}$, ${\mathcal{G}}_2$, $P_{\mathrm{gr}}u$, and the gauge-fixing derivative $\mathcal{F}$ below, are contravariant tensor densities. When ${\mathcal{L}}_{\xi }$ is written on either side of such an operator, it acts in the type of its argument. The two-jet law of xe:ward is the expansion $F^{\ast }(\eta +\kappa h_{\mathrm{cl}}+{\kappa }^2{k}_{\mathrm{cl}})=\eta +\kappa h_{\mathrm{cl}}^{\prime }+{\kappa }^2{k}_{\mathrm{cl}}^{\prime }+O({\kappa }^3)$ for $F=\exp (\kappa \xi +{\kappa }^2\zeta )$.