Lemma: Exact second-order covariance identity

For every smooth vector field $\xi$ and smooth symmetric $u$,

where ${\mathcal{L}}_{\xi }$ acts on $G_{\mathrm{lin}}[u]$ as on a contravariant tensor density and on $u$ as on a covariant tensor.

Proof. Both sides of (eq:cov2) are bidifferential expressions in $(\xi ,u)$ with constant coefficients, so the identity at a point $x$ involves only the germs of $\xi$ and $u$ at $x$; it suffices to prove it for compactly supported $\xi$. Let $F_s$ be the flow of such a $\xi$; $F_s$ is the identity outside a compact set, $F_s^{\ast }\eta =\eta +sK\xi +O(s^2)$, and $\frac{d}{ds}{F}_s^{\ast }T{|}_{s=0}={\mathcal{L}}_{\xi }T$ for every tensor density. By Lemma lem:natural with $q=\eta +\lambda u$,

Expand both sides by xe:einstein with $\kappa h=sK\xi +\lambda u+O(s^2)$, and ${\kappa }^{2}k$ collecting the $O(s^2)$ remainder of $F_s^{\ast }\eta$ and the $s\lambda$ term:

The $s{\lambda }^0$ coefficient gives $G_{\mathrm{lin}}[K\xi ]=0$. The $s\lambda$ coefficient gives $G_{\mathrm{lin}}[{\mathcal{L}}_{\xi }u]+2{\mathcal{G}}_2(K\xi ,u)={\mathcal{L}}_{\xi }{G}_{\mathrm{lin}}[u]$.