Theorem: Coordinate intertwiner through second order

For every $(h,k)$ and every $F\in {\mathcal{G}}_U$,

Equivalently, with $h^{\prime },k^{\prime }$ as in (eq:metricpull),

Proof. By Lemma lem:plateaucoord, ${\mathsf{D}}_{\epsilon }[F^{\ast }q](F^{\ast }{\mathsf{X}}_q)=F_{\mathrm{dens}}^{\ast }({\mathsf{D}}_{\epsilon }[q]{\mathsf{X}}_q)=0(\mathrm{mod}{\kappa }^3)$. Since $F$ is the identity on a neighbourhood of $\{t\le \epsilon \}$, $F^{\ast }{\mathsf{X}}_q={\mathsf{X}}_q\circ F$ has the Cauchy data of ${\mathsf{X}}_q$, namely $(x^{\mu },{{\delta }^{\mu }}_0)$. Uniqueness in Proposition prop:coordeq gives (eq:intertwiner). Expanding with (eq:scalarpull),

and $X_{\xi }{x}^{\mu }={\xi }^{\mu }$, ${X_{\xi }}^2{x}^{\mu }={\xi }^{\nu }{\partial }_{\nu }{\xi }^{\mu }$.