Lemma: Formal naturality

For $F=F_{\kappa }$ as in Lemma lem:group and every formal scalar $\psi$, the following hold modulo ${\kappa }^3$:

Here $F^{\ast }$ on a contravariant tensor density is $(F^{\ast }\mathfrak{A}{)}^{\alpha \beta }(x)=\det DF(x)(DF(x{)}^{-1}{)}^{\alpha }{}_{\gamma }(DF(x{)}^{-1}{)}^{\beta }{}_{\delta }{\mathfrak{A}}^{\gamma \delta }(F(x))$.

Proof. For each fixed point $x$, choose a compact neighbourhood $K$ of $\{x\}\cup F_{\kappa }(x)$ for $|\kappa |$ small. On $K$ and for $|\kappa |$ small, $q(\kappa )=\eta +\kappa h+{\kappa }^{2}k$ is a nondegenerate Lorentzian metric, and the identities hold as genuine tensor identities for the genuine diffeomorphism $F_{\kappa }$: the density of a pulled-back metric is the pulled-back density, the pulled-back volume factor is the density pullback, the divergence of a vector density and the contraction of a density with a covector are natural. Both sides are smooth in $\kappa$ and their Taylor coefficients through order two are polynomial differential expressions in $(h,k,\xi ,\zeta ,{\psi }_j)$; equality for all small $\kappa$ gives equality of the coefficients.