Pure-gauge and exact-family perturbations solve the linearized electrovacuum system

Let ${\mathrm{\Phi }}_{ϵ}:V\to W$ be an orientation-preserving local flow map. Define its pulled-back connection by $({\mathrm{\Phi }}_{ϵ}^{\ast }\mathrm{\nabla }{)}_{X}Y:={\mathrm{\Phi }}_{ϵ\ast }^{-1}({\mathrm{\nabla }}_{{\mathrm{\Phi }}_{ϵ\ast }X}{\mathrm{\Phi }}_{ϵ\ast }Y)$. The coordinate Lie-bracket identity in [F3] and the chain rule give $[{\mathrm{\Phi }}_{ϵ\ast }X,{\mathrm{\Phi }}_{ϵ\ast }Y]={\mathrm{\Phi }}_{ϵ\ast }[X,Y]$, so this connection is torsion-free. For vector fields $X,Y,Z$, differentiating $({\mathrm{\Phi }}_{ϵ}^{\ast }g)(Y,Z)=g({\mathrm{\Phi }}_{ϵ\ast }Y,{\mathrm{\Phi }}_{ϵ\ast }Z)\circ {\mathrm{\Phi }}_{ϵ}$ and using $\mathrm{\nabla }g=0$ shows it is compatible with ${\mathrm{\Phi }}_{ϵ}^{\ast }g$. Uniqueness in [F3] identifies it with ${\mathrm{\nabla }}^{{\mathrm{\Phi }}_{ϵ}^{\ast }g}$. Applying this definition twice in the curvature formula and using the same bracket identity gives $R[{\mathrm{\Phi }}_{ϵ}^{\ast }g](X,Y)Z={\mathrm{\Phi }}_{ϵ\ast }^{-1}(R[g]({\mathrm{\Phi }}_{ϵ\ast }X,{\mathrm{\Phi }}_{ϵ\ast }Y){\mathrm{\Phi }}_{ϵ\ast }Z)$; the trace defining Ricci is unchanged under conjugation by $D{\mathrm{\Phi }}_{ϵ}$, hence $\mathrm{Ric}[{\mathrm{\Phi }}_{ϵ}^{\ast }g]={\mathrm{\Phi }}_{ϵ}^{\ast }\mathrm{Ric}[g]$. The formula for $Q$ uses only tensor products, the metric inverse and contractions; $({\mathrm{\Phi }}_{ϵ}^{\ast }g{)}^{-1}({\mathrm{\Phi }}_{ϵ}^{\ast }\alpha ,{\mathrm{\Phi }}_{ϵ}^{\ast }\beta )={\mathrm{\Phi }}_{ϵ}^{\ast }(g^{-1}(\alpha ,\beta ))$ by the inverse linear map, so $Q({\mathrm{\Phi }}_{ϵ}^{\ast }g,{\mathrm{\Phi }}_{ϵ}^{\ast }\mathcal{F})={\mathrm{\Phi }}_{ϵ}^{\ast }Q(g,\mathcal{F})$. In the defining Hodge identity, pullback commutes with alternating tensor evaluation and the inverse-metric pairing; it also carries ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ to ${\mathrm{v}\mathrm{o}\mathrm{l}}_{{\mathrm{\Phi }}_{ϵ}^{\ast }g}$ because the flow preserves orientation and the volume density transforms by the Jacobian determinant. Uniqueness of the Hodge map therefore gives ${\star }_{{\mathrm{\Phi }}_{ϵ}^{\ast }g}{\mathrm{\Phi }}_{ϵ}^{\ast }\mathcal{F}={\mathrm{\Phi }}_{ϵ}^{\ast }({\star }_g\mathcal{F})$. Finally, write a $k$-form as $\omega ={\sum }_I{\omega }_{I}d{x}^I$. By [F4], $d\omega ={\sum }_{I}d{\omega }_I\wedge d{x}^I$. The chain rule gives the terms where a derivative hits ${\omega }_I$; terms where it hits a coordinate derivative of ${\mathrm{\Phi }}_{ϵ}$ cancel in pairs because second partial derivatives commute while the exterior product is alternating. Hence $d{\mathrm{\Phi }}_{ϵ}^{\ast }\omega ={\mathrm{\Phi }}_{ϵ}^{\ast }d\omega$. Thus every exact residual of the pulled-back pair is the pullback of the corresponding zero residual.

For a smooth exact family $(g_{\lambda },{\mathcal{F}}_{\lambda })$ on a common open set, each of its three residuals in [F1] is identically zero as a smooth function of $(\lambda ,x)$. Differentiating in $\lambda$ commutes with the finite coordinate derivatives and contractions in those residuals, so their derivatives at any ${\lambda }_0$ are exactly the linearized residuals of $({\mathrm{\partial }}_{\lambda }g,{\mathrm{\partial }}_{\lambda }\mathcal{F}){\mathrm{\mid }}_{{\lambda }_0}$ and vanish.

Differentiating the residual identities of step 1.1 at $ϵ=0$ gives zero linearized metric and Maxwell residuals; in the Hodge residual the varied tensor is the explicit $\delta (\star \mathcal{F})$ of [F5]. By the definition of the Lie derivative through the local flow, the derivatives of the pulled-back fields are precisely $({\dot{\mathrm{\Phi }}}_0^{\ast }g,{\dot{\mathrm{\Phi }}}_0^{\ast }\mathcal{F})$, so this is the asserted pure-gauge solution.

The RN and KN data of [F6] are smooth in their displayed parameters on any fixed exterior domain satisfying the relevant strict chart inequalities. Restricting the parameter interval around the chosen values if necessary preserves those inequalities on the common domain. Both verification results establish the exact equations with $\mathrm{\Lambda }=0$, so step 1.2 applies to each such parameter derivative in that fixed sector. The parameter derivatives are stationary because the displayed metric and field coefficients are independent of $x^0$. No inference is made that an arbitrary linearized solution lies on one of these families.