Linearized geometric electrovacuum operator and gauge equivalence

Assume the countable-choice background ${\mathrm{A}\mathrm{C}}_{\omega }$ carried by the stated Lorentzian geometry suppliers. Let $(M,g,N,o)$ be an oriented, time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$, with metric-selected connection and orientation as in the geometric electrovac convention. In a length-coordinate chart, metric components are dimensionless and a geometric-normalized real two-form has components of unit ${\mathrm{m}}^{-1}$.

For a smooth real two-form $\mathcal{F}$, put $Q_{ab}(g,\mathcal{F}):={\mathcal{F}}_{ac}{\mathcal{F}}_b{}^c-\frac{1}{4}{g}_{ab}{\mathcal{F}}_{cd}{\mathcal{F}}^{cd},$ where all raised indices use $g^{-1}$. Its length-coordinate components have unit ${\mathrm{m}}^{-2}$. Fix $\mathrm{\Lambda }\in \mathbb{R}$ with unit ${\mathrm{m}}^{-2}$. An exact geometric electrovacuum background on $U\subseteq M$ is a smooth pair $(g,\mathcal{F})$ on the same open set satisfying ${\mathrm{Ric}}_{ab}[g]-\mathrm{\Lambda }{g}_{ab}=2Q_{ab}(g,\mathcal{F}),d\mathcal{F}=0,d({\star }_g\mathcal{F})=0.$ For $\mathrm{\Lambda }=0$ this is the geometric normalization of Geometric electrovac equations and flux normalization. The symbol $\mathcal{F}$ here is geometric-normalized; it is not the SI-normalized field tensor used in Electrovacuum tensor Cauchy system.

Fix such a background on $U$. A linearized perturbation is a pair $(h,f)$ consisting of a smooth symmetric covariant two-tensor $h$ and a smooth real two-form $f$ on $U$. Its linearized metric residual is $\delta E_{(g,\mathcal{F})}[h,f]:={\frac{d}{dϵ}\mid }_{ϵ=0}(\mathrm{Ric}[g_{ϵ}]-\mathrm{\Lambda }{g}_{ϵ}-2Q(g_{ϵ},{\mathcal{F}}_{ϵ})),$ and its linearized Maxwell residuals are $df,d\delta (\star \mathcal{F}),\delta (\star \mathcal{F}):={\frac{d}{dϵ}\mid }_{ϵ=0}{\star }_{g_{ϵ}}{\mathcal{F}}_{ϵ}\mathrm{.}$ Here $g_{ϵ}=g+ϵh+O({ϵ}^2)$ and ${\mathcal{F}}_{ϵ}=\mathcal{F}+ϵf+O({ϵ}^2)$ are smooth local families on the fixed domain, with dimensionless parameter $ϵ$. The derivatives are the coordinatewise derivatives of these smooth finite-jet operators; the chain rule makes them depend only on $(h,f)$, not on the higher-order parts of the chosen families. The linearized electrovacuum system is the homogeneous system $\delta E_{(g,\mathcal{F})}[h,f]=0$, $df=0$, and $d\delta (\star \mathcal{F})=0$.

For a smooth vector field $X$, let ${\mathrm{\Phi }}_{ϵ}$ be its local flow and define the pure gauge perturbation by $(h,f)={\frac{d}{dϵ}\mid }_0({\mathrm{\Phi }}_{ϵ}^{\ast }g,{\mathrm{\Phi }}_{ϵ}^{\ast }\mathcal{F})$. A parameter perturbation is the derivative of a smooth one-parameter family of exact geometric electrovacuum backgrounds on one common open domain. Neither definition asserts that every solution of the linearized system is pure gauge or is a parameter perturbation.

For a local diffeomorphism $\mathrm{\Phi }:V\to W$ and a covariant $r$-tensor $T$ on $W$, its pullback is $({\mathrm{\Phi }}^{\ast }T{)}_p(v_1,\dots ,v_r):=T_{\mathrm{\Phi }(p)}(D{\mathrm{\Phi }}_p{v}_1,\dots ,D{\mathrm{\Phi }}_p{v}_r)$; this fixes the pullback convention in the pure-gauge definition.

If a potential perturbation $a\in {\mathrm{\Omega }}^1(V)$ is chosen on an open set $V\subseteq U$ with $f=da$, then $a$ and $a+d\chi$ represent the same field-strength perturbation for every smooth scalar $\chi$ on $V$. No global exactness of $f$ or global potential is assumed.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Geometric electrovac equations and flux normalization

Electrovacuum tensor Cauchy system

Lorentzian exterior calculus and Hodge star

The stress variation displays the first-order coupling. Writing $h^{ab}=g^{ac}{g}^{bd}{h}_{cd}$, the identity $\delta g^{ab}=-h^{ab}$ and the product rule give $\delta Q_{ab}=f_{ac}{\mathcal{F}}_b{}^c+{\mathcal{F}}_a{}^c{f}_{bc}-{\mathcal{F}}_{ac}{h}^{cd}{\mathcal{F}}_{bd}-\frac{1}{4}{h}_{ab}{\mathcal{F}}_{cd}{\mathcal{F}}^{cd}-\frac{1}{4}{g}_{ab}(2{\mathcal{F}}^{cd}{f}_{cd}-2h^{ce}{\mathcal{F}}_{cd}{\mathcal{F}}_e{}^d)\mathrm{.}$ Thus a nonzero background field contributes terms bilinear in $\mathcal{F}$ and $f$, and terms linear in $h$ and quadratic in $\mathcal{F}$; the metric and electromagnetic perturbations are coupled at first order. At $\mathcal{F}=0$, every term in $\delta Q$ vanishes, so the electromagnetic stress supplies no first-order source in the metric equation. These are algebraic consequences of the displayed definition of $Q$.

The definition and the finite-slab estimate below do not assert that a linearized solution integrates to a nonlinear family. On backgrounds with Killing fields and compact Cauchy slices, second-order linearization-stability conditions are a separate integrability obligation. No asymptotic decay, nonlinear stability, or global-in-time conclusion is included.