Linearized coupled equations and gauge identities

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.

Let $g_{ϵ}=g+ϵh+O({ϵ}^2)$ be a smooth variation of an oriented Lorentzian metric on a four-manifold, and let ${\omega }_{ϵ}=\omega +ϵ\tau +O({ϵ}^2)$ be a smooth variation of a $k$-form, $0\le k\le 4$. Define $h_a{}^b:=g^{bc}{h}_{ac}$, ${\mathrm{tr}}_{g}h:=g^{ab}{h}_{ab}$, and $(h^{\mathrm{♯}}\text{\hspace{0.17em}⁣}\cdot \omega {)}_{a_1\dots a_k}:={\sum }_{j=1}^k{h}_{a_j}{}^b{\omega }_{a_1\dots a_{j-1}b{a}_{j+1}\dots a_k},$ with $h^{\mathrm{♯}}\text{\hspace{0.17em}⁣}\cdot \omega :=0$ for $k=0$. Then ${\frac{d}{dϵ}\mid }_0{\star }_{g_{ϵ}}{\omega }_{ϵ}={\star }_g(\tau +\frac{1}{2}({\mathrm{tr}}_{g}h)\omega -h^{\mathrm{♯}}\text{\hspace{0.17em}⁣}\cdot \omega )\mathrm{.}$ In particular, for the geometric background $\mathcal{F}$ and perturbation $f$ of Linearized geometric electrovacuum operator and gauge equivalence, $\delta (\star \mathcal{F})={\star }_g(f+\frac{1}{2}({\mathrm{tr}}_{g}h)\mathcal{F}-h^{\mathrm{♯}}\text{\hspace{0.17em}⁣}\cdot \mathcal{F})\mathrm{.}$ If $(h,f)$ satisfies the linearized Maxwell equation, then $d\delta (\star \mathcal{F})=0$; this last condition is exactly the derivative of $d({\star }_{g_{ϵ}}{\mathcal{F}}_{ϵ})=0$, not a consequence of the variation formula alone.

Linearized geometric electrovacuum operator and gauge equivalence

Lorentzian exterior calculus and Hodge star

Let $(g,\mathcal{F})$ be a smooth exact geometric electrovacuum background on an open domain $U$ as in Linearized geometric electrovacuum operator and gauge equivalence. If $X$ is a smooth vector field on $U$ and ${\mathrm{\Phi }}_{ϵ}:V\to U$ is an orientation-preserving local flow it generates, where $V\subseteq U$ is open, ${\mathrm{\Phi }}_0$ is the inclusion, and ${\mathrm{\partial }}_{ϵ}{\mathrm{\Phi }}_{ϵ}=X\circ {\mathrm{\Phi }}_{ϵ}$ wherever defined, then the pure-gauge perturbation $({\dot{\mathrm{\Phi }}}_0^{\ast }g,{\dot{\mathrm{\Phi }}}_0^{\ast }\mathcal{F})$ solves the linearized electrovacuum system on every common flow domain. More generally, let $\lambda \mapsto (g_{\lambda },{\mathcal{F}}_{\lambda })$ be a smooth one-parameter family of exact backgrounds on a common domain, with every member satisfying the equations for one fixed value of $\mathrm{\Lambda }$. Its derivative at any parameter value solves the linearized system at that member. In the $\mathrm{\Lambda }=0$ sector, fixed-domain Reissner–Nordström and Kerr–Newman parameter families provide stationary parameter modes, meaning that their displayed metric and field components are independent of $x^0$, on exterior domains admitted by their verification results, when the parameters vary in an interval on which one such domain remains common. These conclusions do not classify the kernel of the linearized operator or determine which of its solutions arise as derivatives of exact solution families (integrable perturbations).

Linearized geometric electrovacuum operator and gauge equivalence

Full dyonic Reissner-Nordström electrovac verification

Fully verified electric Kerr-Newman electrovacuum

The map-level naturality used in step 1.1 is recorded by the following commutative square. It is a summary of the tensor, connection, Hodge and exterior-derivative calculations in that step; the proof above does not rely on the diagram.

\begin{tikzcd}
(g,\mathcal F) \arrow[r,"\Phi_\epsilon^*"] \arrow[d,"(E,d\mathcal F,d\star_g\mathcal F)"'] &
(\Phi_\epsilon^*g,\Phi_\epsilon^*\mathcal F) \arrow[d,"(E,d\mathcal F,d\star_g\mathcal F)"] \\
(0,0,0) \arrow[r,"\Phi_\epsilon^*"'] & (0,0,0)
\end{tikzcd}

Let $T>0$ and let $(g,\mathcal{F})$ be a smooth exact geometric electrovacuum background on the product collar $[-T,T]\times {\mathbb{T}}_{2\pi }^3$, where ${\mathbb{T}}_{2\pi }^3:=(\mathbb{R}\mathrm{/}2\pi \mathbb{Z}{)}^3$; exactness means ${\mathrm{Ric}}_{ab}[g]-\mathrm{\Lambda }{g}_{ab}=2Q_{ab}(g,\mathcal{F})$ and $d\mathcal{F}=d({\star }_g\mathcal{F})=0$, with $Q_{ab}(g,\mathcal{F}):={\mathcal{F}}_{ac}{\mathcal{F}}_b{}^c-\frac{1}{4}{g}_{ab}{\mathcal{F}}_{cd}{\mathcal{F}}^{cd}$. Here $x^0=t$ is a length coordinate, $n={\mathrm{\partial }}_t$ is the future unit normal, and $g=-d{t}^2+{\gamma }_{ij}(t,\theta )d{\theta }^{i}d{\theta }^j$ with $\gamma$ a smooth positive spatial metric. The fixed cosmological parameter $\mathrm{\Lambda }$ has units ${\mathrm{m}}^{-2}$. Fix the harmonic reference metric $\bar{g}$ to equal the background and define $H^a[g_{ϵ}]:=g_{ϵ}^{bc}({\mathrm{\Gamma }}^a{}_{bc}[g_{ϵ}]-{\bar{\mathrm{\Gamma }}}^a{}_{bc})$, with $H_b[g_{ϵ}]:=g_{ϵbc}{H}^c[g_{ϵ}]$. Thus the background gauge vector vanishes and the linearized reduced metric equation is the derivative of $\mathrm{Ric}[g_{ϵ}]-{\mathrm{\nabla }}_{(a}^{g_{ϵ}}{H}_{b)}[g_{ϵ}]-\mathrm{\Lambda }{g}_{ϵab}-2Q_{ab}(g_{ϵ},{\mathcal{F}}_{ϵ})=0$. Let $L_0>0$ be the fixed spatial length scale and, on the covering space ${\mathbb{R}}^3$, write $X^i=L_0{\theta }^i$. Use the global frame descended from the standard frame of ${\mathbb{R}}^3$ on ${\mathbb{T}}_{2\pi }^3$, denoted ${\mathrm{\partial }}_{{\theta }^i}$, and let $({\mathsf{e}}_1,{\mathsf{e}}_2,{\mathsf{e}}_3)$ be its Gram–Schmidt orthonormalization for $\gamma$, oriented so that $(n,{\mathsf{e}}_1,{\mathsf{e}}_2,{\mathsf{e}}_3)$ is positive for the spacetime orientation; write ${\epsilon }_{123}=1$. For the geometric-normalized two-form perturbation $f={\dot{\mathcal{F}}}_0$ from Linearized geometric electrovacuum operator and gauge equivalence, define $e_i:=f({\mathsf{e}}_i,n),b_i:=\frac{1}{2}{\epsilon }_i{}^{jk}f({\mathsf{e}}_j,{\mathsf{e}}_k),$ where indices on ${\epsilon }_{ijk}$ are raised with ${\delta }_{ij}$. The variation parameter is dimensionless, so these components have the geometric field-strength units of $\mathcal{F}$ (length-coordinate components have unit ${\mathrm{m}}^{-1}$). The first-order state is the $56$-component periodic array $u=(h_{ab},p_{ab},q_{abi},e_i,b_i)$, where $h$ is the metric perturbation, $p_{ab}={\bar{\mathrm{\nabla }}}_0{h}_{ab}$, and $q_{abi}={\bar{\mathrm{\nabla }}}_i{h}_{ab}$. For a periodic $56$-component array, define $\widehat{u}(\mathrm{\ell }):=(2\pi {)}^{-3}{\int }_{[0,2\pi {)}^3}u(\theta )e^{-i\mathrm{\ell }\cdot \theta }d\theta ,\mathrm{\parallel }u{\mathrm{\parallel }}_{H^k}^2:={\sum }_{\mathrm{\ell }\in {\mathbb{Z}}^3}(1+\mathrm{\mid }\mathrm{\ell }{\mathrm{\mid }}^2{)}^k\mathrm{\mid }\widehat{u}(\mathrm{\ell }){\mathrm{\mid }}^2\mathrm{.}$ Write $x=\theta$ in the system below and absorb the fixed factor $L_0^{-1}$ from ${\mathrm{\partial }}_{X^i}=L_0^{-1}{\mathrm{\partial }}_{{\theta }^i}$ into its spatial matrices. For a smooth linearized perturbation state $u:[-T,T]\times {\mathbb{T}}_{2\pi }^3\to {\mathbb{R}}^{56}$ and smooth source $s:[-T,T]\times {\mathbb{T}}_{2\pi }^3\to {\mathbb{R}}^{56}$, write the inhomogeneous first-order system in the chosen trivialization as $A^0(t,x){\mathrm{\partial }}_{t}u+{\sum }_{i=1}^3{A}^i(t,x){\mathrm{\partial }}_{i}u+C(t,x)u=s(t,x),$ where $A^0,A^1,A^2,A^3,C:[-T,T]\times {\mathbb{T}}_{2\pi }^3\to {\mathbb{R}}^{56\times 56}$ are smooth periodic matrix fields, each $A^i$ and $A^0$ is symmetric, and constants $0<a\le b<\mathrm{\infty }$ satisfy $aI\le A^0(t,x)\le bI$ throughout the slab. In the normal-flow chart, the harmonic metric reduction has time matrix $\mathrm{diag}(I,I,{\gamma }^{ij}I)$ and symmetric spatial matrices by [F1], while the six Maxwell field-strength variables have time matrix $I_6$ and symmetric spatial matrices by [F2]. When the Maxwell equations are linearized, derivatives of the perturbed lapse, frame and spatial metric occur through $p_{ab}$ and $q_{abi}$, which are components of the state and hence contribute to $Cu$, not to spatial derivatives of the state. The variation of the Maxwell stress in the metric equation is algebraic in $(h,f)$ by [F3]. Thus the full principal matrices are the direct sums of the metric and Maxwell principal matrices, and the remaining couplings are smooth zeroth-order terms. Compactness of the slab then supplies uniform positive bounds for $A^0$.

For every integer $k\ge 0$, define ${\mathcal{E}}_k(t):={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le k}{\int }_{{\mathbb{T}}_{2\pi }^3}({\mathrm{\partial }}^{\alpha }u{)}^T{A}^0(t,x){\mathrm{\partial }}^{\alpha }udx\mathrm{.}$ There is a constant $C_k$, depending only on $a,b,T$ and the coefficient derivatives through order $k+1$, such that $\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H^k}\le C_k{e}^{C_k\mathrm{\mid }t\mathrm{\mid }}(\mathrm{\parallel }u(0){\mathrm{\parallel }}_{H^k}+{\int }_{\min (0,t)}^{\max (0,t)}\mathrm{\parallel }s(\tau ){\mathrm{\parallel }}_{H^k}d\tau ),\mathrm{\mid }t\mathrm{\mid }\le T\mathrm{.}$ This is a finite-time conditional estimate, not an existence theorem or a global decay statement.

For the unforced linearized system, the Maxwell subsidiary constraints are the spatial restrictions of $df$ and $d\delta (\star \mathcal{F})$ to a slice. If the normal-slot linearized Maxwell evolution equations hold and both initial restrictions vanish, they remain zero on the slab. Define ${\dot{H}}^a:={\frac{d}{dϵ}\mid }_0{H}^a[g_{ϵ}]$ and ${\dot{H}}_b:=g_{bc}{\dot{H}}^c$ at the background; since $H^a[g]=0$, ${\dot{H}}_b$ is also the variation of $H_b[g_{ϵ}]$. If the linearized reduced Einstein and Maxwell equations hold and ${\dot{H}}_b={\mathrm{\partial }}_t{\dot{H}}_b=0$ initially, then ${\dot{H}}^a=0$ throughout the slab. These are the Maxwell Gauss and harmonic-gauge subsidiary identities; no decay of the physical perturbation is asserted.

Linearized geometric electrovacuum operator and gauge equivalence

The harmonic gauge vector, the reduced Einstein equation and the Einstein residual

Positive symmetric Maxwell field-strength reduction

The commutator of covariant derivatives on a vector field and its contracted component identity

Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction

The second Bianchi identity and the vanishing divergence of the Einstein tensor

The estimate is local in time and remains conditional on the smooth coefficient and positive-time-matrix hypotheses. It does not imply asymptotic decay on a black-hole exterior, uniform bounds as $T\to \mathrm{\infty }$, nonlinear stability, or integrability of every linearized solution. Giorgi's cited theorem has distinct small-charge, asymptotically flat and gauge-normalization assumptions and is not used as a supplier for any of those claims.