Effective nonlinear electromagnetism and semiclassical consistency

If $K=\mathrm{\varnothing }$, the form is zero and the stated zero value needs no charts. Otherwise consider the family of all pairs of concentric coordinate balls $B_i\subset V_i$ in positive charts, with $\overline{B_i}$ compact and contained in $V_i$. Every point of $K$ lies in one such inner ball: openness of a chart image permits two sufficiently small concentric Euclidean balls, whose closed inner ball is compact by [F1]. This forms an open cover without selecting a chart separately at every point. Compactness of $K$ gives finitely many such pairs whose inner balls cover $K$. Apply [F1]'s bump construction for each of these finitely many compact inner balls inside its chart image, pull back and extend by zero, to get smooth ${\rho }_i:M\to [0,1]$ with compact support inside $V_i$ and ${\rho }_i=1$ on $B_i$. Only finitely many choices are made. Set $B={\sum }_i{\rho }_i$, so $B\ge 1$ on $K$. Let $u(t)=e^{-1\mathrm{/}t}$ for $t>0$ and $u(t)=0$ for $t\le 0$, and define $q(t)=u(t-1\mathrm{/}4)\mathrm{/}(u(t-1\mathrm{/}4)+u(1\mathrm{/}2-t))$. Its denominator is positive for every real $t$; by [F6], $u$ is smooth and all its derivatives vanish at zero, so $q$ is smooth, lies in $[0,1]$, is zero for $t\le 1\mathrm{/}4$ and is one for $t\ge 1\mathrm{/}2$. Put $\eta =q(B)$ and define ${\chi }_i=\eta {\rho }_i\mathrm{/}B$ on $\{B>0\}$, extending by zero elsewhere. Since $\eta =0$ for $B\le 1\mathrm{/}4$, the extension is smooth; its support is a closed subset of the compact support of ${\rho }_i$ and remains inside $V_i$. Their sum is $\eta$, hence is one on the open neighborhood $\{B>1\mathrm{/}2\}$ of $K$. These are the required finite cutoffs.

The support of ${\chi }_i\omega$ is a closed subset of $K\cap \mathrm{supp}{\chi }_i$, hence compact and contained in its chart. Its coordinate coefficient is continuous there. The coordinate support is compact inside the open chart image, so the zero extension is continuous at artificial chart edges: outside that compact set a neighborhood has identically zero coefficient. On a closed rectangle containing this compact coordinate support the extension is therefore integrable by [F2]. The same fact and [F2]'s bounding-rectangle independence make each summand in the Definition a finite well-defined real number. The rectangle estimate in [F4] and the finite sum give the displayed bound for $\mathrm{\mid }{\mathcal{I}}_M(\omega )\mathrm{\mid }$. Applied to coefficient differences with fixed cutoffs and rectangles, this bound proves the stated uniform-limit property.

Suppose a continuous top form $\zeta$ has compact support contained in two positive charts $(U,\varphi )$ and $(V,\psi )$. On their overlap put $g=\psi \circ {\varphi }^{-1}$; this is a smooth injective diffeomorphism between open Euclidean sets, and $\det Dg>0$ because both charts are positive. If $f_{\psi }$ is its target coefficient, the defining tensor transformation law gives $f_{\varphi }=(f_{\psi }\circ g)\det Dg$. The zero-extended target coefficient is continuous with compact support contained in the target overlap, by the argument of step 2.1. All hypotheses of [F3] hold, and the transformed coefficient extended by zero is exactly the source coefficient extended by zero. Consequently their Euclidean integrals agree. This proves the required chart comparison for continuous coefficients directly, without applying the smooth-only claim in [F5] to $\zeta$.

Let $({\chi }_i,U_i)$ and $({\tau }_j,V_j)$ be two finite cutoff systems for $K$. Every ${\chi }_i{\tau }_j\omega$ has compact support contained in $U_i\cap V_j$. Near $K$ each cutoff sum is one, and outside $K$ the form vanishes, so ${\chi }_i\omega ={\sum }_j{\chi }_i{\tau }_j\omega$ and ${\tau }_j\omega ={\sum }_i{\chi }_i{\tau }_j\omega$. All sums are finite. Apply linearity [F4] and the chart comparison of step 3.1 to each product term. Both proposed integrals equal the same finite double sum, proving independence of cutoff systems and charts. Step 2.1 already proves bounding-rectangle independence. For two compactly supported forms take cutoffs near the union of their supports, which is compact; [F4] then proves linearity in the forms and scalar coefficients.

If an open $V\subseteq M$ contains $K$, the construction of step 1.1 may use only coordinate balls with compact closure inside $V$. The same finite coefficient integrals compute ${\mathcal{I}}_V(\omega {\mathrm{\mid }}_V)$ and ${\mathcal{I}}_M(\omega )$, so step 4.1 proves locality. Now suppose $\omega$ is smooth. Choose a chart-partition system $({\tau }_j,U_j)$ permitted by the smooth integral in [F5]. Only finitely many ${\tau }_j\omega$ are nonzero near $K$, as part of that integral contract. Comparing this system with the finite cutoffs ${\chi }_i$ uses the same product identities as in step 4.1: each ${\chi }_i{\tau }_j\omega$ is supported compactly in the two chart domains, and step 3.1 identifies its two coordinate integrals. The finite double sum consequently equals both ${\mathcal{I}}_M(\omega )$ and the existing smooth chart-partition integral on $M$. Countable Choice enters this comparison only through the explicit convention of [F5]; the finite construction and comparison in steps 1.1–4.1 did not assume full Choice. Thus the continuous integral extends the exact previously defined smooth integral, with the stated locality and linearity.

Each contraction defining $X$ and $Y$ is a scalar under changes of coordinates, because $g^{-1}$ and ${\star }_g$ are tensorial. Since $F$, $g$, and ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ are smooth, $X,Y$ are smooth; $L_X(X,Y)$ and $L_Y(X,Y)$ are $C^1$. Therefore $H$ is a globally defined $C^1$ section of ${\mathrm{\Lambda }}^{2}TM$, and the displayed action density is a globally defined $C^1$ top form.

For a permitted compactly supported variation $a$, [F3] gives $F_{ϵ}-F=ϵda$, which vanishes off $\mathrm{supp}a$. Hence the difference of the two $L$ densities is compactly supported. Its coefficient is at least continuous, so the integral ${\mathcal{I}}_M$ in [F4] is finite without requiring a smooth density. On finitely many fixed chart rectangles near that support, the chain rule gives continuous parameter derivatives; their uniform convergence of difference quotients on compact rectangles allows passage to the integrals by the rectangle bound in [F4]. Hence its first variation is well-defined for $L\in C^2$, with no stronger regularity assumption. Thus the Definition specifies a relative action and stationarity predicate without assuming the unvaried action is finite or that a stationary connection exists.

At fixed $g$, variation of the contractions gives $\delta X=-\frac{1}{2}{F}^{ab}\delta F_{ab}$. By [F2], $F_{ab}({\star }_g\delta F{)}^{ab}=({\star }_{g}F{)}^{ab}\delta F_{ab}$: in an oriented orthonormal coframe this follows from the component formula for ${\star }_g$ and the positive sign of exchanging the two antisymmetric index pairs of the volume tensor. Hence $\delta Y=-\frac{1}{2}({\star }_{g}F{)}^{ab}\delta F_{ab}$. The chain rule gives $\delta L=-\frac{1}{2}{H}^{ab}\delta F_{ab}$. For a connection variation $a_b$, [F1] gives $\delta F_{ab}=2{\mathrm{\nabla }}_{[a}{a}_{b]}$; antisymmetry of $H$ reduces the varied density to $-H^{ab}{\mathrm{\nabla }}_a{a}_b$.

The connection-curvature construction in [F1] is locally $F=d{A}_i$ in every bundle chart. The exterior derivative satisfies $d^2=0$, so $dF=d^2{A}_i=0$ in each chart and hence globally. This equation is a curvature identity, not a second consequence of stationarity.

Put $K:=\mathrm{supp}a$ and $W^a:=H^{ab}{a}_b$. Then $W$ is a compactly supported $C^1$ vector field, not necessarily smooth. If $K$ is empty the variation is zero, so assume $K\mathrm{\ne }\mathrm{\varnothing }$. Cover $K$ by smaller relatively compact coordinate balls $U_i$ and choose finitely many larger positively oriented coordinate balls $V_i$ with $\overline{U_i}\subset V_i$; compactness gives a finite subcover. In each chart choose a nonnegative smooth bump ${\rho }_i$ supported compactly in $V_i$ and equal to $1$ on $U_i$, which is obtained from the standard smooth Euclidean bump by a radial rescaling in the chart. For $B:={\sum }_i{\rho }_i$, one has $B\ge 1$ on $K$. Choose a smooth function $q:\mathbb{R}\to [0,1]$ with $q(s)=0$ for $s\le 1\mathrm{/}4$ and $q(s)=1$ for $s\ge 1\mathrm{/}2$ (such a step is obtained from $u(s)=e^{-1\mathrm{/}s}$ for $s>0$ and $u(s)=0$ for $s\le 0$); set $\eta :=q\circ B$. Then $\eta =1$ on the open neighborhood $V:=\{B>1\mathrm{/}2\}$ of $K$, its support is compact and contained in $\{B>0\}$, and ${\chi }_i:=\eta {\rho }_i\mathrm{/}B$ on $\{B>0\}$, extended by zero elsewhere, are smooth, compactly supported in $V_i$, and satisfy ${\sum }_i{\chi }_i=1$ on $V$. This construction of local cutoffs uses compactness only; the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ condition for the global integral is stated in [F4]. In a positive chart put $w:=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}>0$. Contracting the Christoffel formula gives ${\mathrm{\Gamma }}^a{}_{ac}=\frac{1}{2}{g}^{ab}{\mathrm{\partial }}_c{g}_{ab}={\mathrm{\partial }}_c\log w$, where the last equality is the determinant derivative formula. Hence, for $W_i:={\chi }_{i}W$, ${\mathrm{\nabla }}_a{W}_i^a={\mathrm{\partial }}_a{W}_i^a+{\mathrm{\Gamma }}^a{}_{ac}{W}_i^c=w^{-1}{\mathrm{\partial }}_a(w{W}_i^a)$. Each $w{W}_i^a$ is $C^1$ and compactly supported inside its chart, so extend it by zero to a containing coordinate rectangle. Its coordinate derivative is continuous and each one-variable section of $w{W}_i^a$ is continuously differentiable, so [F3] applies: Fubini reduces the integral of each derivative to an interval integral, and the fundamental theorem makes that integral zero because the section vanishes near the interval endpoints. Thus ${\int }_{V_i}{\mathrm{\nabla }}_a{W}_i^a{\mathrm{v}\mathrm{o}\mathrm{l}}_g=0$. Since ${\sum }_i{\chi }_i=1$ on the open neighborhood $V$ of $K$ and $W$ vanishes outside $K$, ${\sum }_i{\mathrm{\nabla }}_a{W}_i^a={\mathrm{\nabla }}_a{W}^a$ everywhere. The finite cutoffs near $K$ compute the continuous compact-support integral by [F4], so ${\int }_M{\mathrm{\nabla }}_a{W}^a{\mathrm{v}\mathrm{o}\mathrm{l}}_g={\sum }_i{\int }_{V_i}{\mathrm{\nabla }}_a{W}_i^a{\mathrm{v}\mathrm{o}\mathrm{l}}_g=0$. The relative-action density difference is $C^1$ in the parameter and supported in $K$. Multiplying it by the same ${\chi }_i$ gives finitely many compactly supported chart densities; on a fixed coordinate rectangle their parameter derivatives are continuous and hence uniformly continuous on a compact parameter interval. [F3] makes these derivatives integrable, and the mean-value theorem shows their difference quotients converge uniformly to the derivatives at $ϵ=0$. The Darboux definition gives $\mathrm{\mid }{\int }_{Q}f\mathrm{\mid }\le \mathrm{vol}(Q){\sup }_Q\mathrm{\mid }f\mathrm{\mid }$, so these uniform limits pass to the integrals and the finite sum differentiates termwise. Integrating the product identity ${\mathrm{\nabla }}_a{W}^a=({\mathrm{\nabla }}_a{H}^{ab})a_b+H^{ab}{\mathrm{\nabla }}_a{a}_b$ now yields $\delta S_L[a]={\int }_M({\mathrm{\nabla }}_a{H}^{ab})a_b{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$ No boundary term at infinity is used because the varied integrand has compact support. The finite-cover integration-by-parts argument applies to $C^1$ $W$ and does not use a global partition of unity; the overall compact-action convention still has the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis in [F4].

Put $P^b={\mathrm{\nabla }}_a{H}^{ab}$, which is continuous by [A1]. If $P^{b_0}$ were nonzero at a point in a chart, continuity would give a closed nondegenerate coordinate rectangle $Q$ around that point, with $Q$ contained in the chart, on which $P^{b_0}$ has one strict sign. Choose a nonnegative smooth bump $\varphi$ compactly supported in the interior of $Q$ and positive near the point, set $a_{b_0}=\varphi$ and all other coordinate components to zero, and extend by zero. Such a bump is obtained as a product of one-dimensional smooth bumps, each built from $u(s)=e^{-1\mathrm{/}s}$ for $s>0$ and $u(s)=0$ for $s\le 0$. If units are attached, multiply this test one-form by a fixed positive potential-component unit so it has the units required by [F1]; this does not change its sign. After multiplying by the sign of $P^{b_0}$, the coefficient $h:=P^{b_0}\varphi w$ is nonnegative on $Q$ and bounded below by some $\eta >0$ on a smaller nondegenerate rectangle $R\subseteq Q$ around the point. By [F3], a grid of $Q$ with $R$ as one cell has a strictly positive lower sum, so the local integral over $Q$ is positive. Its support lies in $Q$, and [F4] identifies this with the global integral in step 2.1. This is nonzero, contradicting stationarity. Thus $P^b=0$ in every chart. Conversely, $P=0$ makes the variation in step 2.1 zero for every allowed $a$.

For the displayed polynomial, $L_X=1+2\alpha X$ and $L_Y=2\beta Y$. Substitution into the definition of $H$ in [F1] gives the stated constitutive tensor. The proof derives no characteristic polynomial or principal symbol, so no hyperbolicity or causal-cone conclusion is available.

For each chosen state, $j_{\mathcal{R}}$ is a vector field, so ${\mathrm{\nabla }}_a{j}_{\mathcal{R}}^a$ is a scalar field. The divergence ${\mathrm{\nabla }}^a{T}_{\mathcal{R},ab}$ and contraction $F_{bc}{j}_{\mathcal{R}}^c$ are both covector fields. Their equalities are therefore well-typed tensor equations on $M$.

Tensor contraction and covariant differentiation are coordinate-independent, so the two conditions have the same truth value in every chart. Requiring them for each element of the supplied state class defines a predicate on the supplied renormalization and source-assignment data. This proves the contract is mathematically well-defined without constructing any such data or asserting that the predicate is satisfied.

The first two field equations in the Statement are exactly the hypotheses of [F1]. Applying that supplier gives ${\mathrm{\nabla }}_a{j}^a=0$. This is a model deduction conditional on the sourced Maxwell equation, not a measured conservation law.

Take ${\mathrm{\nabla }}^a$ of the Einstein equation. By [F2], ${\mathrm{\nabla }}^a{G}_{ab}=0$; metric compatibility gives ${\mathrm{\nabla }}^a(\mathrm{\Lambda }{g}_{ab})=0$ because $\mathrm{\Lambda }$ is constant. Therefore $0=\kappa ({\mathrm{\nabla }}^a{T}_{ab}^{\mathrm{E}\mathrm{M}}+{\mathrm{\nabla }}^a{T}_{\mathrm{r}\mathrm{e}\mathrm{n},ab})\mathrm{.}$ Since $\kappa \mathrm{\ne }0$, the two stress divergences sum to zero.

Substitute [F3] into step 1.2: $0=-F_{bc}{j}^c+{\mathrm{\nabla }}^a{T}_{\mathrm{r}\mathrm{e}\mathrm{n},ab}$. Rearranging gives ${\mathrm{\nabla }}^a{T}_{\mathrm{r}\mathrm{e}\mathrm{n},ab}=F_{bc}{j}^c$, which is the second contract-admissibility condition in [F4]. Together with step 1.1 this proves both claimed identities. No existence, uniqueness, renormalization prescription, state regularity, or evolution theorem is used.