Constitutive field equation from compact variations

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.