Compact variations of the quadratic field functional: first variation and the coefficient equation

Given: The Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$; a bounded open set $U\subseteq {\mathbb{R}}^4$; an open $N\supseteq \bar{U}$; a real $\kappa >0$; smooth maps $A,J:N\to V={\mathbb{R}}^4$; the Minkowski form $\eta$ and conventions of [F1]; $F_{\mu \nu }:={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$, $F^{\mu \nu }:={\eta }^{\mu \alpha }{\eta }^{\nu \beta }{F}_{\alpha \beta }$, $L:=-\frac{1}{4\kappa }{F}_{\mu \nu }{F}^{\mu \nu }+J^{\nu }{A}_{\nu }$ and $S[A]:={\int }_{U}Ld\lambda$; and, for parts (a)-(c), a fixed $h\in C_c^{\mathrm{\infty }}(U;V)$ with $h_{\nu }:={\eta }_{\nu \mu }{h}^{\mu }$ and $G_{\mu \nu }:={\mathrm{\partial }}_{\mu }{h}_{\nu }-{\mathrm{\partial }}_{\nu }{h}_{\mu }$.

Minkowski form and index conventions: $V={\mathbb{R}}^4$ has the ordered basis ${\mathrm{e}}_0,\dots ,{\mathrm{e}}_3$, the form $\eta (v,w)=-v_0{w}_0+v_1{w}_1+v_2{w}_2+v_3{w}_3$ is symmetric bilinear with matrix $\mathrm{diag}(-1,1,1,1)$, that matrix is its own inverse, and nondegeneracy and the tensor-component transformation are as stated there (The Minkowski affine four-space: the form $\eta$, causal cones, frames and Lorentz maps).

Componentwise smoothness, compact support and zero extension: a map is smooth exactly when all its components are, a member of $C_c^{\mathrm{\infty }}(U;V)$ has compact support in the open $U$ and extends by zero to a smooth compactly supported map on $N$, and constant-coefficient linear combinations of components preserve smoothness and compact support ($C^k$ Euclidean maps and diffeomorphisms, Test function space d of an open set). Hence $h\mapsto (h_{\nu }{)}_{\nu }$ is a bijection of $C_c^{\mathrm{\infty }}(U;V)$ onto $C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$ with inverse $k\mapsto m^{\mu }:={\eta }^{\mu \nu }{k}_{\nu }$ by [F1].

Cutoff functions, in ZF: for compact $K\subseteq \mathrm{\Omega }\subseteq {\mathbb{R}}^4$ with $\mathrm{\Omega }$ open there is $\chi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $0\le \chi \le 1$ and $\chi =1$ on a neighborhood of $K$ (Test function cutoffs and euclidean localization).

Compactness of the closure: $\bar{U}$ is closed and, since $U$ is bounded, bounded; a closed bounded subset of ${\mathbb{R}}^4$ is compact, with no choice principle (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Integrability of continuous compactly supported functions: let $F:{\mathbb{R}}^4\to \mathbb{R}$ be continuous with compact support $K$. If $K=\mathrm{\varnothing }$ then $F=0$ and $\int Fd\lambda =0$; otherwise $M:={\max }_K\mathrm{\mid }F\mathrm{\mid }$ is a finite real number (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value), $K$ is bounded (A compact subset of a metric space is closed and bounded) and hence contained in a closed box $Q$ with $\lambda (Q)<+\mathrm{\infty }$ (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included), and $\mathrm{\mid }F\mathrm{\mid }\le M{\chi }_Q$ gives $\int \mathrm{\mid }F\mathrm{\mid }d\lambda \le \int M{\chi }_{Q}d\lambda =M\lambda (Q)<+\mathrm{\infty }$ (Monotonicity and nonnegative homogeneity of the nonnegative integral, The nonnegative integral agrees with the simple integral on simple functions, The integral of a nonnegative simple function), so $F$ is integrable (Integrable real and complex functions, and their integrals). Under ${\mathrm{A}\mathrm{C}}_{\omega }$ every continuous map between Euclidean spaces is Borel measurable (Continuous functions on Euclidean spaces are Borel measurable), the Borel sigma-algebra is generated by the open sets (The Borel sigma-algebra of a topological space) and every Borel set is Lebesgue measurable (Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable); consequently, for a continuous $f$ on an open $G\subseteq {\mathbb{R}}^4$, the zero extension $f{\chi }_G$ is Borel measurable, because for $a\ge 0$ one has $\{f{\chi }_G>a\}=G\cap \{f>a\}$, open as $f$ is continuous on the open $G$, while for $a<0$ it is the union of the closed set ${\mathbb{R}}^4\setminus G$ and that open set, and the rays $(a,+\mathrm{\infty })$ generate the Borel sigma-algebra.

Signed integrals, monotonicity and linearity: for measurable real $f$ with at most one of $\int f^{+}$, $\int f^{-}$ infinite one sets $\int fd\lambda :=\int f^{+}d\lambda -\int f^{-}d\lambda$, and $f$ is integrable when both are finite; for nonnegative measurable $f$ the integral over a measurable set $E$ is ${\int }_{E}fd\lambda =\int f{\chi }_{E}d\lambda$; if $0\le f\le h$ are nonnegative measurable then $\int fd\lambda \le \int hd\lambda$, and if $c>0$ then $\int cfd\lambda =c\int fd\lambda$, while the zero function has integral $0$; on $L^1$ the integral is linear, $\int (\alpha f+\beta g)d\lambda =\alpha \int fd\lambda +\beta \int gd\lambda$ (Integrable real and complex functions, and their integrals, Integral over a measurable subset, Monotonicity and nonnegative homogeneity of the nonnegative integral, The Lebesgue integral is linear on $L^1(\mu )$).

Partial derivatives and their linearity: for a map on an open subset of ${\mathbb{R}}^4$, ${\mathrm{\partial }}_{\mu }$ is the directional derivative in the standard direction ${\mathrm{e}}_{\mu }$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$), and differentiation is linear, $D(f+cg)=Df+cDg$ for a real constant $c$ (Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives), so ${\mathrm{\partial }}_{\mu }(f+cg)={\mathrm{\partial }}_{\mu }f+c{\mathrm{\partial }}_{\mu }g$ componentwise.

Weak derivatives and the test identity, under ${\mathrm{A}\mathrm{C}}_{\omega }$: a real function $u$ of class $C^1$ on an open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ has each classical partial derivative ${\mathrm{\partial }}_{\mu }u$ locally integrable and equal to its weak derivative (Classical derivatives agree with weak derivatives), so for every test function $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ the defining test identity of the weak derivative gives ${\int }_{\mathrm{\Omega }}u{\mathrm{\partial }}_{\mu }\psi d\lambda =-{\int }_{\mathrm{\Omega }}({\mathrm{\partial }}_{\mu }u)\psi d\lambda$ (Weak derivative of a locally integrable function).

Polynomial derivative: for real $a_0,a_1,a_2$ the polynomial function $P(s)=a_0+a_{1}s+a_2{s}^2$ is differentiable at every real $c$, and $P^{\mathrm{\prime }}(c)=a_1+2a_{2}c$; in particular $P^{\mathrm{\prime }}(0)=a_1$, the term $a_0$ contributing $0$ and the term $a_2{s}^2$ contributing $2a_{2}c$ (For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, clause 4).

Fundamental variation lemma, under ${\mathrm{A}\mathrm{C}}_{\omega }$: if a continuous $g:U\to {\mathbb{R}}^m$ on an open $U\subseteq {\mathbb{R}}^n$ satisfies ${\int }_U⟨g,k⟩d\lambda =0$ for every $k\in C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^m)$, then $g=0$ on $U$ (The fundamental lemma of the calculus of variations for continuous coefficients).

Euclidean inner product: $⟨x,y⟩={\sum }_{i<m}{x}_i{y}_i$ on ${\mathbb{R}}^m$, so ${\sum }_{\nu }{C}^{\nu }{h}_{\nu }=⟨C,k⟩$ for the arrays $C=(C^0,\dots ,C^3)$ and $k=(h_0,\dots ,h_3)$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).