Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional

The laboratory region, its restricted inertial chart $x$ with coordinates $x=(x^0,x^1,x^2,x^3)$, $x^0=ct$, the spatial slices and the fixed-time laboratory fields are those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields; the field components, the Minkowski form $\eta =\mathrm{diag}(-1,1,1,1)$, the potential dictionary and the index conventions are those of The electromagnetic two-form, its observer decomposition and the four-current; the source fields, the charge parameter and the four-current convention $j=(c\rho ,J)$ are those of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current; the units and the constants $c>0$, ${\mu }_0>0$, ${\epsilon }_0:=({\mu }_0{c}^2{)}^{-1}$ are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values; and worldlines, four-velocities and proper times are those of Worldlines, four-velocity, chart readings and the frame/observer distinction. For the field sector (a)-(b) assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) exactly as inherited from Compact variations of the quadratic field functional: first variation and the coefficient equation, whose functional and finiteness statement this item instantiates; the definition uses no further choice. The finite-interval Riemann particle sector (c) uses the choice-free curve lemma and does not require that field-sector measure convention.

(a) [Field action] Let $U\subseteq x(\text{laboratory region})=cI\times \mathrm{\Omega }\subseteq {\mathbb{R}}^4$ be a nonempty bounded open set whose closure is a compact subset of that chart image, let $N\subseteq {\mathbb{R}}^4$ be an open neighbourhood of $\bar{U}$, and let $A:N\to {\mathbb{R}}^4$, $x\mapsto (A^0,A^1,A^2,A^3)$, and $j:N\to {\mathbb{R}}^4$ be smooth ($C^k$ Euclidean maps and diffeomorphisms); $A$ is a four-potential in the sense of clause (c) of The electromagnetic two-form, its observer decomposition and the four-current and $j=(c\rho ,J)$ is the four-current of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current. Write $A_{\nu }:={\eta }_{\nu \mu }{A}^{\mu }$, $F_{\mu \nu }:={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$, $F^{\mu \nu }:={\eta }^{\mu \alpha }{\eta }^{\nu \beta }{F}_{\alpha \beta }$ and $j^{\nu }{A}_{\nu }$ for the indicated contraction, with ${\mathrm{\partial }}_{\mu }$ the coordinate partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, and define the field action by

$S[A]:=c^{-1}{\int }_U[-\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu }+j^{\nu }{A}_{\nu }]d\lambda ,$

$d\lambda$ the four-dimensional Lebesgue measure in the chart coordinates, the integral being that of Integral over a measurable subset. Then $S[A]$ is a finite real number, in units of $\mathrm{J}\mathrm{s}$: the density $-(4{\mu }_0{)}^{-1}{F}_{\mu \nu }{F}^{\mu \nu }+j^{\nu }{A}_{\nu }$ carries $\mathrm{J}{\mathrm{m}}^{-3}$, $d\lambda$ carries ${\mathrm{m}}^4$ and $c^{-1}$ carries $\mathrm{s}{\mathrm{m}}^{-1}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values). With $\kappa :={\mu }_0$, $J:=j$ and the same bounded open $U$, one has $S[A]=c^{-1}{S}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}[A]$, where $S_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}$ is the real functional of Compact variations of the quadratic field functional: first variation and the coefficient equation; the first variation of $S$ is $c^{-1}$ times that of $S_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}$, and since $c^{-1}>0$ their stationarity predicates are equivalent.

(b) [Admissible field variations] The field variations are the real maps $h\in C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$, the real subspace of the test space of Test function space d of an open set, extended by zero to $N$; for every real $s$ the field $A+sh$ is smooth on $N$, and $s\mapsto S[A+sh]$ is the quadratic polynomial of clause (a) of Compact variations of the quadratic field functional: first variation and the coefficient equation with both coefficients multiplied by $c^{-1}$. The field $A$ is stationary for $S$ if $\frac{d}{ds}{\mid }_{s=0}S[A+sh]=0$ for every such $h$.

(c) [Particle functional] Let $m>0$ and $q\in \mathbb{R}$ (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (a)), let $[a,b]$ be a nondegenerate compact interval, let $X:[a,b]\to A$ be a $C^2$ future timelike massive worldline (Worldlines, four-velocity, chart readings and the frame/observer distinction (a)), and let $A_{\mu }$ be a $C^2$ covector field on a neighbourhood of the image of $X$. The particle functional is the one of Fixed-endpoint variation of the relativistic particle functional,

$S_{\mathrm{p}\mathrm{t}}[X]:=-mc{\int }_a^b\sqrt{-\eta (\dot{X},\dot{X})}ds+q{\int }_a^b{A}_{\mu }(X){\dot{X}}^{\mu }ds,$

with admissible variations the fixed-endpoint $C^2$ variations $h:[a,b]\to V$, $h(a)=h(b)=0$, of that lemma; $m$ is the external rest-mass parameter and $q$ the charge parameter of the framework, and $S_{\mathrm{p}\mathrm{t}}$ is a finite real number on the declared class, both summands being finite-interval Riemann integrals of continuous functions on the compact $[a,b]$: the radicand $-\eta (\dot{X},\dot{X})$ is continuous and strictly positive, and $s\mapsto A_{\mu }(X(s)){\dot{X}}^{\mu }(s)$ is continuous because $X$ and $A_{\mu }$ are $C^2$.

(d) [Scope] This item defines only the two functionals, their variation classes and the stationarity predicate; it asserts no stationarity of any configuration, no existence or uniqueness of stationary fields or curves, no boundary conditions, no convergence statement for unbounded regions, no coupled particle-field action and no dynamics of singular sources. Smooth $A$ and $j$ on a neighbourhood of $\bar{U}$ are required, and point or sheet sources are excluded (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (b)). The physical adoption of stationarity is the postulate recorded later on this page (Adopt stationary Maxwell action as alternative physical formulation).

The action density $-(1\mathrm{/}4)F_{\mu \nu }{F}^{\mu \nu }$ with the source coupling $j^{\nu }{A}_{\nu }$ and the expansion to ${\epsilon }_0\mathrm{\mid }E{\mathrm{\mid }}^2\mathrm{/}2-\mathrm{\mid }B{\mathrm{\mid }}^2\mathrm{/}(2{\mu }_0)$ follow Tong §5.5.3 and Binney §§3.2, 3.6, in each case converted to the framework's $(-,+,+,+)$ dictionary, SI constants and $j=(c\rho ,J)$; the particle functional $-mc\int \sqrt{-\eta (\dot{X},\dot{X})}ds+q\int A_{\mu }(X){\dot{X}}^{\mu }ds$ follows Tong §5.5.1–§5.5.2, Landau–Lifshitz §16 and Binney §3.1 with the Gaussian-unit forms re-expressed in the same dictionary. Binney §3.2, printed pp. 26-27, removes the boundary term using variations vanishing on the initial and final hypersurfaces and at spatial infinity. The bounded open $U$ and compact-support variation class here are a local specialization proved by the mathematical field-variation lemma; they are not the spatial domain of Binney's calculation. Binney eq. (3.13) and Tong eqs. (5.42)-(5.43) supply the normalization $c^{-1}$; the units $\mathrm{J}\mathrm{s}$ follow from the framework's SI bookkeeping, and the well-definedness statements are those of the two cited mathematical suppliers.