Alternative action formulation

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.

Adopt as a physical formulation principle of this framework, as an alternative to postulating the field equations and the force law directly, the following assumption about the functionals, variation classes and stationarity predicate of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional.

(i) [Field sector] A smooth field $A:N\to {\mathbb{R}}^4$ on the region $U$ of clause (a) of that definition is dynamically admissible exactly when it is stationary for the field action $S[A]$ under all compactly supported smooth variations $h\in C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$ in the sense of clause (b) of that definition: $\frac{d}{ds}{\mid }_{s=0}S[A+sh]=0$ for every such $h$.

(ii) [Particle sector] For a prescribed external four-potential $A_{\mu }$ on a neighbourhood of the image of a $C^2$ future timelike massive worldline $X$, the motion is dynamically admissible exactly when $X$ is stationary for the particle functional $S_{\mathrm{p}\mathrm{t}}$ of clause (c) of that definition under all fixed-endpoint $C^2$ variations.

This is an adopted assumption of the model, not a theorem, not a derivation from the remaining postulates and not an empirical result. It asserts stationarity only: it does not assert that a stationary configuration is a local or global minimum, that stationary configurations exist or are unique, that the action functional is uniquely determined or forced by symmetry, that no other action formulation is possible, or that the two sectors can be coupled into a single well-posed action for singular sources. No Noether theorem, no symmetry-to-conservation machinery and no matter dynamics beyond the prescribed potential are assumed.

The conditional deductions of this pair are: on the declared smooth classes, (i) is equivalent to the inhomogeneous tensor field equation ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }$ and, for potential-represented fields, to the smooth-branch Maxwell equations, and (ii) is equivalent to the framework's adopted four-force equation $mdU\mathrm{/}d\tau =q\mathrm{\Phi }(U)$; both deductions are recorded separately on this page and in Action stationarity implies the inhomogeneous Maxwell equations and Relativistic prescribed-field particle action yields the Lorentz force, and this postulate asserts neither.

The cited treatments present the action principle of classical electrodynamics as adopted, not derived: Tong §5.5 introduces the action and obtains the field equations from its variation, and Landau–Lifshitz §16 states in the footnote on p. 44 that the form of the interaction term cannot be fixed on general considerations alone. The Gaussian-unit expressions of the monograph are read only for the structure of the principle; the functionals adopted here are exactly the SI functionals of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional. No measurement, no empirical claim and no uniqueness of the formulation is contained in this item.

Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), inherited from Compact variations of the quadratic field functional: first variation and the coefficient equation. Let $U\subseteq cI\times \mathrm{\Omega }$, $N$, the smooth fields $A,j$ on $N$, the components $A_{\nu }$, $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$, $F^{\mu \nu }$, the source convention $j=(c\rho ,J)$, the constants $c,{\mu }_0,{\epsilon }_0$ and the field action $S[A]=c^{-1}{\int }_U[-(4{\mu }_0{)}^{-1}{F}_{\mu \nu }{F}^{\mu \nu }+j^{\nu }{A}_{\nu }]d\lambda$ be as in clauses (a)-(b) of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

Then the following are equivalent: (i) $A$ is stationary for $S$ under all $h\in C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$, i.e. the adopted field equation of Adopt stationary Maxwell action as alternative physical formulation (i); (ii) ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }$ at every point of $U$ for every $\nu \in \{0,1,2,3\}$.

Consequently, if $A$ is stationary then:

(a) [Inhomogeneous tensor equations] ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }$ on $U$;

(b) [Homogeneous tensor equations] ${\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }=0$ on $U$, since $A$ is smooth and its mixed partial derivatives commute;

(c) [Laboratory Maxwell equations] define $\phi :=-c{A}_0$, the spatial potential vector $\mathbf{A}:=(A_1,A_2,A_3)$, $E:=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_t\mathbf{A}$ and $B:=\mathrm{curl}\mathbf{A}$ by clause (c) of The electromagnetic two-form, its observer decomposition and the four-current, and $(\rho ,J)$ by $j=(c\rho ,J)$ (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current); then on $U$ the four smooth-branch Maxwell equations hold, equivalently the eight scalar equations (TM1)-(TM2) of Tensor Maxwell equations equal the four laboratory Maxwell equations hold with these fields and sources, i.e. $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B=0$, $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$ and $\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$.

The statement concerns a given smooth pair $(A,j)$ on the declared region: it asserts no existence of a stationary field, imposes no boundary or decay condition, and excludes the weak, distributional, singular-support and non-potential-represented branches. The equivalence is conditional on the adopted stationary-action postulate and makes no empirical claim.

The variation computation producing the inhomogeneous field equations from the action is Tong §5.5.3 (printed pp. 125-126), Landau–Lifshitz §30 eq. (30.1) and Binney §3.6, each in its own conventions; the conversion to the framework's $(-,+,+,+)$ dictionary, the source coupling $+j^{\nu }{A}_{\nu }$, the normalisation $c^{-1}$ and the boundary term's disappearance through compact support are carried out here rather than assumed from the sources. The homogeneous equations are the closedness of $F=dA$, the structural remark of Binney §3.6 (printed pp. 30-31) and the 'only half the Maxwell equations' remark of Tong (printed p. 126).

Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), inherited through the componentwise fundamental-variation step of The fundamental lemma of the calculus of variations for continuous coefficients and through Distributional differentiation is continuous and commutes. Let $U,N,A,j,F,S$ and the constants $c,{\mu }_0,{\epsilon }_0$ be as in clauses (a)-(b) of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional: $U$ bounded open with $\bar{U}$ compact in the laboratory chart image $cI\times \mathrm{\Omega }$, $N$ an open neighbourhood of $\bar{U}$, $A$ and $j$ smooth on $N$, $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$, $j=(c\rho ,J)$ with the source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, and $S[A]=c^{-1}{\int }_U[-(4{\mu }_0{)}^{-1}{F}_{\mu \nu }{F}^{\mu \nu }+j^{\nu }{A}_{\nu }]d\lambda$. For $\chi \in C_c^{\mathrm{\infty }}(U;\mathbb{R})$ write ${\mathrm{\partial }}^{\mathrm{♯}}\chi$ for the vector field with contravariant components $({\mathrm{\partial }}^{\mathrm{♯}}\chi {)}^{\mu }:={\mathrm{\partial }}^{\mu }\chi ={\eta }^{\mu \nu }{\mathrm{\partial }}_{\nu }\chi$ (Test function space d of an open set, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). It is a compactly supported smooth field variation. Thus $A^{\mu }\mapsto A^{\mu }+{\mathrm{\partial }}^{\mu }\chi$ lowers to $A_{\nu }\mapsto A_{\nu }+{\mathrm{\partial }}_{\nu }\chi$; $A+{\mathrm{\partial }}^{\mathrm{♯}}\chi$ denotes this vector addition.

(a) [Exact gauge difference, no boundary term] For every such $\chi$ one has $F_{\mu \nu }(A+{\mathrm{\partial }}^{\mathrm{♯}}\chi )=F_{\mu \nu }(A)$ on $U$, because ${\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }\chi -{\mathrm{\partial }}_{\nu }{\mathrm{\partial }}_{\mu }\chi =0$ for the smooth $\chi$ (Clairaut--Schwarz theorem for continuous second partial derivatives); consequently the field term of $S$ is unchanged and

$S[A+{\mathrm{\partial }}^{\mathrm{♯}}\chi ]-S[A]=c^{-1}{\int }_U{j}^{\mu }{\mathrm{\partial }}_{\mu }\chi d\lambda =-c^{-1}{\int }_U({\mathrm{\partial }}_{\mu }{j}^{\mu })\chi d\lambda ,$

the second equality being the distributional integration-by-parts identity for the locally integrable fields $j^{\mu }$ and the smooth compactly supported $\chi$ (Distributional derivative, Regular distribution from a locally integrable function, Distributional differentiation is continuous and commutes); no boundary term arises precisely because $\chi$ is compactly supported in $U$.

(b) [Equivalence for smooth sources] If ${\mathrm{\partial }}_{\mu }{j}^{\mu }=0$ on $U$ (the source is conserved), then $S[A+{\mathrm{\partial }}^{\mathrm{♯}}\chi ]=S[A]$ for every $\chi \in C_c^{\mathrm{\infty }}(U;\mathbb{R})$. Conversely, if $S[A+{\mathrm{\partial }}^{\mathrm{♯}}\chi ]=S[A]$ for every such $\chi$, then ${\mathrm{\partial }}_{\mu }{j}^{\mu }=0$ pointwise on $U$: the function ${\mathrm{\partial }}_{\mu }{j}^{\mu }$ is continuous on the open $U$ and pairs to zero against every compactly supported smooth $\chi$, so The fundamental lemma of the calculus of variations for continuous coefficients applies. Hence for smooth prescribed sources the invariance of the field action under all compact gauge variations is equivalent to the conservation of the four-current.

(c) [Maxwell-compatible sources] Suppose in addition that the field pair $(E,B)$ represented by $A$ through the potential dictionary of clause (c) of The electromagnetic two-form, its observer decomposition and the four-current and the source fields $(\rho ,J)$ satisfy the smooth branch of the vacuum Maxwell equations on $U$ (equivalently, by clause (a) of Tensor Maxwell equations equal the four laboratory Maxwell equations, the tensor system ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }$ together with the cyclic identity). Then clause (a) of Charge continuity as a compatibility condition of the vacuum Maxwell equations gives ${\mathrm{\partial }}_t\rho +\mathrm{div}J=0$ on $U$ (Divergence and curl of a $C^1$ vector field), which in the chart reads ${\mathrm{\partial }}_{\mu }{j}^{\mu }={\mathrm{\partial }}_0(c\rho )+\mathrm{div}J=c^{-1}{\mathrm{\partial }}_t(c\rho )+\mathrm{div}J=0$; hence $S[A+{\mathrm{\partial }}^{\mathrm{♯}}\chi ]=S[A]$ for every compact gauge variation.

(d) [Distributional reading] If $j^{\mu }$ is a distribution on $U$ (Distribution), the gauge-difference pairing $c^{-1}⟨j^{\mu },{\mathrm{\partial }}_{\mu }\chi ⟩$ is defined for every test function $\chi \in C_c^{\mathrm{\infty }}(U;\mathbb{R})$ and equals $-c^{-1}⟨{\mathrm{\partial }}_{\mu }{j}^{\mu },\chi ⟩$ by the signed definition of the distributional derivative; hence the pairing vanishes for every compact gauge variation if and only if ${\mathrm{\partial }}_{\mu }{j}^{\mu }=0$ as a distribution. This is a statement about the pairing only: the smooth $j$ of the definition is retained whenever the functional $S$ itself is used, and no products of distributions are formed.

Scope: the item computes the gauge variation of the field action and its compatibility with source conservation; it does not derive the field equations from gauge invariance, does not assert that conservation of a source implies stationarity, does not treat global or topological obstructions, and uses no Noether machinery.

The gauge-shift computation is Tong §5.5.3 (printed pp. 125-126), which computes $S\to S-\frac{1}{c}\int \chi {\mathrm{\partial }}_{\mu }{J}^{\mu }$ and reads invariance as ${\mathrm{\partial }}_{\mu }{J}^{\mu }=0$; Landau–Lifshitz §§28-29 and Binney §§3.6-3.7 supply the four-current, continuity and interaction-term context. The exact boundary-free formulation here uses the framework's own compact-support hypothesis and the published distribution interface; the converse direction replaces the sources' informal 'arbitrary gauge direction at a point' step by the fundamental variation lemma, and no Noether formalism is adopted.

The finite-interval Riemann particle-sector calculation is choice-free: the field-sector Countable Choice convention of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional is not used here. Let worldlines, four-velocities and proper times be those of Worldlines, four-velocity, chart readings and the frame/observer distinction, let $F$ be the electromagnetic two-form with the component and potential dictionary of The electromagnetic two-form, its observer decomposition and the four-current, and let $m>0$ and $q\in \mathbb{R}$ be the framework's external rest-mass and charge parameters (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 (clause (a) of Worldlines, four-velocity, chart readings and the frame/observer distinction), let $A_{\mu }$ be a $C^2$ covector field - the prescribed external four-potential - on a neighbourhood of the image of $X$, and let $S_{\mathrm{p}\mathrm{t}}$ be the particle functional of clause (c) of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional, i.e. the functional $-mc{\int }_a^b\sqrt{-\eta (\dot{X},\dot{X})}ds+q{\int }_a^b{A}_{\mu }(X){\dot{X}}^{\mu }ds$ of Fixed-endpoint variation of the relativistic particle functional.

Then, under the adopted stationary-action model for the particle sector (Adopt stationary Maxwell action as alternative physical formulation (ii)), for every fixed-endpoint $C^2$ variation $h$ with $h(a)=h(b)=0$ the curve $X+\epsilon h$ is future timelike for all sufficiently small $\mathrm{\mid }\epsilon \mathrm{\mid }$, and the following are equivalent: (i) $X$ is stationary for $S_{\mathrm{p}\mathrm{t}}$ under all such variations; (ii) $md{U}_{\nu }\mathrm{/}d\tau =q{F}_{\nu \mu }{U}^{\mu }$ on $[a,b]$, where $F=dA$ and $\tau$ is the proper time of $X$; (iii) the four-force equation $mdU\mathrm{/}d\tau =q\mathrm{\Phi }(U)$ of the framework's adopted covariant coupling The four-force equation reproduces the coordinate Lorentz force and power, where $\mathrm{\Phi }$ is the raised-index field operator $\eta (\mathrm{\Phi }(u),w)=F(w,u)$. In particular the stationary-action formulation of the particle sector is equivalent, for prescribed external potentials, to the adopted Lorentz coupling; the coordinate equations $\frac{dp}{dt}=q(E+v\times B)$ and $\frac{d(\gamma m{c}^2)}{dt}=qE\cdot v$ of The four-force equation reproduces the coordinate Lorentz force and power then hold wherever the laboratory reading is $C^2$.

Scope: the potential, hence the field, is prescribed and external to the particle; the particle's own field, radiation reaction and self-force are excluded, as are magnetic monopoles, zero or variable mass, coupled particle-field existence and any approximation of the trajectory over a time interval. The equivalence is conditional on the adopted postulates and makes no empirical claim.