Retarded fields and radiation

Adopt as a physical assumption of this framework the following solution-selection rule for the classical vacuum model on the laboratory region $U\cong I\times \mathrm{\Omega }$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with the constants and SI units of Framework-local SI units, the vacuum constants and the exact/measured status of their values (so $c>0$, ${\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$).

(a) [Formulation] In the potential formulation of Maxwell's equations in potential form: the Lorenz divergence, the decoupled wave equations and residual generators with the conventions of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, for a $C^2$ potential pair in Lorenz gauge the sourced Maxwell system of The microscopic vacuum Maxwell equations: smooth and weak branches is equivalent to the decoupled sourced wave equations $W\phi =\rho \mathrm{/}{\epsilon }_0$, $WA={\mu }_{0}J$, where $W=c^{-2}{\mathrm{\partial }}_t^2-\mathrm{\Delta }$. The wave expression $W$ is invariant under $t\mapsto -t$; where explicit retarded or advanced constructions for a prescribed history exist, they give distinguished particular solutions built from the corresponding forward or backward fundamental kernels. Every other solution differs from a chosen particular solution by a source-free addition; retarded and advanced particular solutions also differ by such an addition. This describes the affine free-solution freedom, not a claim that arbitrary histories have exactly two solutions or that either construction always exists.

(b) [Adopted rule] The framework adopts the retarded branch with no free field added: the physical potential pair of the model is the retarded one, that is, the one whose support and propagation lie in the forward light cones of the source and which carries no field arriving from the past independently of the source. For the zero-incoming model the selected sourced solution has no independently prescribed homogeneous addition. If nonzero Cauchy data are instead prescribed, their compatible homogeneous addition is a separate initial-data model; it is not equivalent to setting every free addition to zero. The advanced construction and independent incoming free fields are excluded by the present retarded selection.

(c) [Status and scope] This is an adopted extra temporal condition carrying the model's arrow-of-time content. It is not a theorem, not a consequence of the time-symmetric wave or Maxwell equations (which admit advanced and free solutions as mathematical objects), and not a deduction from experiment; it asserts no existence, uniqueness, regularity, support or decay for any source class, all of which must be supplied separately by theorems of this page. It selects a solution only of the models to which it is applied; it introduces no product of distributions, no source model, no self-force, no radiation reaction, no measured signal and no statistical claim. The retarded compact-source branch to which this selection applies is developed on this page as a conditional theorem, and the point-source branch is developed later on this page; neither existence statement is part of this postulate. For the separately declared point-source branch assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as in its exact source/field interfaces, and adopt the same no-free-addition retarded selection in the weak equations of Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources, for the locally integrable field and proper point-current types of Distributional Maxwell equations with point and stationary-sheet sources. This extends the selection's model scope explicitly; neither existence nor an arbitrary convolution of noncompact distributions is postulated.

The classical treatments cited above display the retarded and advanced Green functions of the wave operator and then identify the choice of the retarded branch as additional structure imposed on the time-reversal-symmetric equations; they support the formulation of clauses (a)-(b) and the honest non-theorem status recorded in clause (c). They are not evidence for the physical correctness of the choice, and this postulate reports no experiment, measurement or statistical inference. The exact interfaces of the wave form, the gauge conventions, the smooth Maxwell system and the separately adopted weak singular-source equations are those of the cited framework items; the weak point-source extension is an explicit physical selection on the already-adopted singular-source equations, not a deduction from the smooth-only potential theorem, and a consuming theorem must construct its actual history integrals and prove weak admissibility.

Let the laboratory region, inertial chart, spatial Euclidean structure and SI units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, in particular $c>0$, ${\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$. A retarded point-geometry datum consists of a real charge parameter $q\in \mathbb{R}$ in coulombs (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current(a)) and a prescribed map $z:\mathbb{R}\to {\mathbb{R}}^3$ of class $C^3$ ($C^k$ Euclidean maps and diffeomorphisms) together with a constant $0\le v_{\ast }<c$ such that $\mathrm{\mid }{z}^{\mathrm{\prime }}(s)\mathrm{\mid }\le v_{\ast }$ for every $s\in \mathbb{R}$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); the spacetime curve $s\mapsto (z(s),s)$ is the laboratory reading of a prescribed massive future timelike worldline of the physical model (Worldlines, four-velocity, chart readings and the frame/observer distinction), and $v_{\ast }$ is a declared bound on its coordinate speed, not a measured quantity. Write $\mathrm{\Gamma }:=\{(z(s),s):s\in \mathbb{R}\}$ and $D:=\{(x,t)\in {\mathbb{R}}^3\times \mathbb{R}:x\mathrm{\ne }z(t)\}$ for the open off-worldline observation domain. Then:

(a) [Retarded root] For every observation event $(x,t)\in D$ there is exactly one $s=s(x,t)<t$ with $t-s=\mathrm{\mid }x-z(s)\mathrm{\mid }\mathrm{/}c$; the map $s:D\to \mathbb{R}$ is of class $C^3$, and with the data of (b) below one has ${\mathrm{\partial }}_{t}s=1\mathrm{/}\kappa$ and ${\mathrm{\nabla }}_{x}s=-n\mathrm{/}(c\kappa )$ (Global existence, uniqueness and differentiability of the retarded root, whose $C^3$ branch applies because $z$ is $C^3$ and uniformly subluminal).

(b) [Retarded data] For $(x,t)\in D$ define $R:=x-z(s(x,t))$, $r:=\mathrm{\mid }R\mathrm{\mid }>0$, $n:=R\mathrm{/}r$, $\beta :=z^{\mathrm{\prime }}(s(x,t))\mathrm{/}c$, $\kappa :=1-n\cdot \beta$, $\dot{\beta }:=z^{\mathrm{\prime }\mathrm{\prime }}(s(x,t))\mathrm{/}c$; then $r=c(t-s)$ and $\kappa \in [1-v_{\ast }\mathrm{/}c,1+v_{\ast }\mathrm{/}c]\subseteq (0,\mathrm{\infty })$. On $D$ the objects $r$ and $n$ are of class $C^3$, $\beta$ and $\kappa$ are of class $C^2$, and $\dot{\beta }$ is of class $C^1$ (each is a composition of the $C^3$ root with $z$ or its derivatives).

(c) [Emission-event data and emitted spheres] At an emission event $s_0\in \mathbb{R}$ and a direction $n\in S^2$ put $x_r:=z(s_0)+rn$, $t_r:=s_0+r\mathrm{/}c$ for $r>0$, and ${\beta }_0:=z^{\mathrm{\prime }}(s_0)\mathrm{/}c$, ${\kappa }_0:=1-n\cdot {\beta }_0$. Then $(x_r,t_r)\in D$ and $s(x_r,t_r)=s_0$ for every $r>0$: the sphere $\{x_r\}$ at observation time $t_r$ is emitted at the single event $s_0$.

(d) [Branches and scope] Only the all-time uniformly subluminal branch (a)-(c) is defined by this datum. For a curve prescribed on a finite interval $J$ the same formulas define retarded data only at observation events for which the root equation has a solution $s\in J$; each consumer must assume that cone intersection explicitly. Pointwise subluminality $\mathrm{\mid }{z}^{\mathrm{\prime }}(s)\mathrm{\mid }<c$ for every finite $s$ does not imply (a): the companion example on the examples page exhibits a smooth curve with $\mathrm{\mid }{z}^{\mathrm{\prime }}\mathrm{\mid }<c$ pointwise for which the retarded equation has no root at some event, so the uniform bound $v_{\ast }<c$, or an explicit cone-intersection hypothesis, is part of the datum. This definition is kinematic: it fixes the root, the retarded data and the notation, and asserts no field, no dynamics, no force law, no self-force and no empirical content. The identification of these data with the emission data of the physical field is made by the selection postulate Adopted retarded solution selection: the physical branch carries no incoming free field together with the theorems of this page.

Clauses (a)-(b) are exactly the all-time uniformly subluminal $C^3$ branch of Global existence, uniqueness and differentiability of the retarded root: that lemma supplies, for every event of ${\mathbb{R}}^3\times \mathbb{R}$, existence and uniqueness of the root, the identity of the equality case $s=t$ with $x=z(t)$, the strict inequality $s<t$ on $D$, the derivative formulas with $1-v_{\ast }\mathrm{/}c\le \kappa \le 1+v_{\ast }\mathrm{/}c$, and the $C^k$ regularity of the root on $D$ for a $C^k$ trajectory; no new analysis is introduced here. Clause (c) is the elementary root check $t_r-s_0=\mathrm{\mid }{x}_r-z(s_0)\mathrm{\mid }\mathrm{/}c=r\mathrm{/}c$ combined with the uniqueness in clause (a); no boundedness of $z^{\mathrm{\prime }\mathrm{\prime }}$ is used. Clause (d) records the branch boundaries honestly: the finite-history and nonuniform cases are left to consumers, and the failure of existence under mere pointwise subluminality is exhibited on the examples page rather than asserted here. The phrase 'prescribed worldline' is licensed by Worldlines, four-velocity, chart readings and the frame/observer distinction(a) and (c), which relate chart readings $t\mapsto r(t)$ of massive worldlines to the condition $\mathrm{\mid }{r}^{\mathrm{\prime }}(t)\mathrm{\mid }<c$. The definition introduces no physical field, so no Maxwell, self-force or empirical assertion is made; the cross-reference to Adopted retarded solution selection: the physical branch carries no incoming free field records that the retarded root is the kinematic object that the adopted selection turns into an emission time. No choice principle is used in the root lemma or in these kinematic constructions; its higher-regularity branch is explicitly choice-free.

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), as carried by the surface-measure interface below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $\sigma$ be the polar surface measure on the unit sphere $S^2\subseteq {\mathbb{R}}^3$ (The polar surface set function on the unit sphere), let $e_3:=(0,0,1)$ and write $n_3=n\cdot e_3$ for $n\in S^2$. Then:

(a) [Zonal identity] for every continuous $f:[-1,1]\to \mathbb{R}$, ${\int }_{S^2}f(n_3)d\sigma (n)=2\pi {\int }_{-1}^{1}f(u)du\mathrm{.}$

(b) [Linear weighted moment] for every continuous $f$ and every $a\in {\mathbb{R}}^3$, ${\int }_{S^2}(n\cdot a)f(n_3)d\sigma (n)=2\pi a_3{\int }_{-1}^{1}uf(u)du\mathrm{.}$

(c) [Quadratic weighted moment] for every continuous $f$ and every $a\in {\mathbb{R}}^3$, ${\int }_{S^2}(n\cdot a{)}^{2}f(n_3)d\sigma (n)=2\pi a_3^2{\int }_{-1}^1{u}^{2}f(u)du+\pi (a_1^2+a_2^2){\int }_{-1}^1(1-u^2)f(u)du\mathrm{.}$

(d) [Moments] $\sigma (S^2)=4\pi$, ${\int }_{S^2}nd\sigma =0$, ${\int }_{S^2}{n}_i{n}_{j}d\sigma =(4\pi \mathrm{/}3){\delta }_{ij}$, and for every $a\in {\mathbb{R}}^3$ ${\int }_{S^2}(n\cdot a{)}^{2}d\sigma =\frac{4\pi }{3}\mathrm{\mid }a{\mathrm{\mid }}^2,{\int }_{S^2}\mathrm{\mid }n\times a{\mathrm{\mid }}^{2}d\sigma =\frac{8\pi }{3}\mathrm{\mid }a{\mathrm{\mid }}^2,$ where ${\delta }_{ij}$ is the Kronecker symbol and $\times$ the cross product of The cross product in ${\mathbb{R}}^3$.

(e) [Scope] (a)-(d) are identities for the polar surface measure of the unit sphere with only the stated continuity of $f$; they assert no physical field, flux or radiation statement. The measure is finite, so all displayed integrals are absolutely convergent; $f$ need not be nonnegative. For the uses on this page $f$ is a rational function with denominator bounded below by $1-b>0$, hence continuous on $[-1,1]$.

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) as carried by the retarded-kernel and Cauchy suppliers below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let the laboratory region, chart and units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, so that $c>0$, ${\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ and $W:=c^{-2}{\mathrm{\partial }}_t^2-\mathrm{\Delta }$ is the wave expression of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression. Let $a>0$, $t_0<T$ and let $\rho :{\mathbb{R}}^3\times \mathbb{R}\to \mathbb{R}$, $J:{\mathbb{R}}^3\times \mathbb{R}\to {\mathbb{R}}^3$ be source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current that are jointly of class $C^{\mathrm{\infty }}$ in the sense of $C^k$ Euclidean maps and diffeomorphisms, vanish identically outside the compact set ${\bar{B}}_a(0)\times [t_0,T]$, and satisfy the continuity equation ${\mathrm{\partial }}_t\rho +\mathrm{div}J=0$ everywhere --- the compatibility condition that Charge continuity as a compatibility condition of the vacuum Maxwell equations shows to be forced by the Maxwell system, here assumed for the prescribed source. Define the retarded potentials by the explicit retarded integrals

$\phi (x,t):=\frac{1}{4\pi {\epsilon }_0}{\int }_{{\mathbb{R}}^3}\frac{\rho (y,t-\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}c)}{\mathrm{\mid }x-y\mathrm{\mid }}dy,A(x,t):=\frac{{\mu }_0}{4\pi }{\int }_{{\mathbb{R}}^3}\frac{J(y,t-\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}c)}{\mathrm{\mid }x-y\mathrm{\mid }}dy,$

and put $E:=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$, $B:=\mathrm{\nabla }\times A$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, The electric and magnetic fields as fixed-time spatial vector fields). Then:

(a) [Smoothness and wave equations] $\phi$ and each component of $A$ are of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times \mathbb{R}$, and $W\phi =\rho \mathrm{/}{\epsilon }_0$, $WA={\mu }_{0}J$ pointwise.

(b) [Lorenz gauge] With $G:=\mathrm{div}A+c^{-2}{\mathrm{\partial }}_t\phi$ one has $G=0$ on ${\mathbb{R}}^3\times \mathbb{R}$: the retarded pair is in Lorenz gauge, the continuity hypothesis supplies this identity and the total-charge conclusion (f).

(c) [Maxwell fields] $E$ and $B$ are of class $C^{\mathrm{\infty }}$ and the tuple $(E,B,\rho ,J)$ satisfies the smooth branch of the vacuum Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches on the whole of ${\mathbb{R}}^3\times \mathbb{R}$: $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B=0$, $\mathrm{\nabla }\times E=-{\mathrm{\partial }}_{t}B$, $\mathrm{\nabla }\times B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$.

(d) [No field before the source] $E=B=0$ at every event with $t<t_0$ (indeed $\phi =A=0$ there): the selected retarded solution carries no field from before the source acted.

(e) [Zero-data Cauchy identification] For every $t_1<t_0$ the pair $(\phi ,A)$ restricted to ${\mathbb{R}}^3\times [t_1,\mathrm{\infty })$ is the unique $C^2$ extension solution of the wave Cauchy problem of The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support at time $t_1$ with zero Cauchy data and sources $\rho \mathrm{/}{\epsilon }_0$, ${\mu }_{0}J$.

(f) [Total charge] The total charge $Q(t):={\int }_{{\mathbb{R}}^3}\rho (y,t)dy$ is zero for every $t$: a source that is smooth, compactly supported in space and restricted to a compact time interval and satisfies continuity carries zero total charge, so this branch does not model an eternal isolated nonzero charge.

(g) [Scope] This is the compact smooth branch with an adopted causal selection; it asserts nothing about point, line or sheet sources, about weak or distributional branches, about finite-energy or decay properties, about radiation reaction or self-force, and no empirical claim. In particular the fields of a prescribed isolated point charge with an eternal worldline are not covered by this theorem and are supplied separately by the point-source items of this page.