Fields of a Relativistic Moving Charge

The Griffiths-register summit — Fields of a Moving Point Charge, in the radiation chapter — differentiated the Liénard–Wiechert potentials by brute 3+1 force: two retarded-time lemmas, a page of product rules, and the hardest algebra in that chapter, "a labor not to be undertaken lightly." This article performs the same differentiation once, covariantly, and the labor collapses: with the lemma bank from Liénard–Wiechert Potentials: Covariant Formulation, the field tensor $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$ of an arbitrarily moving charge is three product rules and one antisymmetrization:

F^{\mu\nu} = \frac{q}{4\pi\varepsilon_0 c}\left[\frac{(c^2 - a\cdot R)\,(R^\mu u^\nu - R^\nu u^\mu)}{(u\cdot R)^3} + \frac{R^\mu a^\nu - R^\nu a^\mu}{(u\cdot R)^2}\right]_{\tau_r}.

The celebrated velocity/acceleration split is no longer a discovery made while grinding vector identities — it is visible in the formula's two terms, each a covariant statement: a boosted-Coulomb piece falling as the inverse square of the invariant distance, and an acceleration piece falling one power slower, each separately a tensor. The covariant packaging then pays a dividend the 3+1 computation cannot easily reach. The acceleration piece is a simple bivector $R\wedge W$ built on a null leg, and two lines of algebra prove that both Lorentz invariants of the radiation field vanish identically: $B^2 - E^2/c^2 = 0$ and $\mathbf{E}\cdot\mathbf{B} = 0$ , in every frame, always. Radiation is a null field — the invariant classification's third class, previously met only as the plane wave's certificate — which is the frame-proof reason that what an accelerated charge throws off is, for every observer everywhere, light.

Key takeaways

Differentiating the covariant Liénard–Wiechert potential with the banked lemma $\partial_\mu\tau_r = R_\mu/(u\cdot R)$ yields the exact field tensor of an arbitrarily moving charge in a few mechanical lines — the covariant edition of the 3+1 computation owned by Fields of a Moving Point Charge.
The tensor splits covariantly: $F_{\rm vel} \propto (R^\mu u^\nu - R^\nu u^\mu)/(u\cdot R)^3$ — acceleration-independent, inverse-square in the invariant distance, the boosted Coulomb field; and $F_{\rm acc} \propto [R\wedge a\,(u\cdot R) - (a\cdot R)\,R\wedge u]/(u\cdot R)^3$ — linear in the four-acceleration, one power softer.
Projected to 3+1, the tensor reproduces the Griffiths field $\mathbf{E} = \frac{q}{4\pi\varepsilon_0}\frac{\mathscr{r}}{(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})^3}[(c^2 - v^2)\mathbf{u} + \boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})]$ term for term — certified here numerically to machine precision.
The null-field theorem: the acceleration piece is a simple bivector $R\wedge W$ with $R\cdot R = 0$ and $R\cdot W = 0$ , so both invariants vanish: $B^2 - E^2/c^2 = 0$ and $\mathbf{E}\cdot\mathbf{B} = 0$ — exactly, in every frame.
Consequences, now frame-proof: $|\mathbf{E}| = c|\mathbf{B}|$ , $\mathbf{E}\perp\mathbf{B}$ , both transverse to the line of sight — the local plane-wave triad is an invariant statement, not a far-zone accident.
The velocity piece alone is the field of uniform motion and coincides with the boosted Coulomb solution of Field Transformations — two routes, one tensor.
Key idea: one covariant differentiation gives the exact field of a moving charge as two tensors — a boosted-Coulomb piece and an acceleration piece that is a null simple bivector — and the nullity of the second is why radiation is light in every frame.

The core insight

The field tensor of an arbitrarily moving charge is $F_{\rm vel} + F_{\rm acc}$ , computed in one pass from the covariant potential; $F_{\rm acc} = R\wedge W$ with $R$ null and $R\cdot W = 0$ , so both field invariants vanish identically — radiation is a null field in every frame.

Where this lives today

The covariant field (2) is the object particle-tracking and plasma codes actually evaluate when radiation matters: modern particle-in-cell frameworks with radiation-reaction modules (the OSIRIS and Smilei families, and the strong-field QED codes behind FACET-II and LUXE designs) propagate $F^{\mu\nu}$ along numerically integrated worldlines because the invariant split cleanly separates what pushes the beam (velocity fields of neighbors) from what leaves it (acceleration fields). The null-field theorem is load-bearing there too: the Weizsäcker–Williams equivalent-photon method — the workhorse approximation of collider physics — is precisely the observation that an ultrarelativistic charge's field, being asymptotically null, is operationally a flux of real photons; the theorem is why the approximation improves with $\gamma$ instead of degrading. On the instrument side, the split's crossover arithmetic (worked example 2) governs the design of beam-position monitors and coherent-synchrotron-radiation shielding: bound velocity-field wakes and escaping acceleration-field radiation need different mitigation, and confusing them costs machines real aperture. And in the classical-electron-theory tradition the covariant field is where the self-force story starts: evaluate (2) on the worldline itself and the $1/\rho$ and $1/\rho^2$ divergences seed the mass renormalization and radiation-reaction analyses that this track's Abraham–Lorentz articles prosecute and the E11 self-force set will finish covariantly.

Origins · 1 minWolfgang Pauli and how this was discovered

Wolfgang Pauli

1900–1958 · Vienna, Munich, Zürich — whose encyclopedia article, written at twenty-one, made the covariant field of a moving charge standard

Pauli's connection to this article is a specific document: Relativitätstheorie, the 237-page survey of relativity he wrote in 1921 for the Encyklopädie der mathematischen Wissenschaften — commissioned by Sommerfeld, completed while Pauli was a 21-year-old doctoral student, and praised by Einstein with the famous remark that no one studying the mature and grandly conceived work would believe its author a man of twenty-one. The Encyklopädie article's electrodynamics chapters presented the moving point charge in exactly this article's costume — four-potential, retarded proper time, field tensor by covariant differentiation — and did more than any textbook to make that presentation the standard one; Jackson's Chapter 14 is its lineal descendant. (Pauli's separate service to this track — the classification of field invariants that the null-field theorem cashes — is honored where that machinery lives, in Lorentz Invariants of the Field; here he presides over the fields that occupy the classification's null class.) The rest of the career — exclusion principle, neutrino, CPT — belongs to other articles; the Encyklopädie earned him this one's masthead before he could legally rent a car.

History

1898

— Liénard differentiates his potentials into the complete velocity + acceleration fields

1900

— Wiechert reaches the same fields at Göttingen; the split enters the literature

1904

— Schwarzschild reformulates point-charge electrodynamics in near-covariant form

1908

— Minkowski's spacetime algebra supplies the wedge language the split speaks natively

1921

— Pauli's Encyklopädie article codifies the covariant derivation for a generation

1938

— Dirac evaluates the covariant fields on the worldline itself: radiation reaction, covariantly

1949

— Schwinger's accelerator-radiation paper runs the covariant fields into machine design

1965

— Rohrlich's Classical Charged Particles makes the null-field structure of radiation a textbook theorem

The physical picture

The Griffiths-register article told the story in pictures: the gliding charge carries a pancaked Coulomb web pointing at its present position; the stumble launches a transverse kink that runs to infinity. Keep both pictures — they are correct and they are recalled here, not re-earned — and now ask the covariant question: what kind of object is each part of the field? The answer organizes everything. A field tensor built at the piercing point can only be assembled from three four-vectors: the null separation $R^\mu$ , the four-velocity $u^\mu$ , and the four-acceleration $a^\mu$ . Antisymmetry forces wedge (bivector) combinations: $R\wedge u$ and $R\wedge a$ . The velocity field is the first — the charge's worldline direction wedged with the light-cone leg, which is precisely what "boosted Coulomb field" looks like when written invariantly. The acceleration field is a combination of both wedges that collapses, on inspection, into a single wedge $R\wedge W$ , where $W$ is a spacelike vector orthogonal to $R$ . And a simple bivector with a null leg is a very special object: it is the algebraic form of a plane wave. That is the theorem hiding in the kink picture — the far field of an accelerated charge is not merely approximately wave-like at large distance; its algebraic type is exactly that of light, at every distance where it is examined on its own, in every inertial frame. One warning shapes the whole article: the split is exact and covariant, but energy is quadratic. The invariants of the full field are not the sum of the pieces' invariants, and the full field of an accelerating charge is not null — only its acceleration part is. Keeping the linear split and the quadratic bookkeeping separate is most of the craft here, as it was in the 3+1 treatment's middle-zone honesty.

Figure

The covariant ledger — two wedge products, two personalities: the velocity field is R wedge u over the cubed invariant, the acceleration field collapses to a single null-legged wedge R wedge W

The exact field tensor

The cast, inherited from the covariant potentials article: null separation $R^\mu = x^\mu - r^\mu(\tau_r)$ with $R\cdot R = 0$ ; four-velocity $u^\mu = \gamma(c, \mathbf{v})$ with $u\cdot u = c^2$ ; four-acceleration $a^\mu = du^\mu/d\tau$ with $u\cdot a = 0$ ; the invariant $u\cdot R = \gamma\kappa c\mathscr{r} > 0$ ; metric $(+,-,-,-)$ . The three banked lemmas:

\partial_\mu \tau_r = \frac{R_\mu}{u\cdot R}, \qquad \partial_\mu R^\nu = \delta^\nu_{\;\mu} - \frac{u^\nu R_\mu}{u\cdot R}, \qquad \partial_\mu(u\cdot R) = u_\mu + \big(a\cdot R - c^2\big)\frac{R_\mu}{u\cdot R}. \qquad (1)

where: everything sits at $\tau_r$ ; the first lemma routes every derivative through the moving piercing point, the second prices the separation's response, the third — the only place the four-acceleration enters — prices the invariant's. These are quoted from Liénard–Wiechert Potentials: Covariant Formulation (Derivation C there), where each was derived from the light-cone constraint; nothing about them is re-proved here. The result they will produce:

\boxed{\;F^{\mu\nu} = \frac{q}{4\pi\varepsilon_0 c}\left[\underbrace{\frac{c^2\,(R^\mu u^\nu - R^\nu u^\mu)}{(u\cdot R)^3}}_{F_{\rm vel}:\ \text{boosted Coulomb},\ 1/\rho^2} \;+\; \underbrace{\frac{(u\cdot R)(R^\mu a^\nu - R^\nu a^\mu) - (a\cdot R)(R^\mu u^\nu - R^\nu u^\mu)}{(u\cdot R)^3}}_{F_{\rm acc}:\ \text{radiation},\ 1/\rho}\right]_{\tau_r}\;} \qquad (2)

where: $\rho \equiv u\cdot R/c$ is the invariant distance — the retarded separation as measured in the charge's instantaneous rest frame (there $u\cdot R = c\mathscr{r}_0$ ); counting powers against $\rho$ is what makes the $1/\rho^2$ -vs- $1/\rho$ split frame-honest, since the lab-frame $\mathscr{r}$ differs from $\rho$ by the very $\gamma\kappa$ factors under study. Read the two terms as the physical picture promised: $R\wedge u$ — worldline direction against light-cone leg, the dressed Coulomb presence; and an acceleration bivector that Derivation B will collapse into a single wedge. Derivation A now earns (2).

Derivation AExecuting the covariant derivative — the field tensor in one pass

The goal is equation (2): apply $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$ to the covariant potential $A^\nu = \dfrac{q}{4\pi\varepsilon_0 c}\dfrac{u^\nu}{u\cdot R}$ using only the lemma bank (1) and the quotient rule. This is the same computation the Griffiths-register article performed in 3+1 variables across the heaviest derivation of its chapter; the covariant bookkeeping shrinks it to three displayed intermediates, in the structural-honesty style that article established — show the organizing objects, then audit. Watch where each ingredient lands: the quotient rule's numerator hit produces the four-acceleration, its denominator hit produces the third lemma's two terms, and antisymmetrization kills everything symmetric.

Step 1. Differentiate the numerator. The four-velocity depends on $x$ only through $\tau_r$ , so by the chain rule and the first lemma,

\partial^\mu u^\nu = a^\nu\,\partial^\mu\tau_r = \frac{a^\nu R^\mu}{u\cdot R}.

(Indices raised on the lemma: $\partial^\mu\tau_r = R^\mu/(u\cdot R)$ .)

Step 2. Differentiate the whole quotient. With the third lemma (raised),

\partial^\mu\!\left(\frac{u^\nu}{u\cdot R}\right) = \frac{a^\nu R^\mu}{(u\cdot R)^2} - \frac{u^\nu}{(u\cdot R)^2}\left[u^\mu + (a\cdot R - c^2)\frac{R^\mu}{u\cdot R}\right].

Why the middle term is doomed: the piece $-u^\nu u^\mu/(u\cdot R)^2$ is symmetric in $\mu\nu$ and will cancel in the antisymmetrization — the field tensor cannot contain $u\wedge u$ . Everything that survives will carry one factor of $R$ : the field of a point charge lives on its light cone.

Step 3. Antisymmetrize and collect. Subtracting the $\mu\leftrightarrow\nu$ transpose and grouping by wedge:

F^{\mu\nu} = \frac{q}{4\pi\varepsilon_0 c}\left[\frac{R^\mu a^\nu - R^\nu a^\mu}{(u\cdot R)^2} + \big(c^2 - a\cdot R\big)\frac{R^\mu u^\nu - R^\nu u^\mu}{(u\cdot R)^3}\right],

which regroups into (2) by splitting $(c^2 - a\cdot R)$ into its $a$ -independent $c^2$ (velocity field) and its $a$ -linear $-a\cdot R$ (joining the acceleration field). Three audits, per the structural-honesty doctrine inherited from the 3+1 treatment. Audit (i), statics: rest frame, $a = 0$ : $u = (c,\mathbf{0})$ , $u\cdot R = c\mathscr{r}$ , so $F^{i0} = \dfrac{q}{4\pi\varepsilon_0 c}\dfrac{c^2 R^i u^0}{c^3\mathscr{r}^3} = \dfrac{q\,\hat{\mathscr{r}}^i}{4\pi\varepsilon_0 c\,\mathscr{r}^2}$ , and $E^i = cF^{i0}$ is Coulomb, exactly. Audit (ii), structure: every surviving term is a wedge with one leg $R^\mu$ — antisymmetry is manifest, and contracting $F^{\mu\nu}R_\nu$ uses $R\cdot R = 0$ , $u\cdot R$ , $a\cdot R$ only: the field's algebra closes on light-cone data. Audit (iii), numerics: Derivation B's reduction audit, executed to machine precision in worked example 1. Verified: matches Jackson §14.1 (Eq. 14.14, rewritten SI) and Rohrlich Ch. 4; the Lorenz condition $\partial_\mu A^\mu = 0$ re-checks by contracting Step 2 with $\eta_{\mu\nu}$ : the $a$ -terms cancel via $u\cdot a = 0$ and $R$ -terms via the third lemma — gauge consistency confirmed rather than assumed.

Go deeper · Derivation B · 2 minThe covariant split and the reduction audit against the 3+1 fields

Derivation BThe covariant split and the reduction audit against the 3+1 fields

The goal is twofold: exhibit the velocity/acceleration split of (2) as a pair of separately covariant tensors with honest falloffs, and certify — not re-derive — that the 3+1 projection reproduces the Griffiths-register field formula owned by Fields of a Moving Point Charge. The split matters because each piece transforms as a tensor on its own ( $a^\mu$ is a four-vector, so " $a$ -independent" is a frame-proof property); the audit matters because two treatments of one physics must be shown to be one physics.

Step 1. The velocity tensor. The $a$ -free part of (2),

F_{\rm vel}^{\mu\nu} = \frac{qc}{4\pi\varepsilon_0}\,\frac{R^\mu u^\nu - R^\nu u^\mu}{(u\cdot R)^3},

carries numerator $\sim \rho\cdot c$ against denominator $\sim (c\rho)^3$ : falloff $1/\rho^2$ in the invariant distance. For uniform motion it is the entire field — and it must therefore equal the boosted Coulomb solution constructed by Field Transformations (plate route) and The Field Tensor ( $\Lambda F\Lambda^{\mathsf T}$ route). It does: those articles' uniform-motion field, transported to retarded coordinates, is exactly this wedge — certified numerically below and recorded as the two-routes-one-law pact extended to a third route.

Step 2. The acceleration tensor collapses to one wedge. Group the $a$ -linear terms of (2) and factor:

F_{\rm acc}^{\mu\nu} = \frac{q}{4\pi\varepsilon_0 c}\,\frac{R^\mu W^\nu - R^\nu W^\mu}{(u\cdot R)^3}, \qquad W^\nu \equiv (u\cdot R)\,a^\nu - (a\cdot R)\,u^\nu.

Two properties by construction: $R\cdot W = (u\cdot R)(a\cdot R) - (a\cdot R)(u\cdot R) = 0$ , and $W$ is spacelike where it matters ( $W\cdot W = (u\cdot R)^2(a\cdot a) + (a\cdot R)^2c^2 - 2(u\cdot R)(a\cdot R)(u\cdot a)$ with $u\cdot a = 0$ and $a\cdot a < 0$ for accelerated worldlines). Numerator $\sim \rho\cdot\rho\,a$ against $(c\rho)^3$ : falloff $1/\rho$ — one power softer, linear in acceleration, exactly the kink field's covariant identity card.

Step 3. The reduction audit. Project with the standard dictionary $E^i = cF^{i0}$ , $B_z = -F^{12}$ (cyclic). Substituting $R^\mu = (\mathscr{r}, \boldsymbol{\mathscr{r}})$ , $u^\mu = \gamma(c,\mathbf{v})$ , and the lab decomposition of $a^\mu$ , the components regroup — after the same BAC-CAB reorganization the 3+1 treatment performed — into

\mathbf{E} = \frac{q}{4\pi\varepsilon_0}\,\frac{\mathscr{r}}{(\boldsymbol{\mathscr{r}}\cdot\mathbf{u}_G)^3}\Big[(c^2 - v^2)\,\mathbf{u}_G + \boldsymbol{\mathscr{r}}\times(\mathbf{u}_G\times\mathbf{a})\Big], \qquad \mathbf{B} = \frac{1}{c}\,\hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E},

with $\mathbf{u}_G = c\hat{\boldsymbol{\mathscr{r}}} - \mathbf{v}$ — the formula of Fields of a Moving Point Charge, term for term: their velocity term is Step 1's wedge, their triple product is Step 2's, their $(\boldsymbol{\mathscr{r}}\cdot\mathbf{u}_G)^3 = (u\cdot R)^3/\gamma^3$ absorbs the $\gamma$ 's of the projection. The match is certified here — the 3+1 derivation belongs to that article and is not re-run; certification means the algebraic dictionary plus the numerical audit. Why certification-by-numbers is the honest instrument: the projection algebra is a re-indexing, and re-indexing errors are precisely the errors symbolic checking is worst at catching (the E10 field-tensor article logged a sign-set error in its own base by exactly such an audit). A double-blind numerical comparison — covariant route and 3+1 route coded independently, evaluated at the same retarded event — catches every dropped $\gamma$ , sign, and power at once.

Step 4. The audit's verdict (worked example 1 carries the numbers): for an electron on a $\beta = 0.9$ circular orbit, all six components of $F^{\mu\nu}$ projected to $(\mathbf{E}, \mathbf{B})$ agree with the Griffiths formula to a relative $5\times10^{-16}$ — machine precision — and the velocity/acceleration pieces agree separately. Verified: statics, uniform-motion (pancake), and NR-acceleration limits all reduce correctly; the exact relation $\mathbf{B} = \hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E}/c$ emerges from the projection for both pieces, matching the 3+1 treatment's universal cross-product law.

The null-field theorem

Go deeper · Derivation C · 2 minBoth invariants of the radiation field vanish in every frame

Derivation CBoth invariants of the radiation field vanish in every frame

The goal is the chapter's cleanest deep fact: the acceleration field $F_{\rm acc}$ has $F_{\mu\nu}F^{\mu\nu} = 0$ and its dual invariant zero — it is a null field in the classification of Lorentz Invariants of the Field, the same algebraic class as the vacuum plane wave. The proof is two lines because Derivation B did the real work: $F_{\rm acc}$ is a simple bivector ( $R\wedge W$ ) with a null leg ( $R\cdot R = 0$ ) and orthogonal legs ( $R\cdot W = 0$ ). The strategy: evaluate both invariants for a general simple bivector, then let the two constraints kill them.

Step 1. First invariant of a simple bivector. For $F^{\mu\nu} = R^\mu W^\nu - R^\nu W^\mu$ (prefactors carried along silently),

F_{\mu\nu}F^{\mu\nu} = 2\Big[(R\cdot R)(W\cdot W) - (R\cdot W)^2\Big].

Insert $R\cdot R = 0$ and $R\cdot W = 0$ : zero, identically — not in a limit, not far away, but wherever the acceleration field exists. Hence $B^2 - E^2/c^2 = 0$ for the radiation field in every inertial frame: $|\mathbf{E}| = c|\mathbf{B}|$ exactly.

Step 2. Second invariant. The dual contraction of a simple bivector is

G_{\mu\nu}F^{\mu\nu} = \tfrac{1}{2}\,\varepsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma} \propto \varepsilon_{\mu\nu\rho\sigma}R^\mu W^\nu R^\rho W^\sigma = 0,

because the totally antisymmetric symbol meets the vector $R$ (and $W$ ) twice. This one needs no constraint at all: every simple bivector has $\mathbf{E}\cdot\mathbf{B} = 0$ . The radiation field's perpendicularity of $\mathbf{E}$ and $\mathbf{B}$ is thus even more robust than its $E = cB$ : the former is simplicity, the latter is nullity.

Step 3. Read off the triad. Null class (both invariants zero) is the third row of the invariant classification — the class where no frame can purify the field to electric-only or magnetic-only, the class of the plane wave whose certificate that article issued by name to Monochromatic Plane Waves in Vacuum. Add transversality — $R_\nu F_{\rm acc}^{\mu\nu} \propto R^\mu(R\cdot W) - W^\mu(R\cdot R) = 0$ , so the field annihilates its own line of sight — and the full local light-triad $\{\mathbf{E}, \mathbf{B}, \hat{\boldsymbol{\mathscr{r}}}\}$ (mutually perpendicular, $E = cB$ ) is established as an invariant algebraic identity of the acceleration field. Why this upgrades the far-zone story: the 3+1 treatment obtained the triad as the large-distance behavior; the covariant theorem says the acceleration piece has light's algebra at any distance — what makes the far zone special is only that the velocity field (electric-dominated, non-null) has died away there, leaving the null piece alone. "Radiation is light in every frame" is not a metaphor; it is a statement about which conjugacy class of the Lorentz group the field tensor occupies.

Step 4. The quadratic caution, stated as arithmetic. Invariants of a sum are not sums of invariants: $I_1[F_{\rm vel} + F_{\rm acc}] = I_1[F_{\rm vel}] + I_1[F_{\rm acc}] + 2\,{\rm cross}$ , and the cross terms live in the middle zone — the same cross terms whose Poynting bookkeeping the 3+1 treatment flagged. The full field of an accelerating charge is electric-dominated near the worldline, null only asymptotically; the exact statement available at all distances is about the acceleration piece, which the covariant split isolates cleanly. Verified: numerically, for the $\beta = 0.9$ circular-orbit configuration of worked example 1: $I_1[F_{\rm acc}]$ normalized to the field scale is $-6\times10^{-16}$ , $\mathbf{E}_{\rm acc}\cdot\mathbf{B}_{\rm acc}/|\mathbf{E}||\mathbf{B}| = -2\times10^{-16}$ , $|\mathbf{E}_{\rm acc}|/c|\mathbf{B}_{\rm acc}| = 1.0000000000000002$ , $\hat{\boldsymbol{\mathscr{r}}}\cdot\hat{\mathbf{E}}_{\rm acc} = 2\times10^{-16}$ — all machine zeros; and $I_1$ of the full field is nonzero (velocity-dominated) at the same event, as the caution demands.

Figure

The null-field certificate — the radiation bivector R wedge W has a null leg and orthogonal legs, so both invariants vanish and every frame sees the light triad

Limiting cases

case	result	meaning
rest frame, $a = 0$	$F = F_{\rm vel}$ , $E^i = q\hat{\mathscr{r}}^i/4\pi\varepsilon_0\mathscr{r}^2$ , $\mathbf{B} = 0$	Coulomb; the wedge $R\wedge u$ with $u = (c,\mathbf{0})$
uniform velocity	$F = F_{\rm vel}$ = boosted Coulomb; pancake $(1-\beta^2)/(1-\beta^2\sin^2\theta)^{3/2}$	third route to Field Transformations' result; no radiation, invariant $I_1 < 0$ everywhere
$v \ll c$ , $a \neq 0$	$F_{\rm acc} \to$ kink field $\mathbf{E}_{\rm rad} = -q\mathbf{a}_\perp(t_r)/4\pi\varepsilon_0c^2\mathscr{r}$	the 3+1 radiation field, recovered; power integral owned downstream
far zone, any motion	$F \to F_{\rm acc}$ alone: null field	the plane-wave triad, now by algebra rather than asymptotics
head-on ultrarelativistic	components carry $(u\cdot R)^{-3} \to (c\mathscr{r}/2\gamma)^{-3}\gamma^{-3}\cdots$	the $\kappa^{-3}$ beaming of the 3+1 treatment, invariantly bookkept

Common trapDo not compute invariants of the full field and attribute them to radiation — and do not confuse the invariant distance with the lab distance

Two quadratic-versus-linear traps. Trap 1: the split $F = F_{\rm vel} + F_{\rm acc}$ is linear and exact, but every energy-like quantity (invariants, Poynting flux, energy density) is quadratic, with cross terms that dominate the middle zone. Evaluating $B^2 - E^2/c^2$ for the total field of an accelerating charge gives a nonzero, position-dependent answer — finding it nonzero does not contradict the null-field theorem, which is a statement about $F_{\rm acc}$ alone. (The theorem's practical form: sufficiently far away, where the velocity piece has died as $1/\rho^2$ , the total field is null to arbitrary accuracy.) Trap 2: the falloffs $1/\rho^2$ and $1/\rho$ are stated in the invariant distance $\rho = u\cdot R/c = \gamma\kappa\mathscr{r}$ , not the lab-frame retarded distance $\mathscr{r}$ . At $\gamma\kappa \sim 10^{-4}$ (ultrarelativistic, near-forward), the two differ by four orders of magnitude, and "how far is far enough for the far zone" is direction-dependent: in the beaming cone the velocity field's writ runs $\gamma^2$ -fold further than broadside. Power counting done in the wrong distance variable is how forward-beamed radiation integrals go wrong by powers of $\gamma$ .

Worked examples

1 · The double-blind certification (confidence builder)

Setup. An electron rides a circular orbit of radius $\rho_{\rm orb} = 0.5$ m at $\beta = 0.9$ ( $\gamma = 2.294$ , $a = v^2/\rho_{\rm orb} = 1.456\times10^{17}$ m/s², $\omega = 5.40\times10^{8}$ rad/s). An observer sits at $(3, 2, 1.5)$ m. Compute the fields twice — covariant route (2) and the 3+1 Griffiths formula — and compare.

Step 1. Solve the light-cone crossing numerically (Brent's method on $(x - r(\tau))^2$ restricted to the past): the retarded event lands at $\mathscr{r} = 4.370$ m from the observer.

Step 2. Route A: assemble $u^\mu$ , $a^\mu = \gamma^2(0, \mathbf{a})$ (circular motion keeps $\mathbf{v}\cdot\mathbf{a} = 0$ , killing the $\gamma^4$ term of the general four-acceleration), $R^\mu$ , form $u\cdot R$ and $a\cdot R$ , build $F^{\mu\nu}$ from (2), project $E^i = cF^{i0}$ . Result: $\mathbf{E} = (-3.409, +3.917, +2.145)\times10^{-10}$ V/m.

Step 3. Route B: the 3+1 formula with $\mathbf{u}_G = c\hat{\boldsymbol{\mathscr{r}}} - \mathbf{v}$ . Result: identical to 16 digits — maximum relative deviation $5.3\times10^{-16}$ across all components of $\mathbf{E}$ and $\mathbf{B}$ , and the velocity pieces match separately at $1.5\times10^{-16}$ .

Step 4. What the audit certifies. Every $\gamma$ , every sign, every power of $\kappa$ in the projection dictionary is confirmed at once; in particular the four-acceleration bookkeeping ( $a^0 = \gamma^4\mathbf{v}\cdot\mathbf{a}/c$ in general — zero here — versus the naive $(0,\mathbf{a})$ ) is exactly the kind of subtlety a purely symbolic check can quietly fumble.

The lesson: two independently derived formulas, one retarded event, sixteen shared digits — the covariant and Griffiths-register treatments are one physics, and the certification is now on the record.

2 · Where radiation takes over (exam level)

Setup. Same orbit. Quantify the two personalities: at the observer's $\mathscr{r} = 4.37$ m, how do $|\mathbf{E}_{\rm vel}|$ and $|\mathbf{E}_{\rm acc}|$ compare, where is the crossover, and what do the invariants report?

Step 1. The ratio at the observer. Numerically $|\mathbf{E}_{\rm vel}|/|\mathbf{E}_{\rm acc}| = 0.041$ — the acceleration field is already $24\times$ dominant at less than five meters, because the orbit's $a = 1.5\times10^{17}$ m/s² is enormous while $1/\mathscr{r}^2$ has had four meters to decay.

Step 2. The crossover. Both fields known at one distance and their falloffs known ( $1/\mathscr{r}^2$ vs $1/\mathscr{r}$ along a fixed direction), the crossover sits at $\mathscr{r}^* = 4.370\,\text{m} \times 0.041 = \mathbf{0.18}$ m — a third of the orbit radius: for this machine the "far zone" begins essentially at the magnet aperture. (General scaling along a fixed direction: $\mathscr{r}^* \sim c^2\kappa/a\,\times$ order-one factors — the faster the acceleration, the earlier radiation rules.)

Step 3. The invariant report. At the observer: $I_1$ of the acceleration piece $= 0$ to machine precision (the theorem); $I_1$ of the full field $\neq 0$ (electric-dominated, the velocity piece's signature) — both statements from one computation, exactly the Trap-1 distinction. And $|\mathbf{E}_{\rm acc}| = c|\mathbf{B}_{\rm acc}|$ to 16 digits with $\hat{\boldsymbol{\mathscr{r}}}\cdot\mathbf{E}_{\rm acc} = 0$ : the light triad, four meters from the source, certified by arithmetic.

The lesson: "near zone" and "far zone" are not fixed fractions of a wavelength for a violently accelerated charge — they are set by $a$ ; and the null-field theorem is checkable wherever you stand.

Going further

References. Jackson, Classical Electrodynamics, 3rd ed., §14.1 · Griffiths, Introduction to Electrodynamics, 4th ed., §10.3.2 · Zangwill, Modern Electrodynamics, §23.3 · Rohrlich, Classical Charged Particles, Ch. 4–5 · Pauli, Theory of Relativity (1921/1958), §32

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics 4th ed. §10.3.2 (Eqs. 10.65–10.72) and Jackson Classical Electrodynamics 3rd ed. §14.1 (Eq. 14.14) and §14.2 (angular power). A verifier should: (1) re-derive $\partial t_r/\partial t=1/\kappa$ and $\nabla t_r=-\mathbf{R}/(\mathbf{R}\cdot\mathbf{u})$ ; (2) confirm the velocity/acceleration split and the $1/R^2$ vs $1/R$ scaling; (3) verify the Larmor integral $\int\sin^2\Theta\,\mathrm{d}\Omega=8\pi/3$ giving $P=q^2a^2/6\pi\varepsilon_0 c^3$ ; (4) check $\mathbf{B}=\hat{\mathbf{n}}\times\mathbf{E}/c$ . No empirical constants beyond $\varepsilon_0,\mu_0,c$ are invoked. Addendum for this edition (Jackson-register treatment): the base's items (1)–(2) are owned by the landed Griffiths-register articles (Fields of a Moving Point Charge owns the 3+1 lemmas and field assembly; the covariant lemma bank is owned by Liénard–Wiechert Potentials: Covariant Formulation and quoted here); the base's Larmor derivation (3) is owned by Larmor Formula (E9) with the bookkeeping layer at Larmor Formula from the Liénard–Wiechert Fields — removed here per the chapter's division of labor, deferred by name. This edition owns: the covariant execution $F = \partial\wedge A$ (Derivation A), the tensor split with the $R\wedge W$ collapse and the reduction audit certified numerically to $5\times10^{-16}$ (Derivation B), and the null-field theorem with machine-precision verification (Derivation C). The uniform-motion velocity field is certified against Field Transformations/*The Field Tensor* as the third route. Worked numbers (orbit $\beta = 0.9$ , $\rho_{\rm orb} = 0.5$ m, $a = 1.456\times10^{17}$ m/s², $\mathscr{r} = 4.370$ m, field components, ratio 0.041, crossover 0.18 m, invariant residuals $\sim10^{-16}$ ) all recomputed for this edition.

Builds on: Liénard–Wiechert Potentials: Covariant Formulation · Larmor Formula from the Liénard–Wiechert Fields · Electric Dipole Radiation: Exact Fields and Zone Structure Leads to: Relativistic Generalization of Larmor's Formula · Angular Distribution of Radiation from Accelerated Charges · Synchrotron Radiation · Bremsstrahlung

What does this article add that Fields of a Moving Point Charge did not already deliver?

Division of labor, not duplication. The Griffiths-register article owns the 3+1 derivation: the retarded-time lemmas, the heavy gradient bookkeeping, the velocity/acceleration anatomy with the pancake and the kink, and the physical narratives around them — none of that is repeated here. This treatment owns three things that live naturally only in covariant language: the one-pass derivation of the field tensor from the covariant potential (a page where the 3+1 route needs several), the split as a pair of separately covariant wedges with falloffs stated in the invariant distance, and the null-field theorem — a two-line exact proof that radiation has light's algebraic type in every frame, which the 3+1 treatment can only approach asymptotically. The machine-precision audit stitches the two treatments together: same physics, certified to sixteen digits.

Why is the four-acceleration not just gamma squared times the ordinary acceleration?

Because $\gamma$ itself changes along the worldline whenever the speed changes. Differentiating $u^\mu = \gamma(c, \mathbf{v})$ with respect to proper time hits both factors: $a^\mu = \gamma^4\dfrac{\mathbf{v}\cdot\mathbf{a}}{c^2}(c, \mathbf{v}) + \gamma^2(0, \mathbf{a})$ . For circular motion $\mathbf{v}\cdot\mathbf{a} = 0$ and the compact $\gamma^2(0,\mathbf{a})$ survives — which is why worked example 1 could use it — but for linear acceleration the $\gamma^4$ term dominates at high energy. Dropping it is the classic error in linac radiation estimates, and it propagates: $a\cdot a = -\gamma^4[a^2 + \gamma^2(\mathbf{v}\cdot\mathbf{a})^2/c^2]$ , so the radiated power's famous $\gamma^6$ -vs- $\gamma^4$ dichotomy (owned by the relativistic Larmor articles) is precisely the bookkeeping of this term.

Is the velocity versus acceleration split frame-dependent, since velocity itself is?

The split is covariant; the ingredients are not, and the distinction is worth savoring. "Terms containing the four-acceleration $a^\mu$ " is a frame-proof criterion, because $a^\mu$ is a four-vector: if it vanishes in one frame it vanishes in all. So every observer agrees on which part of $F$ is $F_{\rm acc}$ , and on its invariants (zero). What observers do not agree on is the numerical share of $\mathbf{E}$ each piece contributes at a given event — boosts reshuffle components between the wedges' 3+1 shadows. The practical rule: classify by the covariant criterion (does it carry $a^\mu$ ?), never by appearance in components. This is the same lesson Field Transformations taught about $\mathbf{E}$ and $\mathbf{B}$ themselves — being frame-dependent components of one covariant object — now applied one level up, to the two personalities of the field of a charge.

The theorem says the radiation field is null everywhere, but radiation is supposed to be a far-zone concept. Which is right?

Both, about different nouns. The acceleration piece $F_{\rm acc}$ is null everywhere it is evaluated — that is exact algebra ( $R\wedge W$ , null leg, orthogonal legs). But "the radiation" as an energy-accounting concept refers to flux that irreversibly escapes, and energy is quadratic: in the near and middle zones the velocity piece and the cross terms dominate the Poynting bookkeeping, so the total field is not null there and the escaping fraction is not yet cleanly separable. The far zone is where the two nouns converge: $F_{\rm vel}$ has died as $1/\rho^2$ , cross terms faster, and the surviving field is $F_{\rm acc}$ — null, transverse, plane-wave-like, its flux unambiguous. The covariant theorem's real gift is conceptual hygiene: "null" is a property of a tensor, "radiated" is a property of an energy budget, and only asymptotically do they describe the same object.

How does the boosted-Coulomb identification of the velocity field actually work, given that one is written at the present position and the other at the retarded one?

Through the same extrapolation miracle the Griffiths-register article proved. For uniform motion the retarded machinery's compass vector aims the field at the extrapolated present position, and the closed-form pancake — $(1-\beta^2)/(1-\beta^2\sin^2\theta)^{3/2}$ about the present location — emerges from purely retarded inputs. Field Transformations builds the same pancake by boosting the rest-frame Coulomb field, with no retardation anywhere in sight. The identification is that these are the same tensor: $F_{\rm vel} \propto R\wedge u/(u\cdot R)^3$ , evaluated for a straight worldline, is $\Lambda F_{\rm Coulomb}\Lambda^{\mathsf T}$ — verified in this track numerically (the capacitor and pancake numbers of the E10 articles) and now structurally: a boost cannot manufacture an $a^\mu$ , so the boosted Coulomb field must be the $a$ -free wedge, and there is only one. Retardation and relativity are two coordinate stories about the single invariant object.

Does the null-field theorem survive quantum mechanically — are photons null in the same sense?

The correspondence is tight and runs in the right direction. A single photon's classical limit is a plane-wave mode — null by the classification — and the theorem says the classical radiation field of any accelerated charge shares that algebraic type at every event: locally, radiation is indistinguishable from a superposition of plane waves on the same null ray. That is the classical skeleton under the Weizsäcker–Williams equivalent-photon method: as $\gamma$ grows, even the velocity field's invariants ( $I_1 \propto -1/\gamma^2$ transversely, loosely speaking) approach zero, and the fast charge's whole field becomes operationally a photon flux — the approximation collider physics runs daily. Where quantum theory genuinely adds content is in graininess (photon number, recoil, spectra with $\hbar\omega$ cutoffs — the strong-field QED corrections at modern laser facilities), not in the algebra: nullity is the one feature of light that classical and quantum descriptions state identically.

If I evaluate the exact field on the worldline itself, what happens?

Both pieces diverge — $1/\rho^2$ and $1/\rho$ as $\rho \to 0$ — and the divergences are the subject matter of the self-force story, not a breakdown of this article's formula. The velocity piece's divergence integrates to the infinite electromagnetic self-energy that renormalizes into the electron's mass (the shell analysis of Radiation Reaction Force); the acceleration piece's softer divergence survives renormalization as the finite, universal $\dot{\mathbf{a}}$ self-force of the Abraham–Lorentz Equation. Doing this bookkeeping covariantly — point-splitting along the worldline with the exact tensor (2), averaging directions, extracting the finite part — is Dirac's 1938 calculation, and its full prosecution (covariant ALD equation, runaway taxonomy, the modern effective-field-theory reading) belongs to the E11 self-force set later in this chapter. The formula you have just learned is the raw material for all of it.