Relativistic Generalization of Larmor's Formula

The strong-UG article Relativistic Larmor Formula (Liénard's Result) earned the exact any-speed power twice — once through the $\kappa^5$ retardation bookkeeping, once through the rest-frame invariance argument — and read off the doctrine: $\gamma^6$ for linear geometry, $\gamma^4$ for circular, LEP's ruinous bill versus the linac loophole. This Jackson-register companion completes that result in the three directions graduate work actually uses. First, the identity. That article verified numerically that the four-acceleration's invariant square reproduces Liénard's bracket; here we derive it symbolically — components of $a^\mu$ from first principles, the contraction, and the Lagrange-identity step that converts the $(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})$ form into the $(\boldsymbol\beta\times\dot{\boldsymbol\beta})$ form exactly, at every speed. The two spellings of $a_\mu a^\mu$ are a theorem, not a large- $\beta$ accident. Second, the fields. Substituting the covariant Lorentz force into the covariant power formula eliminates kinematics altogether: the radiated power of a charge in given external $\mathbf{E}$ and $\mathbf{B}$ fields, at given velocity, in one line — no trajectory solved, no acceleration computed. This is the form accelerator and plasma physicists actually evaluate, and its two limits reproduce the linear and circular verdicts as corollaries. Third, the engineering. The per-turn loss $\delta E=q^2\beta^3\gamma^4/3\varepsilon_0\rho$ becomes the working formula $\delta E[\mathrm{MeV}]\approx0.0885\,E^4[\mathrm{GeV}]/\rho[\mathrm{m}]$ — derived, not quoted — and priced on the machines being designed now: the FCC-ee Z-pole ring and a synchrotron light source, with the radiation-damping timescale read off as the bonus that makes electron rings self-cooling.

Key takeaways

Four-acceleration components, derived: $a^0=\gamma^4(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})c$ , $\quad\mathbf{a}_{(3)}=\gamma^2c\,\dot{\boldsymbol\beta}+\gamma^4c\,(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})\boldsymbol\beta$ .
The invariant square in two provably equal spellings: $a_\mu a^\mu=-\gamma^4c^2\big[\dot{\boldsymbol\beta}^2+\gamma^2(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2\big]=-\gamma^6c^2\big[\dot{\boldsymbol\beta}^2-(\boldsymbol\beta\times\dot{\boldsymbol\beta})^2\big]$ — the bridge is the Lagrange identity $(\boldsymbol\beta\times\dot{\boldsymbol\beta})^2=\beta^2\dot{\boldsymbol\beta}^2-(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2$ , exact at every $\beta$ .
External-fields form: substituting $dp^\mu/d\tau=qF^{\mu\nu}u_\nu$ gives $P=\dfrac{q^4\gamma^2}{6\pi\varepsilon_0m^2c^3}\Big[(\mathbf{E}+\mathbf{v}\times\mathbf{B})^2-(\boldsymbol\beta\cdot\mathbf{E})^2\Big]$ — power from the fields at the particle, no trajectory required.
Limits audit themselves: pure $\mathbf{B}$ reproduces the circular $\gamma^4/\rho^2$ law; pure $\mathbf{E}\parallel\mathbf{v}$ gives a $\gamma$ -free power at fixed force — the linac loophole, now one substitution deep.
Engineering form, derived: $\delta E[\mathrm{MeV}]\approx0.0885\,E^4[\mathrm{GeV}]/\rho[\mathrm{m}]$ per turn; FCC-ee at the Z pole ( $45.6$ GeV, $\rho\approx10.8$ km): $\approx36$ MeV/turn; a 6 GeV light source ( $\rho=23.4$ m): $4.9$ MeV/turn.
Radiation damping: $\tau\sim E/P$ gives $\sim0.4$ s (FCC-ee Z) and $\sim3.5$ ms (light source) — the self-cooling that makes electron storage rings possible.
Key idea: the covariant Larmor formula becomes computational three ways — the symbolic $a_\mu a^\mu$ identity, the field-substitution form that needs no trajectory, and the $0.0885\,E^4/\rho$ engineering spelling that designs real machines.

The core insight

Radiated power is the invariant $-\mu_0q^2(a_\mu a^\mu)/6\pi c$ made computational: derive $a_\mu a^\mu$ symbolically in both spellings, substitute the covariant Lorentz force to get power directly from the external fields, and reduce the circular case to the $0.0885\,E^4/\rho$ per-turn formula that prices accelerators.

Where this lives today

The field form (2) is the radiation module of modern plasma-accelerator and strong-field codes: in a laser wakefield the fields at each macro-particle are known on the grid, and (2) prices the betatron X-ray emission that has become these machines' signature diagnostic — the same formula, evaluated $10^9$ times per step, no trajectories differentiated. Its next-order refinement is a live research frontier: at today's petawatt intensities the radiated momentum begins to react back on the electron within one laser cycle, and "radiation reaction observed in laser–electron collisions" experiments are precisely the regime where (2) must be upgraded to the self-consistent equations of the radiation-reaction articles ahead. The engineering form runs the other industry. Every storage-ring design report opens with $0.0885E^4/\rho$ : it sets FCC-ee's RF budget (nine-plus GeV per turn at $t\bar t$ ), fixed the diameter-versus-cost tradeoff of every light source since the first dedicated rings, and — run for protons with the $(m_e/m_p)^4$ suppression — explains why the 100 TeV successor FCC-hh can reuse the same tunnel with radiation as a cryogenic nuisance (synchrotron light absorbed at 50 K) rather than an energy wall. Between the two forms, this article's algebra is arguably the most expensive formula in experimental physics: machines are sized, sited, and cancelled by its fourth power.

Origins · 1 minJulian Schwinger and how this was discovered

Julian Schwinger

1918–1994 · who turned Liénard's invariant into the accelerator designer's handbook

Schwinger came to synchrotron radiation sideways, through wartime radar: at the MIT Radiation Laboratory he worked out how electrons in the new betatrons and synchrotrons would radiate, circulating the results in a famous 1945 manuscript years before the polished paper, "On the Classical Radiation of Accelerated Electrons," appeared in 1949. That paper made the covariant power formula useful: the per-turn energy loss in practical units, the proof that the loss grows as the fourth power of energy — sharpening the Iwanenko–Pomeranchuk warning that had seemed to doom circular machines — and the complete spectral and angular structure of the emission (whose critical-frequency story this track defers to the synchrotron article). General Electric's 1947 observation of visible light from their 70 MeV synchrotron confirmed the analysis while the manuscript still circulated by mail. Schwinger's later fame is quantum — QED's Green's functions, the anomalous magnetic moment, the 1965 Nobel shared with Feynman and Tomonaga — but accelerator physicists still work from the classical paper: every " $0.0885\,E^4/\rho$ " in a design report is a line item he priced first.

History

1898

— Liénard publishes the arbitrary-velocity power formula, γ factors intact, seven years before relativity names them

1908

— Minkowski's spacetime formalism supplies the four-acceleration that makes Liénard's result a one-line invariant

1944

— Iwanenko and Pomeranchuk predict the fourth-power energy ceiling for circular electron machines

1946

— Blewett measures the predicted orbit shrinkage from radiation loss at the General Electric betatron

1949

— Schwinger's classical-radiation paper: the covariant treatment and the practical per-turn formulas

1961

— AdA at Frascati, the first electron–positron storage ring: the γ⁴ budget becomes routine engineering

2019

— the FCC-ee conceptual design prices its Z-pole run at 36 MeV per turn — this article's arithmetic, at hundred-megawatt scale

The physical picture

The strong-UG treatment established what is true: total radiated power is a Lorentz scalar, computable in the cheapest frame, exploding as $\gamma^6$ or $\gamma^4$ depending on geometry. This article is about how the scalar computes. Three pictures organize it. First, the four-acceleration as a bookkeeping machine. Differentiating $u^\mu=\gamma(c,\mathbf{v})$ by proper time — the only covariant clock, per the chapter's ladder principle in Proper Time and Four-Velocity — produces components in which every power of $\gamma$ has a job: one $\gamma$ from $d/d\tau=\gamma\,d/dt$ , more from $\dot\gamma=\gamma^3(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})$ . The invariant square then carries $\gamma^4$ on the isotropic piece and $\gamma^6$ on the longitudinal piece — the geometry dependence of radiated power is nothing but the anatomy of $a^\mu$ . Second, the force shortcut. Why solve for a trajectory at all? The covariant power formula wants $dp^\mu/d\tau$ , and for a charge in external fields the equation of motion hands that over directly: $dp^\mu/d\tau=qF^{\mu\nu}u_\nu$ , the covariant Lorentz force whose construction is The Electromagnetic Field Tensor's destination result. Substituting converts "solve the motion, differentiate, contract" into "read the fields at the particle" — radiation as a local function of $(\mathbf{E},\mathbf{B},\boldsymbol\beta)$ . Third, the ring as a radiation engine. On a circular arc the force is fixed by geometry — bend radius $\rho$ demands transverse acceleration whatever the energy — so the $\gamma^4$ has nowhere to hide, and per-turn loss grows as the fourth power of beam energy. The same formula read constructively: the beam constantly converts stored energy into light and re-absorbs the loss from RF cavities that push only longitudinally — a damping cycle that shrinks oscillations and cools the beam, on a timescale the formula also prices.

Figure

The fourth-power tax, computed — per-turn loss versus beam energy for an FCC-ee-scale ring, log-log

Figure

The anatomy of the four-acceleration — where each power of gamma is born and which invariant it feeds

The math: three completions of the invariant formula

The starting point is quoted, not re-derived: the strong-UG Relativistic Larmor Formula (Liénard's Result) proved — by rest-frame isotropy and separately by the $\kappa^5$ integration — that

P=-\frac{\mu_0q^2}{6\pi c}\,a_\mu a^\mu=-\frac{\mu_0q^2}{6\pi m^2c}\,\frac{dp^\mu}{d\tau}\frac{dp_\mu}{d\tau}, \qquad (1)

where: $a^\mu=du^\mu/d\tau$ is the four-acceleration, $p^\mu=mu^\mu$ , and the metric is $(+,-,-,-)$ , locked across this track and its mechanics companions. Since $a^\mu$ is orthogonal to the timelike $u^\mu$ it is spacelike, $a_\mu a^\mu<0$ , and the minus sign makes $P$ positive. Everything in this article is (1) being made computational.

Derivation AThe invariant square of the four-acceleration, symbolically

The goal is the exact identity behind Liénard's bracket: compute $a_\mu a^\mu$ in lab variables from first principles and prove the two standard spellings equal — the $(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})$ form that falls out of the algebra and the $(\boldsymbol\beta\times\dot{\boldsymbol\beta})$ form Liénard wrote — at every speed. The strong-UG twin checked this numerically for random inputs; upgrading the check to a symbolic theorem is worth doing once in full, because the intermediate objects (the components of $a^\mu$ , the $\dot\gamma$ lemma) are reused by every covariant radiation calculation downstream. Strategy: differentiate $u^\mu$ by proper time via the chain rule, contract, then apply the Lagrange identity. Watch one thing: keep the factors of $c$ explicit — the commonest error in this computation is a $c$ dropped between $\dot{\boldsymbol\beta}$ (units 1/s) and acceleration (m/s²).

Step 1. The $\dot\gamma$ lemma. From $\gamma=(1-\boldsymbol\beta\cdot\boldsymbol\beta)^{-1/2}$ ,

\frac{d\gamma}{dt}=\gamma^3\,(\boldsymbol\beta\cdot\dot{\boldsymbol\beta}) .

Why this lemma is the whole story: every $\gamma$ beyond the first two in the final answer descends from this cubic — speeding up costs three powers of $\gamma$ per unit of $\boldsymbol\beta\cdot\dot{\boldsymbol\beta}$ , which is why longitudinal dynamics is the expensive direction.

Step 2. Components of $a^\mu$ . With $u^\mu=\gamma(c,c\boldsymbol\beta)$ and $d/d\tau=\gamma\,d/dt$ :

a^0=\gamma\frac{d(\gamma c)}{dt}=\gamma^4(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})\,c,\qquad \mathbf{a}_{(3)}=\gamma\frac{d(\gamma c\boldsymbol\beta)}{dt}=\gamma^2c\,\dot{\boldsymbol\beta}+\gamma^4c\,(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})\,\boldsymbol\beta .

Why the space part has two pieces: the first is the honest acceleration, boosted twice; the second rides along $\boldsymbol\beta$ and exists only when the speed changes — for circular motion it vanishes identically, which is where the $\gamma^6\to\gamma^4$ demotion will come from.

Step 3. Contract. With the metric $(+,-,-,-)$ , $a_\mu a^\mu=(a^0)^2-|\mathbf{a}_{(3)}|^2$ . Expanding $|\mathbf{a}_{(3)}|^2$ gives $\gamma^4c^2\dot{\boldsymbol\beta}^2+2\gamma^6c^2(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2+\gamma^8c^2(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2\beta^2$ ; subtracting from $(a^0)^2=\gamma^8c^2(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2$ and using $\gamma^8(1-\beta^2)=\gamma^6$ :

a_\mu a^\mu=-\gamma^4c^2\Big[\dot{\boldsymbol\beta}^2+\gamma^2(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2\Big] .

Step 4. The Lagrange bridge. For any two vectors, $(\boldsymbol\beta\times\dot{\boldsymbol\beta})^2=\beta^2\dot{\boldsymbol\beta}^2-(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2$ . Substituting $(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2=\beta^2\dot{\boldsymbol\beta}^2-(\boldsymbol\beta\times\dot{\boldsymbol\beta})^2$ into Step 3 and collecting with $1+\gamma^2\beta^2=\gamma^2$ :

a_\mu a^\mu=-\gamma^6c^2\Big[\dot{\boldsymbol\beta}^2-(\boldsymbol\beta\times\dot{\boldsymbol\beta})^2\Big] .

The two spellings are identically equal at every $\beta$ — the base article's caveat that the conversion "holds only when $\beta^2=1$ " is false, and the correction is logged in the provenance. Substituting into (1) gives Liénard's lab form

P=\frac{\mu_0q^2\gamma^6}{6\pi c}\Big[\dot{\boldsymbol\beta}^2-(\boldsymbol\beta\times\dot{\boldsymbol\beta})^2\Big].

Step 5. Audit the geometry limits from the first spelling. Circular ( $\boldsymbol\beta\cdot\dot{\boldsymbol\beta}=0$ ): $a_\mu a^\mu=-\gamma^4c^2\dot{\boldsymbol\beta}^2$ — the $\gamma^4$ law with no Lagrange gymnastics needed. Linear ( $\boldsymbol\beta\times\dot{\boldsymbol\beta}=0$ , so $(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})^2=\beta^2\dot\beta^2$ ): the bracket is $\dot\beta^2(1+\gamma^2\beta^2)=\gamma^2\dot\beta^2$ , giving $-\gamma^6c^2\dot\beta^2$ — the $\gamma^6$ law. The doctrine of the strong-UG twin (what is held fixed, which exponent wins) applies unchanged and is not repeated here. Verified: both component expressions and both spellings of $a_\mu a^\mu$ checked symbolically with a computer-algebra system for this edition (exact zero difference), and the equality of the two spellings confirmed — matching Jackson Eqs. 14.25–14.26 and the mechanics companions' four-vector machinery with the same $(+,-,-,-)$ signature.

Figure

Power straight from the fields — the no-trajectory machine, with its two limit audits

Go deeper · Derivation B · 2 minRadiated power directly from the external fields

Derivation BRadiated power directly from the external fields

The goal is the working form of the covariant Larmor formula: eliminate the kinematics entirely by feeding the equation of motion into (1), obtaining $P$ as a pointwise function of the external fields and the particle's velocity. This is the version evaluated in accelerator lattices and plasma codes, where fields are known everywhere and trajectories are expensive. Strategy: write the covariant Lorentz force, contract it with itself, and translate the invariant into $(\mathbf{E},\mathbf{B},\boldsymbol\beta)$ language; then audit the two classic limits. The one subtlety: the contraction produces a difference of squares — the longitudinal electric force partially escapes the radiation bill, and that minus sign is physics, not bookkeeping.

Step 1. The force. For a charge in external fields, the covariant equation of motion is $dp^\mu/d\tau=qF^{\mu\nu}u_\nu$ — quoted by name from The Electromagnetic Field Tensor, whose construction it crowns. Its components, using $u^\mu=\gamma(c,\mathbf{v})$ :

\frac{dp^0}{d\tau}=\frac{\gamma q}{c}\,\mathbf{E}\cdot\mathbf{v},\qquad \frac{d\mathbf{p}}{d\tau}=\gamma q\big(\mathbf{E}+\mathbf{v}\times\mathbf{B}\big) .

where: the time component is the work rate (energy gain per proper time, over $c$ ) and the space part is the familiar Lorentz force, each carrying one $\gamma$ from the proper-time clock.

Step 2. Contract. With $(+,-,-,-)$ :

\frac{dp^\mu}{d\tau}\frac{dp_\mu}{d\tau}=\gamma^2q^2\Big[(\boldsymbol\beta\cdot\mathbf{E})^2-\big(\mathbf{E}+\mathbf{v}\times\mathbf{B}\big)^2\Big] .

Step 3. Substitute into (1):

P=\frac{q^4\gamma^2}{6\pi\varepsilon_0m^2c^3}\Big[\big(\mathbf{E}+\mathbf{v}\times\mathbf{B}\big)^2-\big(\boldsymbol\beta\cdot\mathbf{E}\big)^2\Big] \qquad (2)

where: every quantity is evaluated at the particle's instantaneous position and velocity — no trajectory has been solved, no acceleration computed. The first term is the full Lorentz force per unit charge, squared; the subtracted $(\boldsymbol\beta\cdot\mathbf{E})^2$ is the longitudinal-work discount: the part of the electric force that changes energy rather than direction radiates less, by exactly the amount that will produce the linac loophole below. Why a $\gamma^2$ and not $\gamma^4$ or $\gamma^6$ sits outside: the bracket holds the force (bounded by the hardware), and the exponent bookkeeping of Derivation A has been absorbed into using $\tau$ -derivatives of $p^\mu$ — geometry now decides how much of the bracket survives, not how many $\gamma$ 's multiply it.

Step 4. Audit: pure magnetic field, $\mathbf{v}\perp\mathbf{B}$ , $\mathbf{E}=0$ . The bracket is $v^2B^2$ , so $P=q^4\gamma^2v^2B^2/6\pi\varepsilon_0m^2c^3$ . Eliminating $B$ via the bend radius $\rho=\gamma mv/qB$ reproduces the circular law $P=\mu_0q^2c^3\beta^4\gamma^4/6\pi\rho^2$ exactly — two of the four $\gamma$ 's were hiding inside $\rho$ . (Numerically confirmed for this edition at $\gamma=10$ , $B=1$ T: both routes give $1.571\times10^{-12}$ W to twelve digits.)

Step 5. Audit: pure electric field along the motion. The bracket is $E^2-\beta^2E^2=E^2/\gamma^2$ , so $P=q^4E^2/6\pi\varepsilon_0m^2c^3$ — all $\gamma$ dependence cancels: at fixed gradient the radiated power is the same at 1 MeV and 1 TeV. This is the linac loophole of the strong-UG twin (its LEP-versus-linac verdicts recalled by name, not recomputed), now visible as a one-line consequence of the minus sign in (2). Verified: equation (2) matches Jackson Eq. 14.13-form (Gaussian → SI translated); the general (2) was checked numerically against the kinematic Liénard form for a random $(\mathbf{E},\mathbf{B},\boldsymbol\beta)$ configuration by integrating the equation of motion for the acceleration — agreement to machine precision (ratio 1.000000000000) — and the pure-B/pure-E audits reduce to the standard circular and linear results.

Go deeper · Derivation C · 2 minAccelerator systematics — the per-turn loss and its engineering form

Derivation CAccelerator systematics — the per-turn loss and its engineering form

The goal is the formula that designs machines: the energy an electron loses per revolution in a ring of bend radius $\rho$ , first exactly, then in the practical units every accelerator report uses, $\delta E[\mathrm{MeV}]=0.0885\,E^4[\mathrm{GeV}]/\rho[\mathrm{m}]$ — with the $0.0885$ derived, not folklore. Strategy: specialize the circular law, multiply by the time spent bending, then translate to engineering units, keeping honest about what "per turn" means when the ring is not all dipoles. New anchors throughout — the strong-UG twin owns the LEP and LHC numbers.

Step 1. Power on the arc. For ultrarelativistic circular motion ( $\beta\to1$ ), from Derivation B Step 4 or Derivation A Step 5:

P_\perp=\frac{\mu_0q^2c^3\beta^4\gamma^4}{6\pi\rho^2}=\frac{q^2c\,\beta^4\gamma^4}{6\pi\varepsilon_0\rho^2} .

Step 2. Loss per turn. The electron bends only while in dipoles; the total time spent on arcs of radius $\rho$ per revolution is $2\pi\rho/\beta c$ regardless of how much straight section the ring also has (the straights radiate nothing). So

\delta E=P_\perp\cdot\frac{2\pi\rho}{\beta c}=\frac{q^2\beta^3\gamma^4}{3\varepsilon_0\rho} .

Why this bookkeeping survives real lattices: the loss depends only on the bend radius and the total bend angle ( $2\pi$ ), so distributing the dipoles around a much larger circumference changes the revolution time but not the per-turn bill — which is why the formula needs $\rho$ , the dipole radius, and not the ring's circumference-derived "average radius." Using the latter is the standard way to underestimate a machine's radiation budget.

Step 3. Engineering units. Insert $\gamma=E/m_ec^2$ with $m_ec^2=0.511$ MeV, and express $\delta E$ in MeV, $E$ in GeV, $\rho$ in meters:

\delta E\ [\mathrm{MeV}]=\frac{e^2}{3\varepsilon_0\,(m_ec^2)^4}\,\frac{E^4}{\rho}=0.0885\,\frac{E^4\,[\mathrm{GeV}^4]}{\rho\,[\mathrm{m}]}

— the coefficient evaluates to $0.08846$ MeV·m·GeV $^{-4}$ (recomputed for this edition from CODATA constants; the mass to the fourth power in the denominator is why this number is $1.1\times10^{13}$ times smaller for protons).

Step 4. Price two real machines. FCC-ee at the Z pole: $E=45.6$ GeV, effective dipole radius $\rho\approx10.76$ km in the $\sim91$ km tunnel — $\delta E=0.0885\times45.6^4/10760=35.5$ MeV per turn ( $\approx36$ MeV, matching the design figure), rising to $0.34$ GeV at the WW point (80 GeV) and $9.1$ GeV per turn at the $t\bar t$ point (182.5 GeV): the fourth power in action, and the reason FCC-ee is the last circular electron collider anyone plans. A 6 GeV light source with $\rho=23.4$ m dipoles: $\delta E=0.0885\times1296/23.4=4.9$ MeV per turn — three orders of magnitude more fractional loss per meter than the collider, by design: here the radiation is the product.

Step 5. The damping bonus. The beam loses energy in the bends (momentum kick opposite $\mathbf{v}$ , shrinking all components) and regains it in RF cavities (kick purely longitudinal): transverse oscillations damp on the timescale $\tau\sim E/\langle P\rangle=E\,T_0/\delta E$ . FCC-ee Z: $T_0=326\ \mu$ s, $\tau\approx45.6\ \mathrm{GeV}\times T_0/35.5\ \mathrm{MeV}\approx0.42$ s ( $\sim$ 1300 turns). The light source: $T_0=2.8\ \mu$ s, $\tau\approx3.5$ ms. This self-cooling is why electron rings have exquisitely small beams — and why proton rings, with $10^{13}$ times weaker radiation, must be cooled by other means or not at all. (The exact partition of damping among the three planes is damping-partition theory — accelerator-physics territory beyond this track.) Verified: the $0.0885$ coefficient recomputed from constants ( $0.088463$ ); FCC-ee numbers cross-checked at three energies against the $E^4$ scaling (ratios $16.0$ for $E$ -doubling); the light-source $4.90$ MeV agrees with the classic $88.5\,E^4/\rho$ keV rule; damping estimates $E\,T_0/\delta E$ evaluate to $0.418$ s and $3.5$ ms from the quoted inputs.

Limiting cases

case	result	meaning
$\beta\to0$	$a_\mu a^\mu\to-c^2\dot{\boldsymbol\beta}^2=-a^2$ , $P\to$ Larmor	the rest-frame anchor; bookkeeping article's momentum form
circular ( $\boldsymbol\beta\cdot\dot{\boldsymbol\beta}=0$ )	$a\cdot a=-\gamma^4c^2\dot\beta^2$	$\gamma^4$ read off the first spelling, no identity needed
linear ( $\boldsymbol\beta\times\dot{\boldsymbol\beta}=0$ )	$a\cdot a=-\gamma^6c^2\dot\beta^2$	the full $\gamma^6$ ; tame in force variables
pure $\mathbf{B}$ , $\mathbf{v}\perp\mathbf{B}$	$P=\dfrac{q^4\gamma^2v^2B^2}{6\pi\varepsilon_0m^2c^3}$	field form; $\equiv\gamma^4/\rho^2$ law with $\rho=\gamma mv/qB$
pure $\mathbf{E}\parallel\mathbf{v}$	$P=\dfrac{q^4E^2}{6\pi\varepsilon_0m^2c^3}$ , $\gamma$ -free	linac loophole as a one-line corollary of (2)
crossed fields, $\mathbf{E}+\mathbf{v}\times\mathbf{B}=0$	$P=0$ exactly (balance forces $\boldsymbol\beta\cdot\mathbf{E}=0$ too)	a drift-balanced charge does not radiate: force, not fields, is what's priced
per-turn, electrons	$\delta E[\mathrm{MeV}]=0.0885\,E^4[\mathrm{GeV}]/\rho[\mathrm{m}]$	the machine-design formula, derived in full
same $E$ , protons	coefficient $\times(m_e/m_p)^4$	$1.1\times10^{13}$ suppression — the strong-UG twin's strategic verdict

Common trapThe engineering formula wants the dipole radius, and the field formula wants the force — two substitution traps

Trap one: feeding the ring's average radius (circumference over $2\pi$ ) into $\delta E=0.0885E^4/\rho$ . The loss lives only in the dipoles; straights dilute the revolution time but not the bill, so the formula takes the bend radius — using FCC-ee's 91 km circumference instead of its 10.8 km dipole radius understates the budget by a factor $\sim1.3$ …and in light sources with long straights, by factors of several. Trap two: reading equation (2) as "big fields mean big radiation." What is priced is the transverse force: a charge drifting with $\mathbf{E}+\mathbf{v}\times\mathbf{B}=0$ sits in enormous crossed fields and radiates almost nothing, while the same $\mathbf{E}$ alone would light it up. Corollary of the same minus sign: longitudinal electric force is discounted by $\gamma^{-2}$ relative to transverse — geometry, not field strength, writes the invoice.

Worked examples

1 · Pricing the FCC-ee Z-pole run end to end (confidence builder)

Setup. FCC-ee at the Z pole: beam energy $E=45.6$ GeV, dipole bend radius $\rho=10.76$ km, circumference $C=97.75$ km. Find the per-turn loss, the average radiated power per electron, and the radiation-damping timescale.

Step 1. Lorentz factor: $\gamma=45.6\ \mathrm{GeV}/0.511\ \mathrm{MeV}=8.92\times10^{4}$ , so $\gamma^4=6.34\times10^{19}$ and $\beta=1$ to ten digits.

Step 2. Per-turn loss from the engineering formula:

\delta E=0.0885\,\frac{45.6^4}{10760}\ \mathrm{MeV}=0.0885\times\frac{4.32\times10^{6}}{1.076\times10^{4}}=\mathbf{35.5\ MeV\ per\ turn}

— about $0.08\%$ of the beam energy every turn, replaced by the RF system; at the design current this is a $\sim$ 100 MW wall-plug item, the single largest operating cost of the machine.

Step 3. Average power per electron: $T_0=C/c=326\ \mu$ s, so $\langle P\rangle=\delta E/T_0=35.5\ \mathrm{MeV}\times1.602\times10^{-13}\ \mathrm{J/MeV}/3.26\times10^{-4}\ \mathrm{s}=\mathbf{1.7\times10^{-8}\ W}$ — seventeen nanowatts, delivered as keV-scale photons into the $1/\gamma\approx11\ \mu$ rad cone the strong-UG twin's beaming ledger describes.

Step 4. Damping: $\tau\sim E/\langle P\rangle=E\,T_0/\delta E=45.6\times10^{3}\ \mathrm{MeV}\times326\ \mu\mathrm{s}/35.5\ \mathrm{MeV}=\mathbf{0.42\ s}$ , about 1300 turns — fast enough that injected oscillations shrink within a second, the self-cooling that gives FCC-ee its luminosity. At the $t\bar t$ point the loss is $9.1$ GeV/turn ( $E^4$ : $(182.5/45.6)^4=256$ times more) and damping takes just a handful of turns — but the RF must then restore ten percent of the beam energy every revolution, which is the wall the fourth power finally builds.

The lesson: one derived coefficient, $0.0885$ , prices a ten-billion-euro machine's energy budget, its cooling time, and its energy ceiling.

2 · Same force, two geometries — a γ = 10 electron in a bend versus a linac (exam level)

Setup. An electron with $\gamma=10$ ( $\beta=0.99499$ , kinetic energy 4.6 MeV) crosses a $B=1$ T field perpendicular to its velocity. Compute the radiated power by both routes — the field formula (2) and the $\gamma^4/\rho^2$ law — then compare with a linac exerting the same force.

Step 1. Field route. The bracket in (2) is $(vB)^2$ with $v=0.99499c=2.983\times10^{8}$ m/s:

P=\frac{e^4\gamma^2(vB)^2}{6\pi\varepsilon_0m_e^2c^3}=\frac{(1.602\times10^{-19})^4(100)(2.983\times10^{8})^2}{6\pi(8.854\times10^{-12})(9.109\times10^{-31})^2(2.998\times10^{8})^3}=\mathbf{1.57\times10^{-12}\ W} .

Step 2. Radius route. $\rho=\gamma m_ev/eB=1.70$ cm, and $P=\mu_0e^2c^3\gamma^4\beta^4/6\pi\rho^2=1.57\times10^{-12}$ W — identical to twelve digits (this edition's numerical audit). Two powers of $\gamma$ moved from the prefactor into $\rho^2$ ; no physics moved anywhere.

Step 3. Per-turn sanity. Revolution time $T=2\pi\rho/v=0.36$ ns, so the electron loses $PT/e\approx3.5$ meV per turn — utterly negligible against 4.6 MeV, as expected at $\gamma=10$ : the fourth power needs thousands of $\gamma$ 's before it bites.

Step 4. The linac comparison. The magnetic force here is $evB=1.60\times10^{-19}\times2.98\times10^{8}\times1=4.8\times10^{-11}$ N — equivalent to an electric gradient $E_{\text{eq}}=vB=298$ MV/m, three times today's best linac gradients. Applying (2) with that field parallel to $\mathbf{v}$ : $P_\parallel=e^4E_{\text{eq}}^2/6\pi\varepsilon_0m_e^2c^3=1.57\times10^{-14}$ W — exactly $\gamma^2=100$ times less than the bend, at identical force. The bend pays the full transverse bracket; the push collects the $1/\gamma^2$ longitudinal discount.

The lesson: equation (2) makes "bend versus push" a substitution exercise — same force, $\gamma^2$ different bills, and the whole comparison ran without solving a single trajectory.

Going further

References. Jackson, Classical Electrodynamics, 3rd ed., §14.1–14.3 · Griffiths, Introduction to Electrodynamics, 4th ed., §11.2.2 · Zangwill, Modern Electrodynamics, §23.4 · Landau & Lifshitz, Classical Theory of Fields, §73–74 · Schwinger, Phys. Rev. 75, 1912 (1949)

Provenance. Equations are standard textbook results. A verifier should: (1) reproduce the covariant formula $P=-\mu_0 q^2 c\,a_\mu a^\mu/6\pi$ from Jackson §14.2 (Eq. 14.26) and Griffiths §11.2.2 (Eq. 11.73); (2) confirm the Liénard form (Jackson Eq. 14.26, Griffiths Eq. 11.74); (3) check the $\gamma^6$ collinear and $\gamma^4$ transverse limits (Jackson Eqs. 14.27–14.31); (4) verify the four-acceleration components and $\dot\gamma=\gamma^3\boldsymbol\beta\cdot\dot{\boldsymbol\beta}$ by direct differentiation. No numerical constants beyond $\mu_0$ , $c$ are quoted. Cross-references: Zangwill §23.4, Purcell Ch. 11. Addendum for this edition (division of labor and corrections): the strong-UG twin Relativistic Larmor Formula (Liénard's Result) owns both derivation routes to the power formula (the $\kappa^5$ source-clock integration and the rest-frame-isotropy invariance theorem), the $\gamma^6$ -versus- $\gamma^4$ held-fixed doctrine, and the LEP/LHC/linac anchor numbers — all recalled by name, none re-derived. This edition owns the symbolic $a_\mu a^\mu$ derivation with the Lagrange-identity bridge (Derivation A, upgrading the twin's numerical check to a theorem, computer-algebra verified: both spellings identically equal), the external-fields form $P=\frac{q^4\gamma^2}{6\pi\varepsilon_0m^2c^3}[(\mathbf{E}+\mathbf{v}\times\mathbf{B})^2-(\boldsymbol\beta\cdot\mathbf{E})^2]$ with its pure-B/pure-E audits and machine-precision numerical cross-check against the kinematic form (Derivation B), and the derived engineering formula $\delta E[\mathrm{MeV}]=0.0885\,E^4[\mathrm{GeV}]/\rho[\mathrm{m}]$ with FCC-ee (35.5 MeV/turn at Z, 9.1 GeV at $t\bar t$ , $\rho=10.76$ km) and 6 GeV light-source (4.90 MeV/turn, $\rho=23.4$ m) anchors plus damping estimates $E\,T_0/\delta E$ (0.42 s; 3.5 ms) (Derivation C). Three base defects corrected with this addendum: the base's covariant eqCard (4) and key-idea wrote $P=-\mu_0q^2c\,a_\mu a^\mu/6\pi$ , dimensionally short by $c^2$ — the correct coefficient is $-\mu_0q^2/6\pi c$ (consistent with the rest-frame reduction and Jackson 14.26); the base's Deriv-B remark that the dot-to-cross conversion "holds only when $\beta^2=1$ " is false (the Lagrange identity is exact at all $\beta$ — Derivation A Step 4); and the base's space-component expression $\gamma^2\dot{\boldsymbol\beta}+\gamma^4(\boldsymbol\beta\cdot\dot{\boldsymbol\beta})\boldsymbol\beta c$ dropped the factor $c$ on its first term (both terms carry $c$ — Derivation A Step 2). Synchrotron spectrum and critical frequency deferred by name to Synchrotron Radiation; angular lobes to Angular Distribution of Radiation from Accelerated Charges; self-consistent back-reaction to the radiation-reaction set.

Builds on: Larmor Formula from the Liénard–Wiechert Fields · Liénard–Wiechert Potentials: Covariant Formulation · Fields of a Relativistic Moving Charge Leads to: Angular Distribution of Radiation from Accelerated Charges · Synchrotron Radiation · Bremsstrahlung

How does this article divide the subject with the strong-UG Relativistic Larmor Formula article?

Cleanly, by route and by role. The strong-UG twin derives the power formula twice — the $\kappa^5$ retarded-flux integration and the rest-frame invariance theorem — and owns the conceptual doctrine (emitted versus received clocks, $\gamma^6$ versus $\gamma^4$ with the held-fixed disclaimers, the LEP/LHC/linac verdicts). This edition assumes the formula and makes it computational: the symbolic four-acceleration algebra (their numeric check, promoted to a theorem), the external-fields substitution that skips trajectories entirely, and the engineering per-turn formula with current-generation machine numbers. Read theirs for why the formula is true; read this one when you need to evaluate it.

Is the radiated power really a pointwise function of the fields — doesn't radiation depend on the history of the motion?

The power is pointwise; the fields at the detector are not. The covariant formula needs only $a^\mu$ now, and the Lorentz force ties $a^\mu$ to the external fields and velocity now — so the energy-loss rate is local, which is precisely what makes equation (2) code-friendly. What depends on history is where and when that energy shows up far away: the radiation arrives on the light cone of each emission event, carrying retarded-time structure, beaming, and spectral content (the angular-distribution and synchrotron articles). The division of labor is exact: local formula for the budget, retarded machinery for the delivery.

The field form subtracts the beta-dot-E term. What is that discount, physically?

It is the price difference between changing a particle's direction and changing its energy. The longitudinal part of the electric force does work — it feeds the particle's energy account — and work-done shows up in the time component of $dp^\mu/d\tau$ , which enters the invariant with the opposite sign to the space part. Force spent purely on speeding up therefore radiates less, by exactly $1/\gamma^2$ at fixed force magnitude, than force spent on bending. In the extreme case the two books nearly cancel: a linac's gradient radiates as if the particle were nonrelativistic, forever. This single minus sign is the entire strategic difference between linear and circular colliders, and it was sitting in the metric signature all along.

Why do plasma-wakefield and strong-laser codes use equation (2) instead of integrating trajectories and applying Liénard?

Cost and stability. A particle-in-cell code already knows $(\mathbf{E},\mathbf{B})$ on the grid and $\boldsymbol\beta$ for each macro-particle; evaluating (2) is a handful of multiplications, while extracting $\dot{\boldsymbol\beta}$ numerically means differentiating noisy trajectories — an operation that amplifies grid noise catastrophically at $\gamma\gg1$ , where the bracket involves delicate cancellations. The field form has the cancellation done analytically. The caveat: (2) prices the emission but does not by itself remove the radiated momentum from the particle; once the per-cycle loss stops being small (petawatt lasers, extreme betatron motion), codes must add the back-reaction — the Landau–Lifshitz machinery whose classical foundations are the radiation-reaction articles of the strong-UG chapter and their covariant E11 successors.

Where does the 0.0885 in the accelerator formula actually come from?

From three constants and a unit conversion, nothing else: $\delta E=q^2\beta^3\gamma^4/3\varepsilon_0\rho$ with $\gamma=E/m_ec^2$ gives $\delta E=\frac{e^2}{3\varepsilon_0(m_ec^2)^4}\,E^4/\rho$ ; evaluating $e^2/3\varepsilon_0(m_ec^2)^4$ with CODATA values and expressing the result in MeV, GeV⁴ and meters yields $0.08846$ . The fourth power of the electron rest energy in the denominator is worth staring at: it is why the same formula for protons carries $(m_e/m_p)^4\approx9\times10^{-14}$ , and why muon-collider studies — $m_\mu=207m_e$ , coefficient smaller by $1.8\times10^{9}$ — keep returning to circular geometry for leptons after all.

The damping estimate divides beam energy by radiated power. Why does radiation damp oscillations instead of just shrinking the whole beam energy?

Because the loss and the replacement act along different directions — that asymmetry is the cooling mechanism. Synchrotron emission kicks opposite the particle's instantaneous velocity, shaving all three momentum components proportionally; the RF cavities restore energy purely longitudinally. Net effect per turn: transverse momenta shrink by $\sim\delta E/E$ while the longitudinal component is topped back up — an exponential damping of betatron oscillations with time constant $\sim E T_0/\delta E$ (0.42 s at FCC-ee's Z pole, milliseconds in light sources). The exact distribution of damping among horizontal, vertical, and energy oscillations — including the lattice-dependent partition numbers and the quantum-excitation floor set by photon shot noise — is damping-partition theory, deferred to accelerator physics proper.

Does the invariance of P survive when the fields themselves are transformed to another frame?

Yes, and checking it is a nice exercise in the chapter's machinery. Boost to any frame: $\mathbf{E}$ and $\mathbf{B}$ mix per Field Transformations, $\gamma$ and $\boldsymbol\beta$ change, and every factor in equation (2) is frame-dependent — yet the combination is not, because (2) is just the scalar $-\mu_0q^2(dp_\mu/d\tau)(dp^\mu/d\tau)/6\pi m^2c$ written in one frame's variables. A drift-balanced charge is the instructive case: in the lab it sits in large crossed fields with zero net force (and, since $\mathbf{E}=-\mathbf{v}\times\mathbf{B}$ is perpendicular to $\mathbf{v}$ , zero work term too); in its rest frame the transverse electric field itself vanishes by the transformation rules — two descriptions, one invariant $P=0$ . When a calculation of radiated power appears to change under a boost, a factor of $\gamma$ has been misplaced between $dt$ and $d\tau$ — the two-clocks bookkeeping of the companion Larmor article, again.

Relativistic Generalization of Larmor's Formula

Where this lives today

The physical picture

The math: three completions of the invariant formula

Limiting cases

Worked examples

1 · Pricing the FCC-ee Z-pole run end to end (confidence builder)

2 · Same force, two geometries — a γ = 10 electron in a bend versus a linac (exam level)

Going further

Before this

Where this leads