Relativistic Dynamics of a Charged Particle

The chapter has spent eight articles building nouns — four-vectors, the four-current, the potential, the field tensor. This article supplies the verb. How does a charge actually move when $\mathbf{E}$ and $\mathbf{B}$ push it at speeds where Newton's $m\mathbf{a}$ fails? The answer was previewed twice — Relativistic Energy and Four-Momentum and The Electromagnetic Field Tensor both flashed the equation $dp^\mu/d\tau=qF^{\mu\nu}u_\nu$ at statement level — and now it gets executed: motivated as the only four-force the field tensor can build, unpacked into its space and time components, and then solved, three times, for the three field configurations that run the world's accelerators. The solutions are the payoff. In a uniform electric field the trajectory is not Newton's parabola-in-time but a hyperbola in spacetime — Born's "hyperbolic motion" — and the numbers explain every linear accelerator ever built: an electron in a 10 MV/m gradient passes $0.99c$ within 31 centimeters, after which the machine's remaining kilometers buy almost no speed and all energy. In a uniform magnetic field the circle of E4's cyclotron survives, but its clock slows by exactly $\gamma$ : the frequency $qB/\gamma m$ is why cyclotrons die at relativistic energies and why the LHC is a synchrotron whose 8.33-tesla dipoles bend 7-TeV protons on a 2.8-kilometer radius. And in crossed fields the chapter's machinery performs its neatest trick: Lorentz Invariants of the Field classified every field pair by its invariants — here that classification stops being taxonomy and becomes a solution method. Boost to the frame where one field vanishes, solve the easy problem, boost back.

Key takeaways

The relativistic equation of motion is $dp^\mu/d\tau=qF^{\mu\nu}u_\nu$ : the field tensor contracted with the four-velocity is the only four-force linear in $u$ that electromagnetism can build.
Antisymmetry of $F$ makes the force four-orthogonal to the velocity, $u_\mu f^\mu=0$ — so $u\cdot u=c^2$ is preserved and the rest mass never drifts: the law cannot change $m$ .
The space components reduce exactly (no approximation) to $d\mathbf{p}/dt=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ with $\mathbf{p}=\gamma m\mathbf{v}$ ; the time component is the work-energy theorem $dE/dt=q\mathbf{E}\cdot\mathbf{v}$ — magnetic fields do no work, covariantly guaranteed.
Uniform $\mathbf{E}$ : rapidity grows linearly in proper time; the worldline is the hyperbola $x^2-c^2t^2=(c^2/\alpha)^2$ with $\alpha=qE/m$ ; energy gain is exactly $qEx$ ; speed saturates while momentum grows without bound.
Uniform $\mathbf{B}$ : circular motion with $\omega=qB/\gamma m$ and $r=p/qB$ — frequency falls, radius grows with energy; this single $\gamma$ split cyclotrons from synchrotrons.
Crossed $\mathbf{E}\perp\mathbf{B}$ with $E<cB$ : drift at $v_d=E/B$ , then a pure magnetic circle in the drift frame — the field-invariant classification used as a solving tool.
Key idea: one contraction, $qF^{\mu\nu}u_\nu$ , is the complete relativistic Lorentz force — force and power in a single covariant package, solvable exactly in the fields that matter.

The core insight

A charge's four-momentum evolves as $dp^\mu/d\tau=qF^{\mu\nu}u_\nu$ : the unique four-force linear in the four-velocity that the field tensor can supply, whose space part is the exact Lorentz force on $\gamma m\mathbf{v}$ and whose time part is the work-energy theorem.

Where this lives today

Equation (1) is, by volume of computation, one of the busiest equations in physics. Every particle-in-cell plasma code — laser-wakefield accelerators, magnetic-confinement fusion, magnetospheric weather — advances billions of macro-particles per timestep with a discretized $dp^\mu/d\tau=qF^{\mu\nu}u_\nu$ (the workhorse "Boris pusher" and its relativistic descendants are numerical schemes for exactly this contraction). Accelerator lattices from medical synchrotrons to the LHC are designed as solutions of it, rigidity $p/qB$ setting every magnet's job. Astrophysics runs it at extremes no machine reaches: electrons with $\gamma\sim10^7$ spiraling in pulsar and blazar fields, their orbits Derivation C's circles, their energy losses handed to E9's relativistic Larmor machinery. And the crossed-field drift of Derivation C, generalized to slowly varying fields, becomes "guiding-center theory" — the reason a fusion tokamak can be understood at all is that every particle is, to first approximation, a drifting relativistic cyclotron. The equation's conceptual legacy is subtler: it is the classical template for minimal coupling. That the entire influence of electromagnetism on matter funnels through one contraction of $F$ with $u$ — no gradients of the field, no higher powers, no reference to gauge — is the structure the next article derives from a variational principle and quantum theory inherits verbatim. Solve this article's equation and you have, without knowing it, rehearsed the interaction term of quantum electrodynamics.

Origins · 1 minMax Born and how this was discovered

Max Born

1882–1970 · Göttingen and Edinburgh

Born enters this chapter not through the quantum mechanics that made him famous but through his first love: relativistic electrodynamics. His 1909 Habilitation work asked the sharpest available question about equation (1) — what does constant acceleration even mean relativistically? — and answered it with the hyperbolic worldline of Derivation B, "hyperbolic motion," together with the accompanying notion of rigid (Born) motion that exposed how badly Newtonian rigidity fits Minkowski spacetime. The uniformly accelerated charge became his laboratory for the deepest puzzles at the classical–relativistic boundary: its fields, its radiation (a controversy that simmers to this day and hands off to this track's radiation-reaction articles), and its horizon — the asymptote in the worldline figure — which reappears in Rindler coordinates and ultimately in the thermodynamics of the Unruh effect. That the same man later formalized the probabilistic reading of quantum mechanics and mentored Heisenberg is a reminder of how directly the accelerator physics of this article feeds the century that followed it: Born's hyperbola is the classical skeleton on which quantum field theory in accelerated frames was later hung.

History

1905

— Einstein's electrodynamics paper derives the force on a moving charge from transformation arguments — longitudinal and transverse masses appear, and begin their retreat

1908

— Minkowski writes the force covariantly: the contraction of

F

with the four-velocity

1909

— Born solves uniform proper acceleration: hyperbolic motion, the linac worldline

1932

— Lawrence's cyclotron rides the

\gamma\approx1

loophole: fixed frequency, spiral orbit

1944

— Veksler, then McMillan, invent phase-stable synchrotrons — engineering around

\omega=qB/\gamma m

2008

— LHC dipoles hold 7-TeV protons on Derivation C's circle, 2.8 km radius,

\gamma=7460

The physical picture

Start with what a relativistic force law is even allowed to look like. Newton's $\mathbf{F}=m\,d\mathbf{v}/dt$ fails not because the push is wrong but because the bookkeeping is: $d/dt$ is one observer's clock, and $\mathbf{v}$ one observer's slicing. Proper Time and Four-Velocity installed the fix as a doctrine — differentiate four-vectors by proper time, the only clock all observers agree on — so the law must read $dp^\mu/d\tau=f^\mu$ for some four-force $f^\mu$ . What can electromagnetism offer for the right-hand side? The field's local content is $F^{\mu\nu}$ ; the particle's local state is $u^\mu$ (and the scalar $q$ ). The simplest covariant object with one free index, linear in both — linear in the field because doubling the plates doubles the push, linear in the velocity because that is what the magnetic half of the 3+1 force already told us — is the contraction $qF^{\mu\nu}u_\nu$ . There is nothing else on the shelf: no other index arrangement gives a four-vector without extra ingredients. The law writes itself. The geometry of the contraction is the article's central image. Because $F$ is antisymmetric, the force it produces is always four-orthogonal to the velocity: $u_\mu F^{\mu\nu}u_\nu=0$ identically. In Minkowski geometry, "orthogonal to a timelike vector" means the force can only rotate the four-velocity on its hyperboloid $u\cdot u=c^2$ — never lengthen it. Electric fields perform hyperbolic rotations (boosts) of $u$ , magnetic fields perform circular rotations; both are rigid motions of the same hyperboloid. Everything in this article — mass constancy, speed saturation, the $\gamma$ in the cyclotron clock — is a corollary of that one picture: the Lorentz force steers; it never stretches.

Figure

The force that steers but never stretches — the four-velocity pinned to its hyperboloid

Hold on to one economic fact as the solutions unfold: momentum and energy never saturate, but speed does. A linac pushing electrons at fixed force gains almost no velocity after the first half meter — yet every additional meter deposits the same $qE\,dx$ of energy and the same $qE\,dt$ of momentum. Accelerator physics is the business of buying energy long after speed has sold out, and the hyperbolic worldline of Derivation B is its ledger.

The law and its reductions

The equation of motion for a point charge $q$ of mass $m$ :

\frac{dp^\mu}{d\tau}=qF^{\mu\nu}u_\nu, \qquad (1)

where: $p^\mu=mu^\mu$ is the four-momentum (Relativistic Energy and Four-Momentum), $u_\nu$ the covariant four-velocity $(\gamma c,-\gamma\mathbf{v})$ , and $F^{\mu\nu}$ carries the fields in this track's dictionary ( $F^{0i}=-E^i/c$ , $F^{ij}=-\epsilon^{ijk}B_k$ ; The Electromagnetic Field Tensor). Equation (1) is four equations, but only three are independent — the antisymmetry constraint $u_\mu f^\mu=0$ ties the time component to the space components, which is exactly as it should be: energy change is determined by momentum change on the mass shell. Note what (1) is not: it is not derived here from anything deeper. It is the relativistic completion of the experimentally anchored Lorentz force, argued for by uniqueness; its derivation from an action principle — where it drops out of one variation — is the next article's opening act (Relativistic Lagrangian and Action for Charged Particles).

Derivation AUnpacking the covariant force — exact Lorentz force, work-energy theorem, and mass constancy

We extract the physical content of equation (1): the spatial components reproduce the Lorentz force law exactly — on the relativistic momentum, with no $v/c$ corrections left behind — the time component is the work-energy theorem, and the antisymmetry of $F$ guarantees the rest mass is a constant of the motion. The strategy is straight component bookkeeping with the track's dictionary, keeping the $\gamma$ 's honest: every $d/d\tau$ costs one $\gamma$ when converted to lab time, and every contraction with $u_\nu$ supplies one. Watch them cancel in matched pairs — the reason the 3+1 force law comes out clean rather than $\gamma$ -dressed.

Step 1. Mass constancy first, since it licenses everything else. Contract (1) with $u_\mu$ :

u_\mu\frac{dp^\mu}{d\tau}=\frac{m}{2}\frac{d(u\cdot u)}{d\tau}=qF^{\mu\nu}u_\mu u_\nu=0,

the last equality because an antisymmetric tensor meets the symmetric pair $u_\mu u_\nu$ . So $u\cdot u$ stays pinned at $c^2$ (Proper Time and Four-Velocity's normalization is dynamically preserved), and with it $p\cdot p=m^2c^2$ : the field cannot alter the rest mass, only steer the four-momentum around its mass shell.

Step 2. Space components. For $\mu=i$ , expand the contraction with $u_\nu=(\gamma c,-\gamma\mathbf{v})$ :

f^i=q\left(F^{i0}u_0+F^{ij}u_j\right) =q\left[\frac{E^i}{c}(\gamma c)+(-\epsilon^{ijk}B_k)(-\gamma v_j)\right] =q\gamma\left[\mathbf{E}+\mathbf{v}\times\mathbf{B}\right]^i.

Why each sign lands: $F^{i0}=+E^i/c$ is the first column of the dictionary; $u_j=-\gamma v_j$ because lowering a spatial index costs a minus; and $\epsilon^{ijk}v_jB_k=(\mathbf{v}\times\mathbf{B})^i$ assembles the cross product with both minus signs spent.

Step 3. Convert the left side to lab time: $dp^i/d\tau=\gamma\,dp^i/dt$ . The $\gamma$ 's cancel:

\frac{d\mathbf{p}}{dt}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right), \qquad \mathbf{p}=\gamma m\mathbf{v}.

This is exact — the familiar Lorentz force survives relativity untouched, provided it acts on the relativistic momentum. All relativistic novelty hides inside $\mathbf{p}$ 's $\gamma$ .

Step 4. Time component. For $\mu=0$ : $f^0=qF^{0j}u_j=q(-E^j/c)(-\gamma v_j)=\gamma q\,\mathbf{E}\cdot\mathbf{v}/c$ , while $dp^0/d\tau=\gamma\,d(E/c)/dt$ with $E=\gamma mc^2$ the energy. Cancel the $\gamma$ 's and the $c$ :

\frac{dE}{dt}=q\,\mathbf{E}\cdot\mathbf{v}.

Only the electric field appears: magnetic forces do no work, not as a 3+1 curiosity but as a structural fact — $\mathbf{B}$ lives in the space-space block of $F$ , which the time slot of (1) never touches. The two reductions are not independent: Step 1's orthogonality says the time equation is the space equations' shadow on the mass shell, the same "zero-component" logic Relativistic Energy and Four-Momentum used for collisions, now for forces. Verified: contraction recomputed symbolically (sympy) with the full dictionary and arbitrary $\mathbf{E},\mathbf{B},\mathbf{v}$ : spatial slots return $q\gamma(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ , time slot returns $\gamma q\mathbf{E}\cdot\mathbf{v}/c$ , and $u_\mu f^\mu=0$ identically; matches the mech track's Lagrangian-route result (its verification note cross-references this component form) and Griffiths Eq. 12.127.

Before the solutions, the honest caveat that frames them. Equation (1) contains the external field only: an accelerated charge also radiates, and the radiated energy-momentum back-reacts. E9 priced the leak — the Larmor Formula and its relativistic completion (Relativistic Larmor Formula (Liénard's Result)) — and found it utterly negligible for the motions below (a 7-TeV LHC proton loses about 6 keV of its $7\times10^9$ keV per turn to synchrotron radiation). When the leak matters, the equation itself must be amended, and the chapter that tried is E9's radiation-reaction pair; here we solve (1) as it stands.

Figure

The three solved motions — hyperbola in E, slowed circle in B, drifting circle in crossed fields

Go deeper · Derivation B · 2 minUniform electric field — hyperbolic motion, the worldline of every linac

Derivation BUniform electric field — hyperbolic motion, the worldline of every linac

We solve equation (1) exactly for a charge released from rest in a uniform field $\mathbf{E}=E\hat{\mathbf{x}}$ , obtaining Born's hyperbolic motion, and read off the two numbers that govern linear accelerators: the distance scale $c^2/\alpha$ on which speed saturates, and the exact energy ledger $E_{\text{kin}}=qEx$ . The strategy exploits the structure just established: a pure electric field generates a boost of the four-velocity, so in rapidity language the equation must become trivial — rapidity simply grows linearly with proper time. Watch the contrast between the three clocks in the problem: proper time $\tau$ (linear in rapidity), lab time $t$ (exponentially outpaced), and distance $x$ (the honest energy meter).

Step 1. Reduce to two components. With $\mathbf{B}=0$ and motion along $x$ , equation (1) couples $u^0$ and $u^x$ only:

\frac{du^x}{d\tau}=\frac{qE}{mc}\,u^0, \qquad \frac{du^0}{d\tau}=\frac{qE}{mc}\,u^x,

(the first from $f^x=qF^{x0}u_0=q(E/c)(\gamma c)=qE\gamma=qEu^0/c$ ; the second from the time slot). Define the proper acceleration $\alpha\equiv qE/m$ — the acceleration the particle's instantaneous rest frame always measures.

Step 2. Solve by rapidity. The system is the derivative of a hyperbolic rotation, so set $u^0=c\cosh\eta(\tau)$ , $u^x=c\sinh\eta(\tau)$ (automatically on the shell $u\cdot u=c^2$ ). Both equations collapse to one line:

\frac{d\eta}{d\tau}=\frac{\alpha}{c} \qquad\Longrightarrow\qquad \eta=\frac{\alpha\tau}{c}.

Rapidity — the additive velocity variable of Lorentz Transformations — grows linearly forever. Nothing saturates in the natural variable; saturation is an artifact of reading $\eta$ through $v=c\tanh\eta$ .

Step 3. Integrate once more for the worldline (release from $x=c^2/\alpha$ , $t=0$ for cleanliness):

x=\frac{c^2}{\alpha}\cosh\frac{\alpha\tau}{c}, \qquad ct=\frac{c^2}{\alpha}\sinh\frac{\alpha\tau}{c} \qquad\Longrightarrow\qquad x^2-c^2t^2=\left(\frac{c^2}{\alpha}\right)^{\!2}.

The worldline is a hyperbola asymptotic to the light cone — the trajectory Spacetime and Minkowski Geometry used to calibrate its diagrams, now earned dynamically. Newton's parabola $x\approx\tfrac12\alpha t^2$ is the small- $\tau$ osculation; the light-line is the large- $\tau$ asymptote the particle chases forever.

Step 4. The accelerator ledger. From Step 3, $\gamma=\cosh(\alpha\tau/c)=\alpha x/c^2$ (measuring $x$ from the hyperbola's center), i.e. measuring distance $d$ from the release point,

\gamma(d)=1+\frac{\alpha d}{c^2}=1+\frac{qEd}{mc^2} \qquad\Longleftrightarrow\qquad E_{\text{kin}}=qEd\ \text{ exactly},

the time component of (1) integrated: energy gain equals charge times voltage drop, at any speed. Numbers for an electron in a $10$ MV/m gradient: $\alpha=1.76\times10^{18}$ m/s², saturation scale $c^2/\alpha=5.1$ cm; it passes $v=0.99c$ ( $\gamma=7.09$ ) at $d=(\gamma-1)c^2/\alpha=31$ cm, taking $1.20$ ns of lab time but only $0.45$ ns of its own; over SLAC's 3 km the same gradient banks $qEd=30$ GeV ( $\gamma\approx59{,}000$ ) while the speed creeps from $0.99c$ to $0.99999999985c$ . Speed is bought in the first foot; energy is bought by the mile. Verified: $\alpha$ , $c^2/\alpha$ , the $0.99c$ distance/time/proper-time triplet, and the 30 GeV figure recomputed numerically; hyperbola algebra checked against the mech track's free-motion limit and Griffiths §12.2.2 (Gaussian worked example transposed to SI).

Go deeper · Derivation C · 2 minUniform magnetic field, then crossed fields — the cyclotron clock and the invariant-powered drift trick

Derivation CUniform magnetic field, then crossed fields — the cyclotron clock and the invariant-powered drift trick

Two solutions in one because the second is built from the first. We solve (1) in a uniform $\mathbf{B}$ , finding the relativistic cyclotron: same circle as E4, but with the frequency demoted by $\gamma$ — the single fact that forced the synchrotron's invention. Then we solve crossed uniform fields $\mathbf{E}\perp\mathbf{B}$ ( $E<cB$ ) with no new integration at all: Lorentz Invariants of the Field guarantees a frame where the electric field vanishes; we boost there, run the cyclotron solution, and boost back. Watch the division of labor: the invariants article certified that the pure- $\mathbf{B}$ frame exists and which boost reaches it; this derivation spends that certificate as a solution method.

Step 1. Pure $\mathbf{B}=B\hat{\mathbf{z}}$ : the energy is frozen. The time slot of (1) gives $dE/dt=q\mathbf{E}\cdot\mathbf{v}=0$ , so $\gamma$ — and $|\mathbf{p}|$ , and $|\mathbf{v}|$ — are constants. The motion is steering at fixed speed.

Step 2. With $\gamma$ constant, the exact force law of Derivation A becomes linear precession: $\gamma m\,d\mathbf{v}/dt=q\mathbf{v}\times\mathbf{B}$ , i.e. $\mathbf{v}$ rotates about $\hat{\mathbf{z}}$ at

\omega=\frac{qB}{\gamma m}, \qquad r=\frac{v}{\omega}=\frac{p_\perp}{qB}.

where: the radius law $r=p_\perp/qB$ is unchanged from the nonrelativistic one when written in momentum — the magnetic rigidity of accelerator jargon — while the frequency inherits the full $1/\gamma$ . E4's Cyclotron Motion and Magnetic Confinement is the $\gamma\to1$ anchor; its constant-frequency machine breaks exactly when $\gamma$ leaves 1.

Step 3. Numbers that built physics. A classic cyclotron drives protons at fixed $\omega_c=qB/m$ ; by $\gamma\approx1.1$ ( $\sim$ 90 MeV) the particle's actual frequency has slipped 10% out of phase — dead. The fix is to ramp: hold $r$ fixed and sweep $B$ and the RF together as $\gamma$ grows — the synchrotron. At the LHC: $\gamma=7460$ for 7 TeV protons, $p=3.74\times10^{-15}$ kg·m/s, and in the $B=8.33$ T dipoles $r=p/qB=2.80$ km — the machine's actual bending radius. The revolution-scale frequency $qB/\gamma m=1.07\times10^5$ rad/s (versus the nonrelativistic $8.0\times10^8$ ): the $\gamma$ in the clock is a factor of seven thousand.

Step 4. Crossed fields by boost. Take $\mathbf{E}=E\hat{\mathbf{y}}$ , $\mathbf{B}=B\hat{\mathbf{z}}$ with $E<cB$ . The invariants: $\mathcal{I}_2\propto\mathbf{E}\cdot\mathbf{B}=0$ and $\mathcal{I}_1\propto B^2-E^2/c^2>0$ — by the classification theorem of Lorentz Invariants of the Field, a frame with $\mathbf{E}'=0$ exists, reached by boosting at the drift velocity $\mathbf{v}_d=\mathbf{E}\times\mathbf{B}/B^2$ (magnitude $E/B$ ). In that frame the surviving field is pure magnetic with $B'=\sqrt{B^2-E^2/c^2}=B\sqrt{1-(E/cB)^2}$ — the invariant itself, cashed as a field strength. The particle there runs Step 2's circle at $\omega'=qB'/\gamma' m$ ; boosting back superposes the drift: a relativistic cycloid. Numbers: $E=3.0$ MV/m, $B=20$ mT gives $E/cB=0.50$ , drift $v_d=E/B=1.5\times10^8$ m/s $=0.50c$ ( $\gamma_d=1.15$ ), and $B'=17.3$ mT; the lab-frame invariant $B^2-E^2/c^2=3.0\times10^{-4}$ T² equals $B'^2$ to the last digit — the frame-proof receipt. If instead $E>cB$ , no such frame exists (the invariant flips sign); the drift construction fails and the motion is a boosted electric problem — Derivation B's hyperbola in disguise. Verified: LHC $\gamma$ , $p$ , $r=2803$ m, and both frequencies recomputed from CODATA constants; crossed-field numbers ( $v_d=0.500c$ , $\gamma_d=1.155$ , $B'=17.3$ mT, invariant $2.999\times10^{-4}$ T² both frames) recomputed numerically; classification logic quoted by name from the invariants article, not re-derived.

Limiting cases

case	result	meaning
$v\ll c$ , pure $\mathbf{E}$	$x\approx\tfrac12\alpha t^2$	Newton's parabola osculates the hyperbola
$\tau\to\infty$ , pure $\mathbf{E}$	$v\to c$ , $\eta=\alpha\tau/c\to\infty$	speed saturates; rapidity, momentum, energy never do
$\gamma\to1$ , pure $\mathbf{B}$	$\omega\to qB/m$	E4's cyclotron recovered — fixed-frequency machines live here
$E\to cB$ in crossed fields	$v_d\to c$ , $B'\to0$	drift frame runs away to the light cone; null-field borderline
$E>cB$ crossed	pure- $\mathbf{E}$ frame exists instead	hyperbolic motion in a boosted dress
radiation losses included	(1) needs amendment	E9's radiation-reaction pair; negligible here (LHC: $\sim$ 6 keV of 7 TeV per turn)

Common trapThere is no relativistic mass doing the pushing — the gamma lives in the momentum, not in m

The seductive wrong reading of these results is "the particle gets heavier, so it responds less." It leads to real errors: if you write $F=\gamma m\,a$ you get the wrong answer whenever force and velocity are not perpendicular — the honest relation splits, $f_\perp=\gamma m\,a_\perp$ but $f_\parallel=\gamma^3 m\,a_\parallel$ , which is why early literature carried the fossil terms "transverse mass" and "longitudinal mass" before abandoning the idea as unworkable. The clean statement, and the only one this track uses: the mass $m$ is the invariant $p\cdot p=m^2c^2$ , dynamically protected by Derivation A's orthogonality theorem — the field cannot change it. What grows with speed is the momentum $\gamma m\mathbf{v}$ and energy $\gamma mc^2$ ; what the Lorentz force acts on is exactly that momentum, with no correction, $d\mathbf{p}/dt=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ . The cyclotron frequency $qB/\gamma m$ is not "heavier particle, slower circle" — it is "same steering force spread over a momentum $\gamma$ times larger."

Worked examples

Example 1 — the first meter of a linac. An electron starts from rest in a uniform $E=10$ MV/m gradient. Where does it cross $v=0.99c$ , and what has each clock recorded?

Step 1. Proper acceleration: $\alpha=eE/m=(1.602\times10^{-19})(10^7)/(9.109\times10^{-31})=1.76\times10^{18}$ m/s². Saturation length: $c^2/\alpha=5.1$ cm — the machine's entire velocity story happens on this scale.

Step 2. Target $\gamma=1/\sqrt{1-0.99^2}=7.09$ . Distance: $d=(\gamma-1)c^2/\alpha=6.09\times0.0511=\mathbf{0.31\ m}$ . Energy check: $qEd=10\ \text{MV/m}\times0.311\ \text{m}=3.11$ MeV $=(\gamma-1)mc^2=6.09\times0.511$ MeV ✓ — the ledger closes exactly.

Step 3. Clocks: lab time $t=(c/\alpha)\sinh\eta$ with $\eta=\cosh^{-1}(7.09)=2.65$ : $t=1.20$ ns; proper time $\tau=(c/\alpha)\eta=0.45$ ns — the electron ages at less than half the lab rate, averaged over its first foot of life. Over the remaining 3 km at this gradient it banks $qEd=30$ GeV while gaining less than $10^{-7}c$ of speed: all energy, no velocity.

Example 2 — reading an LHC dipole. A 7-TeV proton enters a dipole field $B=8.33$ T. Find its orbit radius and revolution-scale angular frequency, and check the cyclotron-death argument.

Step 1. $\gamma=E/mc^2=7\times10^{12}\ \text{eV}/938.3\times10^6\ \text{eV}=7460$ ; momentum $p=\sqrt{E^2-(mc^2)^2}/c=3.74\times10^{-15}$ kg·m/s (at this $\gamma$ , $pc$ and $E$ agree to $10^{-8}$ ).

Step 2. Radius from the rigidity law: $r=p/qB=3.74\times10^{-15}/(1.602\times10^{-19}\times8.33)=\mathbf{2.80\ km}$ — matching the LHC's actual bending radius of 2803 m; the ring's 27 km circumference is this arc plus straight sections.

Step 3. Frequency: $\omega=qB/\gamma m=(7.98\times10^8)/7460=1.07\times10^5$ rad/s. A fixed-frequency cyclotron driving protons at $\omega_c=qB/m$ would be out of phase by a factor of 7460 — the RF must track $\gamma(t)$ as the beam ramps, which is precisely what a synchrotron's control system does: $B$ from 0.54 T at injection to 8.33 T at top energy, RF locked to the slipping clock the whole way.

Going further

Reading. Griffiths, Introduction to Electrodynamics, 4th ed., §12.2.2 and §12.3.4 — hyperbolic motion and the covariant force. Jackson, Classical Electrodynamics, 3rd ed., §12.1–12.4 — motion in uniform and crossed fields, with the drift construction. Zangwill, Modern Electrodynamics, §22.6. Landau & Lifshitz, Classical Theory of Fields, §§17–22 — the solutions at full speed.

Provenance. Built from the AllQFT base article for this topic (covariant law, both reductions, hyperbolic and cyclotron solutions retained and extended), components locked to this track's $(+,-,-,-)$ dictionary. A verifier should: (1) recompute Derivation A's contraction with the dictionary and confirm the exact (γ-free) 3+1 force law plus $dE/dt=q\mathbf{E}\cdot\mathbf{v}$ ; (2) rerun the linac numbers ( $\alpha=1.76\times10^{18}$ m/s², $c^2/\alpha=5.11$ cm, $d(0.99c)=0.311$ m, $t=1.20$ ns, $\tau=0.451$ ns, 30 GeV over 3 km at 10 MV/m); (3) rerun the LHC numbers ( $\gamma=7460$ , $p=3.74\times10^{-15}$ kg·m/s, $r=2803$ m, $\omega=1.07\times10^5$ rad/s) and the crossed-field set ( $v_d=0.50c$ , $\gamma_d=1.155$ , $B'=17.3$ mT, invariant $3.0\times10^{-4}$ T² frame-independent). Division of labor for this edition: the equation's statement-level previews live in Relativistic Energy and Four-Momentum and The Electromagnetic Field Tensor (delivered here as executed dynamics); its derivation from an action is owned by Relativistic Lagrangian and Action for Charged Particles (next); the drift-frame existence theorem by Lorentz Invariants of the Field (spent here as a method); the nonrelativistic cyclotron by E4's Cyclotron Motion and Magnetic Confinement; radiation losses and their back-reaction by E9's Larmor Formula, Relativistic Larmor Formula (Liénard's Result), and the radiation-reaction pair; Thomas precession and spin dynamics by the atomic/quantum tracks. This article owns the uniqueness argument, both exact reductions with the mass-constancy theorem, and the three solved motions.

Builds on: Four-Vectors and the Metric Tensor · Proper Time and Four-Velocity · Relativistic Energy and Four-Momentum · The Electromagnetic Field Tensor · Lorentz Transformations

Leads to: Relativistic Lagrangian and Action for Charged Particles · The Electromagnetic Stress-Energy Tensor · Conservation Laws and Energy-Momentum Flow · Transformation of Electric and Magnetic Fields

Where does the covariant force law actually come from — is it derived or postulated?

At this article's level it is the relativistic completion of an experimental law: we argued by uniqueness (four-vector, linear in $u$ , built from $F$ and $q$ alone) that $qF^{\mu\nu}u_\nu$ is the only candidate, and checked it reduces to the tested 3+1 Lorentz force. That is strong motivation, not derivation. The genuine derivation arrives in the next article: postulate the interaction action $-q\int A_\mu dx^\mu$ — one gauge-invariant scalar — and equation (1) drops out of the variation, with $F$ born antisymmetric in the process. Deeper still, quantum electrodynamics derives the classical law as the expectation-value dynamics of its own minimal coupling. Physics tends to promote its force laws to consequences; this one has been promoted twice.

Is the magnetic-does-no-work statement really exact, even relativistically?

Exactly exact, and the covariant bookkeeping shows why with unusual clarity: the energy-change slot of equation (1) is $f^0=qF^{0j}u_j$ , and the magnetic field simply does not live in the $F^{0j}$ entries — it occupies the space-space block, which the time slot never contracts with. All the familiar 3+1 puzzles (the crane lifting iron with a magnet "doing work") resolve the same way they did in E4: the energy is delivered by induced electric fields or by the sources maintaining currents, with the magnetic field acting as a broker. What relativity adds is frame honesty — a field purely magnetic for you has an electric part for a moving observer (Transformation of Electric and Magnetic Fields), so who does the work is frame-dependent even though each frame's ledger closes perfectly.

A uniformly accelerated observer never reaches c, but the rapidity grows without bound. What does the particle itself experience?

A perfectly boring constant push — that is the meaning of proper acceleration $\alpha$ . In its instantaneous rest frame the particle always measures the same $qE/m$ ; relativity never tells it "you are near the speed limit." The saturation is entirely in the lab's description, because velocity is a bounded function ( $c\tanh\eta$ ) of the honestly additive variable. Two famous consequences hide in the hyperbola: the asymptote means a sufficiently head-started light ray never catches the particle (a horizon — the Rindler horizon of accelerated frames), and the proper-time integral converges slowly enough that a 1-g "acceleration" sustained for mere decades of proper time formally crosses the visible universe. Both are geometry of Derivation B's worldline, no new physics required.

Why does the LHC use magnetic fields to steer but electric fields to accelerate?

Read the two reductions side by side. Acceleration is energy gain, and only $\mathbf{E}$ appears in $dE/dt=q\mathbf{E}\cdot\mathbf{v}$ — so RF electric fields (in cavities) do all the energizing. Steering at fixed energy is momentum rotation, and there the force comparison is brutal at high speed: a magnetic force $qvB$ with $v\approx c$ and $B=8$ T matches an electric field of $cB\approx2.4\times10^9$ V/m — three orders of magnitude beyond breakdown limits of real hardware. Magnets are simply the only affordable way to bend a stiff beam; the rigidity law $r=p/qB$ then fixes the entire geography of the machine. The division of labor in an accelerator is the two slots of equation (1) turned into engineering departments.

Does a uniformly accelerated charge radiate, and did we cheat by ignoring it?

It radiates — the Larmor Formula machinery applies, and its relativistic completion gives finite power for hyperbolic motion — and no, we did not cheat by much: for every motion solved here the radiated energy is a negligible perturbation (the LHC's 6 keV per 7-TeV turn is the extreme case that still rounds to zero). But the question you are circling is one of physics' famous subtleties: the uniformly accelerated charge's radiation interacts nontrivially with the equivalence principle and with the horizon structure of Born's worldline, and the self-consistent inclusion of the back-reaction is exactly where the innocent equation (1) breaks down. That story — energy budgets, the Schott term, runaway pathologies — is owned by E9's radiation-reaction pair and their E11 deep dives; this article stops where the external-field approximation stops.

How does the crossed-field drift trick generalize when the fields are not uniform?

Into one of plasma physics' load-bearing approximations: guiding-center theory. If the fields vary slowly on the scale of the gyro-radius and gyro-period, the motion decomposes — locally — into Derivation C's picture: a fast circle around a "guiding center" that drifts. The $\mathbf{E}\times\mathbf{B}$ drift survives verbatim; field gradients and curvature add their own drifts ( $\nabla B$ drift, curvature drift), each derivable by the same boost-and-average logic. The magnetic moment of the gyro-circle becomes an adiabatic invariant, which is why magnetic mirrors and tokamaks confine at all. Every fusion device and every radiation-belt model is, at heart, this article's Derivation C plus perturbation theory.

Why is there no Thomas precession or spin in this article?

Because equation (1) governs a structureless point charge: its state is position and four-momentum, nothing else. Real electrons carry spin — an intrinsic angular momentum with a magnetic moment — and a spinning particle's rest frame, followed along the accelerated orbit, rotates relative to the lab by a purely kinematic amount (Thomas precession, a composition-of-boosts effect) on top of the dynamical precession its moment performs in $\mathbf{B}$ . The combined covariant description (the BMT equation) is the natural sequel to this article, but it belongs to the atomic and quantum tracks, where the factor it famously repairs — the electron's spin-orbit coupling, off by 2 without Thomas's kinematics — actually lives. The clean split: orbit here, spin there.

Is the equation still valid for a charge moving through matter rather than vacuum fields?

Equation (1) is exactly as valid as your knowledge of the local microscopic field at the particle: it holds with $F^{\mu\nu}$ the total field from all sources, external and induced. What changes in matter is bookkeeping economy — you rarely know the microscopic field, so you use averaged fields plus effective forces: stopping power for the accumulated kicks of atomic electrons, Cherenkov drag once the charge outruns light's phase velocity in the medium, transition radiation at boundaries. Each is equation (1) statistically processed. The covariant treatment of the averaged fields in moving matter — what replaces $F$ , and how $\mathbf{D}$ and $\mathbf{H}$ join the tensor family — is precisely the business of Tensor Formulation of Fields in Media, three articles ahead.