Illustrated Physics
physics, drawn
Concept

Relativistic momentum — where it appears

Momentum as special relativity defines it, γmv, which grows without limit as a body's speed approaches that of light. Near light speed a large change of momentum is a tiny change of speed, which is why accelerators gain energy without gaining much speed.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Thrown across a uniform field, a fast charge does not draw a parabola. The paths of a charge launched across a uniform electric field (which points up the page) at 0.30, 0.60, 0.90, 0.99 of the speed of light, from the exact relativistic motion (solid), beside the parabolas Newton's law gives for the same launches (dashed); lengths are in units of mc²/qE, the distance over which the field does work equal to the charge's rest energy. Near its launch each exact path is a parabola, but one with the inertia γ₀m in place of m, so it is deflected less than Newton says — 1.05, 1.25, 2.29, 7.09 times less for the four. Further out each turns steeply along the field: one unit across, the 0.3c charge is 11.6 units along where Newton says 5.6, because its sideways speed has fallen while the field kept pushing. Each exact path is a hyperbolic cosine. The field adds to the momentum along it while the sideways momentum stays fixed, and the charge's growing energy is a growing inertia that slows its sideways progress.

The throw a uniform field bends into a chain

Throw a stone sideways across the Earth's gravity and it draws a parabola. Throw a charge sideways across a uniform electric field and, at low speed, it does the same. At high speed it draws a different curve, a hyperbolic cosine — the shape of a chain hanging between two posts, stretched — and the reason is that nothing pushes it sideways yet its sideways speed falls. Its sideways momentum is fixed, the field keeps adding energy, and energy is inertia. The charge gets only logarithmically far across the field however long it flies, and its sideways distance turns out to be a reading of its own clock.

relativity · Relativistic dynamics A longer road against a faster car. For a proton with 0.1% more momentum than the design proton in a ring 628 m round, the change in its time per turn, in picoseconds, against the Lorentz factor: from its longer orbit, which adds about 56 ps a turn at every energy; from its higher speed, which subtracts 605 ps at γ = 2 but only 5.2 ps at γ = 20, because near the speed of light more momentum buys almost no more speed; and the net, computed exactly. The speed wins below γₜ = 6.1 and the path wins above it. In special relativity a body can always be given more momentum, but its speed is capped, and that is the whole cause.

The energy above which faster arrives later

In a racing circuit a faster car laps sooner. In a proton synchrotron it does too, at first — and then, past a definite energy, the faster proton laps later. Extra momentum sends a proton round a slightly longer orbit and makes it slightly faster, and relativity decides which wins: at low energy the speed does, but near the speed of light more momentum buys almost no more speed, and the longer road takes over. At the crossing, called transition, a spread of momenta stops producing a spread of arrival times, the bunch stops oscillating in time, and the accelerating wave has to jump from one slope to the other in a few turns. The first machine to try it did so in 1959.

relativity · Relativistic dynamics

Named alongside it

The objects these essays reach for when they reach for this one.

CatenaryElectron beamHyperbolic functionsInertiaThe Lorentz factorThe Lorentz forceMomentum compactionParticle acceleratorPhase stabilityProper timeSlip factorSynchrotron

All concepts