Illustrated Physics
physics, drawn
Concept

Boost — where it appears

A change to a frame moving at constant velocity relative to the first, mixing the time and space coordinates as a rotation mixes two spatial ones. Quantities built from lengths and energies transform under it; quantities that are counts do not.

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

Every speed there is, on one disc. The whole of velocity space drawn as a disc: the boundary is the speed of light and every possible velocity is a point inside. The rings are equal steps of rapidity — 0.5, 1, 1.5, 2, 2.5 — and they sit at speeds 0.4621, 0.7616, 0.9051, 0.9640, 0.9866 of light. Equal steps of rapidity crowd towards the edge, checked ring by ring, which is the same fact as speeds refusing to add: a boost is a fixed step in rapidity and a shrinking step in speed. The drawing is the Poincaré model, in which angles are true and distances are not — so a shape near the rim is drawn small and is not small, and the boundary is infinitely far away in the geometry although it is a finite circle on the page.

The space that speeds live in

Speeds do not add, and the reason is that the set of all possible velocities is not a flat space. It is a hyperbolic plane of curvature minus one in rapidity — and the rotation two boosts leave behind is exactly the area of the triangle they make in it.

relativity · Velocity addition What a boost leaves alone, and what it does not. How each quantity of a box of blackbody radiation changes when the observer moves at 0.8 of the speed of light, a Lorentz factor of 1.667, on a logarithmic axis with one at the centre. The top four do not change at all, and the reason is the same in each case: they are counts, or logarithms of counts, or invariants built from four-vectors. A number of photons is a number, and every observer arrives at the same number. The rest change, by powers of the Lorentz factor that follow from the first four. And the entry that matters is the last two together: the energy density rises as the square of the factor while the entropy density rises as the factor itself, so the ratio between them that would define a temperature does not stay fixed — which is why the boosted radiation cannot be a blackbody at any temperature at all.

The count that no observer can disagree about

A moving body, it turns out, has no temperature. What it does have is an entropy, and every observer agrees about it — because entropy is the logarithm of a count of arrangements, and a count is a number. That one invariant, with energy and momentum being parts of one object, is enough to compute everything a temperature could not: what happens to the energy density, the entropy density, and the relation between them that having a temperature consists of.

relativity · Relativistic thermodynamics

Named alongside it

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

Charge invarianceCountingCurvatureEnergy densityEntropyEquilibriumFour-vectorGeodesicHolonomyHyperbolic geometryInvarianceThe Lorentz factor

All concepts