Illustrated Physics
physics, drawn
Concept

Hyperbolic geometry — where it appears

The non-Euclidean geometry of a surface curved like a saddle everywhere, in which the angles of a triangle add to less than 180°. It appears in relativity as the geometry of velocities and of light-travel times inside an accelerating rocket.

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 Light inside an accelerating rocket travels on circles. Rays of light leaving one point inside a rocket with constant acceleration, drawn in the rocket's own coordinates: height above the horizon up the page, in units of c²/a (one light-year for one gravity), and distance across. Each ray is a straight line in an inertial frame, transformed event by event. Launched across the rocket (0°) or at +30°, +60°, −30°, −60°, every ray is an exact semicircle centred on the horizon, with radius 1.000, 1.155, 2.000, 1.155, 2.000: the rays bend down towards the horizon and meet it at right angles. A ray sent straight up stays straight. These are the straight lines of the hyperbolic plane, although the spacetime inside the rocket is flat.

The rocket in which light travels on circles

Inside a rocket that accelerates for ever at a constant rate, a beam of light sent across the cabin sags towards the floor, as Einstein argued it must if acceleration and gravity are indistinguishable. Followed further than any cabin, the sag turns out to be exact geometry: every ray inside the rocket is a perfect semicircle centred on the horizon below, and the rays obey the rules of a non-Euclidean plane in which the angles of a triangle add to less than two right angles. The spacetime in the rocket is flat; the curvature belongs to the clocks.

relativity · Accelerated frames

Named alongside it

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

GeodesicBoostCurvatureEquivalence principleFermat's principleGravitational time dilationHolonomyLight deflectionThe Lorentz factorThe Lorentz transformationRapidityReference frames

All concepts