Illustrated Physics
physics, drawn
Concept

Satellite navigation — where it appears

Finding a position from the arrival times of radio signals from satellites whose orbits and clocks are known. It works only because the satellites' clocks are corrected for relativity, and precise positioning also corrects the carrier phase for the rotation of the antennas.

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

The clock that gains going one way and loses going the other. The rate at which a flown clock gains on a clock left at 30° latitude, in nanoseconds per hour, against the aeroplane's ground speed, with east taken as positive. Two terms are drawn and then their sum. Height alone gives 3.5 nanoseconds an hour at 9 km and does not care which way the aircraft is pointed. Motion costs time, and because the ground is already moving eastward at 402 metres a second, flying east adds to that speed and flying west subtracts from it — so the kinematic term is much larger going east and can change sign going west. The sum crosses zero at 180 metres a second eastward, which is the ground speed at which an aeroplane's clock keeps the time of the airfield it left. Over the two flights Hafele and Keating actually made, this simple model gives -61 nanoseconds eastward and +304 westward, against their own predictions of -40 and +275 and their measurements of -59 and +273. The model here uses one average altitude, one average speed and one latitude, where the real prediction integrated the flight logs; getting the signs and the rough sizes out of three lines of arithmetic is the point, and the last twenty per cent is what the logs are for. What no amount of arithmetic supplies is the thing the experiment settled: that the effect is real, that it acts on a caesium clock in a passenger seat, and that a difference of a few hundred nanoseconds after two days is measurable.

The two clocks that flew in opposite directions

Two caesium clocks were flown round the world in 1971, one each way, and came back disagreeing with the clock left behind — one having lost 59 nanoseconds and the other gained 273. Height alone would have made both gain. The sign flip comes from the ground already moving eastward at 400 metres a second before the aircraft took off.

relativity · Time dilation Light whose crests wind round its axis. Cross-sections of three beams carrying orbital angular momentum, with phase exp(iℓφ) round the axis, for ℓ = 1, 2 and 3, at one instant: solid lines where the field is at a crest, dashed where it is at a trough. Along the beam the crests form ℓ interleaved helices, and at a fixed plane the pattern turns about the axis once every ℓ optical periods. Looked at from a frame turning about the axis at a rate Ω, every crest arrives ℓΩ/2π times a second more or less often: the field's frequency is shifted by ℓΩ. That is a Doppler shift with no approach and no recession — the angular counterpart of k·v, with the winding number ℓ standing in for the wavenumber and the rotation rate for the speed. Each photon of such a beam carries ℓħ of orbital angular momentum about the axis.

The Doppler shift with nothing coming closer

Every Doppler shift so far has needed something to approach or recede. Light can be shifted in frequency by something that does neither: a surface spinning in place, or a polarising plate turning in a beam that passes straight through it. A beam whose wavefronts wind round its axis, ℓ times per wavelength, comes back from a surface spinning at Ω shifted by exactly ℓΩ, at every distance from the axis. A circularly polarised beam through a spinning half-wave plate comes out shifted by exactly 2Ω, because the plate takes two units of angular momentum from each photon and does work on it. The shift is tiny and perfectly measurable, and satellite navigation has to correct for it.

waves · Doppler

Named alongside it

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

Angular momentumAtomic clockCircular polarisationDoppler effectEquivalence principleFrequency shiftGeometric phaseGravitational redshiftHafele keatingInterferometryOrbital angular momentumProper time

All concepts