Illustrated Physics
physics, drawn
Concept

Sagnac effect — where it appears

The phase difference between two beams sent in opposite directions around a rotating loop, proportional to the rotation rate times the enclosed area. It is how optical and matter-wave gyroscopes measure rotation, and a neutron interferometer on the rotating Earth must correct for it.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A circumference that is more than 2π times the radius. The ratio of a rotating disc's measured circumference to 2π times its measured radius, against the speed of the rim, together with the rate of a clock carried on the rim. Rulers laid round the rim lie along their own direction of motion and are contracted; rulers laid along a radius lie across it and are not. So the circumference takes more of them than a stationary observer counts and the radius takes the same number, and the ratio is γ: 1.091 at β = 0.4, 1.400 at β = 0.7, 2.294 at β = 0.9. The geometry a rotating observer measures is therefore not Euclidean, and it is not Euclidean by an amount that depends on where on the disc the measurement is made. That is the observation Einstein said set him on the road to describing gravity with curved geometry: here is an accelerated frame, and here is a geometry in it that no choice of Cartesian coordinates can flatten. The rim's clock runs slow by the same factor, so a rotating frame has neither a common time nor a flat space.

The disc that cannot be spun

Set a disc turning and measure its circumference with rulers carried on the rim. They lie along their own direction of motion and are contracted, so more of them fit; rulers along a radius lie across the motion and are not. The ratio of circumference to radius is therefore not two pi, in a frame where nothing is happening but rotation — and Einstein said that was what set him looking for gravity in geometry.

relativity · Length contraction Two paths through a neutron interferometer at two heights. A neutron interferometer cut from one silicon crystal, tilted by 30 degrees about its incoming beam so that one path runs higher than the other. Left: the two paths, split at the first slab, turned at the second and recombined at the third, enclosing 10.1 square centimetres; the heights are drawn exaggerated. Taken as a rectangle of the same area, the upper path runs 15.8 mm higher for 3.2 cm. There a neutron of wavelength 1.445 Å, moving at 2738 m/s, is slower by 56.5 micrometres per second, so its wavelength is longer by 20.6 parts per thousand million. Over 3.2 cm that accumulates 28.7 radians less phase than the lower leg — 4.6 whole fringes — computed by integrating the local wavenumber along both legs and checked against 2πm²gλA sin α / h².

The fall that leaves the mass in the phase

Every body falls the same way whatever its mass, and a neutron is no exception. But a neutron is also a wave, and the phase that wave accumulates while falling depends on the mass — as its square, at a fixed wavelength. Tilt a neutron interferometer so that one path runs a centimetre higher than the other and the neutrons swing between its two detectors, which in 1975 was the first measurement in which gravity and quantum mechanics both had to be right at once.

quantum · Matter waves Two straight worldlines that meet twice. A spacetime diagram of a universe whose space is a circle 10 light-years round, drawn unrolled: the shaded strip is the whole of space, and the strips beside it are the same space again, so the dashed lines are the same stay-at-home twin A seen in the copies. Twin B leaves A at 0.6c and never changes speed or direction. After 16.67 years by A's clock B has gone once round and meets A again — the line reaches A's copy at 10 light-years. Neither twin has accelerated at any moment; both worldlines are straight. A has aged 16.67 years and B 13.33, less by the factor 1/γ = 0.800. The usual resolution of the twin paradox, that the travelling twin is the one who turned round, is not available: here nobody turned.

The twin who never turns round

The twin paradox is usually settled by pointing at the turn: the travelling twin is the one who fired the engines and came back, and so the one whose worldline is not straight. Put the twins in a universe whose space closes on itself, like the surface of a cylinder, and the traveller can come home by going straight on. Neither twin ever accelerates, both worldlines are straight, and the traveller still comes back younger. What decides it is not any motion but the shape of space, which picks out one state of motion as special — and hands every observer a way to measure it.

relativity · Time dilation

Named alongside it

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

Equivalence principleSimultaneityTime dilationAccelerationActionAtom interferometryBorn rigidityEhrenfest paradoxGeneral relativityGeodesicGravimetryLength contraction

All concepts