Illustrated Physics
physics, drawn
Series

Length contraction — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

    The length that depends on when, and is not really about length

    A moving object is measured shorter. The contraction is real, it is not an illusion of light travel time, and it turns out to be a disagreement about simultaneity wearing a different costume.

    part 1 · relativity
  2. Simultaneity at β = 0.866. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

    The pole that fits and does not fit

    A twenty-metre ladder is carried through a ten-metre barn at 0.866 of light speed, and both doors shut behind it. In the ladder's own frame the barn is five metres long and there is plainly no room. Both accounts are correct, and the doors' closings are 57.8 nanoseconds apart in one of them.

    part 2 · relativity
  3. A cube photographed at four speeds. The outline a camera records of a cube passing at β = 0, β = 0.4, β = 0.7, β = 0.9, moving to the right, seen at the moment it passes. The dashed square is what the usual picture shows: the cube flattened along its motion by the Lorentz factor. It is not what the camera gets. Light from the far face left earlier than light from the near face, by the time it needed to cross the extra distance, and the cube moved in that interval — so the far face appears displaced backwards and the side of the cube comes into view. The two effects together are exactly a rotation: at β = 0.4 the cube photographs as one turned 23.6°, at β = 0.7 the cube photographs as one turned 44.4°, at β = 0.9 the cube photographs as one turned 64.2°. The contraction has not gone anywhere — a ruler laid alongside still reads 43.6 per cent of the proper length at the highest speed drawn — but it is a statement about two positions at one time, and a photograph is not that.

    The contraction no photograph shows

    Every textbook picture of a relativistically moving object drawn between 1905 and 1959 showed it squashed. A camera would have shown it turned. The contraction is entirely real in the measurement that defines it, and a photograph is not that measurement.

    part 3 · relativity
  4. Two rockets that keep their distance, and the string that does not. Two rockets 0.5 unit apart in the laboratory, given identical acceleration programmes there, drawn in units where the light speed is one and c²/a is one. Their laboratory separation is constant for ever — the two worldlines are the same curve shifted sideways, and every horizontal line meets them 0.5 apart. The slanted lines are the rockets' own lines of simultaneity, and the distance between the worldlines measured along those is what a string tied between them has to span: at 0.3c it is 0.512, a stretch of 2 per cent; at 0.6c it is 0.557, a stretch of 11 per cent; at 0.8c it is 0.631, a stretch of 26 per cent; at 0.9c it is 0.710, a stretch of 42 per cent. The γL that is always quoted — 0.524, 0.625, 0.833, 1.147 here — is the limit of that measurement for a vanishing gap, and at a gap of 0.5 in these units it overstates the stretch by up to 38.1 per cent; shrinking the gap a hundredfold brings the two within 0.46 per cent. Either way the string is stretched and breaks, while the gap in the laboratory never changes by a millimetre. Length contraction is not something that happens to a rod. It is a statement about which events count as simultaneous, and a rod that is not allowed to contract is a rod that is being pulled apart.

    The string that breaks between two rockets

    Two rockets a metre apart, given identical acceleration programmes, stay a metre apart in the laboratory for ever. A string tied between them breaks anyway. Nothing pulls on it, nothing in the laboratory moves relative to anything else, and the string is stretched — because the distance it has to span is measured on the rockets' slices of simultaneity and not on the laboratory's.

    part 4 · relativity
  5. 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.

    part 5 · relativity

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