Illustrated Physics
physics, drawn
Series

Length contraction — the series

8 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 pole is carried through a ten-metre barn at 0.866 of light speed, and both doors shut behind it. In the pole'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
  6. An orbit that flattens without being told to. The path of a classical electron relative to its nucleus, in the laboratory, before and after the nucleus has been accelerated gently — over 200 orbits — from rest to 0.6 of the speed of light along the horizontal, computed from Maxwell's field of a moving charge and the relativistic law of motion alone. Before, the orbit is a circle of radius 1. After, it is an ellipse 0.810 wide along the motion and 1.012 across it, a ratio of 0.8003 against 1/γ = 0.8000, and one turn takes 1.2504 times as long as before, against γ = 1.2500. Nothing in the calculation assumed a contraction or a slowing clock: the forces between the moving charges produced both.

    The contraction the forces work out for themselves

    Relativity usually introduces length contraction as a consequence of how moving observers disagree about simultaneity, not as something that happens to matter. There is a second way to reach the same number, and it is worth knowing because it is the way the idea was first found. Take an atom, accelerate its nucleus gently, and compute what the electron does using only Maxwell's field of a moving charge and the law of motion. Nobody tells the orbit to contract, and it does, by exactly 1/γ, while each turn takes exactly γ times as long. The forces holding matter together work out the contraction for themselves.

    part 6 · relativity
  7. Two factors of γ between a magnet and an X-ray. An electron of 6 GeV (γ = 11742) crossing an undulator, a row of magnets alternating every λᵤ = 3 cm. Top: the laboratory, where the electron wiggles once per period. Middle: the electron's own frame, where the row is contracted by γ to 2.55 μm and rushes past at nearly the speed of light; to the electron it is a wave of that wavelength, and the electron radiates at it, in the infrared. Bottom: that light, emitted forward and seen back in the laboratory, is Doppler-shortened by a further factor of about 2γ, to λᵤ/2γ² = 1.088 Å; the wiggle's own slowing of the electron adds a factor (1 + K²/2), giving 1.632 Å for a deflection parameter K = 1 — hard X-rays from centimetre magnets. The drawing is schematic: the three periods differ by factors of about twelve thousand and a hundred million, and cannot be drawn to one scale.

    The magnet period that comes out as an X-ray

    A row of permanent magnets alternating every three centimetres, with a beam of six-gigaelectronvolt electrons sent down the middle, shines X-rays with a wavelength of an ångström and a half — two hundred million times shorter than the spacing of the magnets that made them. The factor is γ squared, and the two γ's come from two different frames. In the electron's frame the row of magnets is contracted by γ and rushes past it as a wave; the electron shakes in step and radiates; and that light, sent forward, comes back to the laboratory shortened by another factor of 2γ. Length contraction is half of every X-ray a modern synchrotron makes.

    part 7 · relativity
  8. The photon a fast electron throws back. The highest energy of a laser photon scattered straight back by an electron it meets head-on, against the electron's energy, for a carbon-dioxide laser, an infrared and a green one, on logarithmic axes: dashed, 4γ² times the laser photon's energy, the double Doppler shift of a mirror moving at the electron's speed; solid, the exact edge with the electron's recoil, E·x/(1 + x) with x = 4γε/mc²; dotted, the electron's own energy, which no scattered photon reaches. A 1 GeV electron turns 1.17-eV infrared photons into gamma rays of 17.6 MeV, within 1.8 per cent of the mirror's figure. At 46.6 GeV and 2.35 eV the mirror's figure is 78.2 GeV, more than the electron has to give; the photon actually comes back with 29.2 GeV.

    The photon a fast electron throws back

    Shine an infrared laser at a beam of gigaelectronvolt electrons and gamma rays come back out of the collision, each carrying seventeen million times the energy of the laser photon it started as. The electron acts as a mirror moving at nearly the speed of light, and a moving mirror multiplies the energy of light by two Doppler factors. Push the electron's energy far enough and the mirror arithmetic promises photons more energetic than the electron itself — and the photon's own momentum, which the mirror ignored, steps in to stop it.

    part 8 · relativity

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