Illustrated Physics
physics, drawn
Concept

Length contraction — where it appears

The shortening of a moving object along its motion, which is a consequence of how simultaneity is sliced rather than a squeezing of anything. Measuring a moving length means locating both ends at once, and observers who disagree about at once necessarily disagree about the length.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

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.

relativity · Length contraction The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

Magnetism is electricity seen sideways

The force on a charge moving beside a current-carrying wire is magnetic in the laboratory and purely electrostatic in the charge's own frame. Both calculations give the same answer, and the drift speed that makes them agree corresponds to a Lorentz factor differing from one in the twenty-sixth decimal place.

relativity · Field transformation 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.

relativity · Length contraction 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.

relativity · Length contraction 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.

relativity · Length contraction How far a 1-gravity ship gets, against its own clock. The distance covered by a ship accelerating steadily at 1 gravity for half the trip and braking for the other half, against the time on its own clock, on a logarithmic vertical axis. The curve is a cosine hyperbolic and therefore an exponential once the ship is relativistic, which it is after about a year: Proxima Centauri in 3.5 shipboard years, Sirius in 4.6 shipboard years, the Pleiades in 11.9 shipboard years, the galactic centre in 19.8 shipboard years, Andromeda in 28.6 shipboard years. Nothing about this violates anything. The speed never reaches c — it is the hyperbolic tangent of the rapidity and after 3.5 years it is 0.9495 at turnover — and every one of those journeys takes slightly more than the distance in years as measured from home. What is growing exponentially is not the speed but the length contraction, and the traveller's honest description of the trip is that the distance shrank. The rapidity is what accumulates steadily: it grows by one unit every 0.969 years of shipboard time, for ever, with no ceiling anywhere in the arithmetic. That is the whole reason the numbers come out survivable, and the reason the fuel does not.

The ship that never arrives at c

Accelerate at one gravity and never stop. The speed creeps toward light and never reaches it, and meanwhile the galactic centre is twenty shipboard years away and Andromeda twenty-nine. What makes the journey survivable is that rapidity has no ceiling; what makes it impossible is that the fuel goes as the exponential of the same quantity.

relativity · Accelerated frames 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 Where the second tick actually goes. A spacetime diagram with a second observer's axes at β = 0.6. The hyperbolae are the sets of events one second and one metre from the origin — invariantly, by the interval — and every observer's unit tick is where their own axis crosses them. That is checked here rather than drawn by eye. The moving observer's one-second mark sits 1.458 times further from the origin on the page than the stationary one's, so a ruler laid on this picture reads the two frames on different scales. The picture is not distorted; the page is Euclidean and spacetime is not. The Lorentz factor here is 1.2500.

The diagram a ruler cannot read

A spacetime diagram is drawn on flat paper, and the geometry it depicts is not flat. The tick marking one second on a moving observer's axis sits further from the origin than the stationary observer's, by an amount that is not the Lorentz factor and means nothing at all.

relativity · Spacetime diagram Two things that change and one that does not. How a boost treats a piece of charged matter: the charge density rises by the Lorentz factor because the same charges occupy a contracted length, and the length falls by the same factor. At β = 0.6 the density is 1.250 times what it was and the length is 0.800 times, and their product is one to fourteen decimals across the whole range drawn. So the total charge is the same number in every frame, and it is the only quantity in the transformation that is. Charge density is the time component of a four-vector and transforms like an energy; charge itself is a scalar, and nothing about the observer changes it.

The one quantity a boost leaves alone

Energy, momentum, length, duration, density and field strength all change when the observer moves. Electric charge does not, and the whole of the field-transformation argument rests on it — so it is worth asking what the evidence is.

relativity · Field transformation Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

Charge and current are one thing

The rung below asks what a boost leaves alone and answers charge. That answer forces the next one: a fixed charge in a contracting volume gives a density that transforms like a time component, and a current that transforms like a space one. So charge density and current density are the four parts of one object — and conservation of charge stops being an extra law and becomes the condition that makes the object exist.

relativity · Field transformation

Named alongside it

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

The Lorentz transformationReference framesSimultaneityCharge densityInvariant intervalThe Lorentz factorProper lengthAccelerationCharge invarianceConservation lawsElectric fieldField transformation

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