Illustrated Physics
physics, drawn
Concept

Rigidity — where it appears

The resistance of a structure or a packing to deformation. A granular material acquires it only once the contacts are numerous enough to constrain every motion; below that threshold the assembly is a fluid with no shear modulus at all, and the transition between the two is jamming.

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

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 Where a heap stops flowing. The number of motions that cost nothing, against the mean number of contacts each grain has, for a patch of 37 grains. Every point is the rank of an actual rigidity matrix — one row per contact, one column per coordinate — subtracted from the number of coordinates, so it is a measurement of the network rather than a formula about it. The dashed line is Maxwell's count, two coordinates a grain minus half a contact each, which is what the counting alone predicts. The measured curve reaches its floor of three free motions — the rigid-body ones — at a coordination of 4.11, and Maxwell's line reaches zero at exactly four, which is twice the dimension of the plane. That is the isostatic point and the difference between the two numbers is the boundary of a finite patch, whose outer grains have fewer neighbours than they would in an infinite one. Above the threshold the two curves separate: the measured floor stays at three while the count keeps falling, and the gap is the redundancy — contacts whose forces no equilibrium equation can determine. Below it the pack has genuine mechanisms and will rearrange under any load at all. The transition is in the count and not in the material, which is why a fluid and a solid here are made of exactly the same grains.

The heap that becomes a solid

Sand poured into a jar flows; the same sand shaken down and pressed does not. Nothing about the grains has changed — not their size, their hardness, their friction or their density by more than a per cent — and the thing that changed is a count of contacts, which crosses a threshold set by the dimension of space and by nothing else.

fluids · Granular matter In the rod's frame: a long rod threads through, front first. The same events seen from the rod's own frame, computed by transforming each point's motion from the hole's frame. Here the rod is its full length and the hole is the contracted one, 0.60 of it, sweeping left at 0.8c — a long rod over a short hole. But the moment each point starts to fall is not one moment in this frame: the front end starts at t′ = −1.07, the rear end at −0.27, and each point then falls at γu = 0.50c. While the start of the fall runs back along it the rod is bent, level behind and sloping in front; once every point is falling it is straight again and tilted, front down, by 21.8°. It threads through the hole front first as the hole slides under it, each part dropping below the plate just before the plate's far edge reaches it. It passes through in both frames; only the order of events along it differs.

The rod that is level for only one observer

A metre rod slides at high speed over a metre-wide hole in a thin plate. In the plate's frame the rod is contracted and drops through. In the rod's frame the hole is contracted, the rod is far too long for it, and it drops through anyway — because the instruction 'every point starts to fall now' names a different now for the rod. Its front end starts first, the start runs back along it, and the rod tips into the hole like a spear. Whether a moving thing is level is a question about which events are simultaneous, and so it has an answer only for an observer.

relativity · Simultaneity

Named alongside it

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

Length contractionThe Lorentz transformationSimultaneityWorldlineAccelerationConstraint countingCoordination numberDegrees of freedomForce-chainGranular matterIsostaticJamming

All concepts