The wheel whose top cannot go twice as fast

Speeds that refuse to add found that two velocities in the same direction do not combine by addition but by

$$u = \frac{u' + v}{1 + u'v/c^2},$$

so that a ball thrown forward at $0.5c$ from a train moving at $0.5c$ moves at $0.8c$ relative to the track, and nothing put together from slower parts ever reaches $c$. The formula is usually illustrated with trains, rockets and particles fired from moving sources: two bodies, one carried by the other.

A rolling wheel is a better illustration, because it carries every case at once. Each point of its rim has a velocity relative to the axle — the same speed, in a different direction at every point — and the axle has a velocity relative to the ground, and the point’s velocity relative to the ground is the composition of the two. At the bottom they cancel; at the top they add; in between they meet at every angle. Everyday life knows the answer for slow wheels exactly: the bottom of the wheel is momentarily at rest, which is what rolling without slipping means, and the top moves at twice the axle’s speed. Photographs of moving bicycles show it, with spokes sharp near the ground and blurred at the top.

Push the wheel towards the speed of light and the two facts part company. One of them is a statement about relativity’s composition law and survives exactly. The other contradicts it.

The bottom stays still and the top slows down #

Measured in the axle’s frame, a rim point at angle $\alpha$ from the bottom moves with speed $v$ along the rim — the rim’s speed equals the axle’s, which is what makes the contact point stationary in the ground’s frame. Its velocity in the ground’s frame is that rim velocity composed with the axle’s velocity $v$, using the full relativistic rule for velocities in any direction.

At the bottom the rim velocity is $-v$ and the composition gives $(-v + v)/(1 - v^2/c^2) = 0$. The contact point is still at rest, exactly, for any speed: the composition of a velocity with its own reverse is zero in relativity as in everyday mechanics, so rolling without slipping is the same condition in both. It could hardly be otherwise. Whether the rubber at the contact point slides over the road is a question about two pieces of matter touching at one place, and either they move relative to each other there or they do not; no change of observer can turn a skid into a grip. Relativity changes how velocities measured in different frames combine, never whether two things at the same place are moving relative to each other. At the top the rim velocity is $+v$ and the composition gives

$$u_{\text{top}} = \frac{2v}{1 + v^2/c^2}.$$

For an axle at half the speed of light the top moves at $0.8c$, not $c$. At $0.8c$ it moves at $0.976c$; at $0.95c$, $0.9987c$. The everyday formula, $2v$, reaches $c$ when the axle reaches half of it and then goes past; the relativistic one approaches $c$ only as the axle does.

The shape of the curve changes too. At everyday speeds the rim’s speed rises smoothly from zero at the bottom to its peak at the top, as $2v\sin(\alpha/2)$. At relativistic speeds it climbs steeply from the bottom and then flattens into a broad plateau just under $c$, so that most of the upper half of the rim moves at nearly the same speed. A wheel at $0.95c$ has three-quarters of its rim moving at more than $0.95c$. The fast part of the wheel has been pressed up against the speed limit.

For any real wheel the difference is invisible. The correction factor $1 + v^2/c^2$ differs from one by about $10^{-14}$ for a car on a motorway and by about $10^{-12}$ for the fastest wheels anyone has built, the rotors of ultracentrifuges and the turbines of jet engines, whose rims move at a few hundred metres per second. Relativity says the top of a car’s tyre is slower than twice the car’s speed by a few millionths of a millimetre per hour.

A wheel caught at one instant #

Asking where every part of the wheel is at one instant is a harder question than it sounds, because the instant has to be chosen in one frame and the wheel’s motion is described in another.

In the axle’s frame the wheel is a circle with straight spokes, turning steadily. In the ground’s frame the natural first guess is that it is the same wheel, contracted along the direction of motion by $\gamma = 1/\sqrt{1 - v^2/c^2}$: an upright ellipse with straight spokes. That guess takes the axle frame’s picture at one moment and squeezes it, and the squeeze is right for the rim. It is wrong for the spokes, for the reason that the length that depends on when found behind every relativistic length: events that are simultaneous in one frame are not simultaneous in another.

The ground’s “now” cuts across the axle’s history at a slant. Points of the wheel ahead of the hub are caught at an earlier moment of the axle’s time, and points behind it at a later one, by an amount proportional to their distance ahead or behind: $t' = -vx'/c^2$. Since the wheel is turning, an earlier moment means less rotation and a later moment more. A spoke that points straight forward at the axle’s moment zero is caught, in the ground’s snapshot, before it has turned that far, so it appears tilted back up toward the top; a spoke that points straight back is caught after it has turned further, so it too appears tilted up. Spokes in between bend accordingly, each point along a spoke caught at its own moment.

The rim, by contrast, comes out as exactly the contracted ellipse. Every rim point lies on the axle frame’s circle whenever it is caught, so collecting them at different moments still traces the circle, and the ground’s coordinates squeeze it by $\gamma$. What differs from the naive picture is not the outline but which material point is where along it — which is invisible on a plain rim and conspicuous on spokes.

The bending grows quickly with speed, because the spread of moments the ground’s instant cuts across grows as $vR/c^2$ while the wheel turns through an angle $v/R$ per unit time, and the two multiply.

At $0.95c$ the wheel’s rim is squeezed to under a third of its width and the spokes are swept into long arcs, the forward and backward ones climbing nearly to the top of the rim. The ground’s instant has cut across a large fraction of a turn: a point on the front of the rim is caught more than a quarter of a revolution earlier than the matching point on the back. A wheel with many spokes would show them crowded into the upper part of the ellipse, as if the wheel’s material had been swept upward — which in the ground’s frame, at that instant, it has.

A clock carried on the rim makes the same point in time rather than space. In the ground’s frame the clock moves at a speed that varies round the turn, from zero at the bottom to nearly $c$ at the top, so it runs at its normal rate while it touches the ground and slows almost to a stop as it passes over the top. Its total time for one turn is what the axle frame gives it, $2\pi R/(\gamma v)$, because in that frame it moves at a steady $v$; the ground’s frame reaches the same total by integrating a rate that changes all the way round. Two frames, two very different accounts of where the clock lost its time, and one number at the end.

This is the wheel’s shape in the ground’s frame, the arrangement of its parts at one moment of the ground’s time. It is not what a camera would show. The contraction no photograph shows found that light from the far side of a fast object leaves earlier than light from the near side to arrive together, and that a camera records objects rotated rather than squashed. A photograph of the wheel would combine that effect with this one, and both with the Doppler shifts that turn the forward-moving top blue and the stationary bottom unchanged.

The ground receives the rim’s own length #

The most surprising fact about the fast wheel is how far it goes.

The traced path is the relativistic counterpart of the cycloid, the curve the curve that does not ask where it started found as the path of a point on a rolling circle and as the shape of a pendulum’s isochronous track. At low speed it is that cycloid: an arch from the ground to twice the radius and back, one circumference long. At $0.8c$ the arch is stretched: the point rises to the same height, since nothing perpendicular to the motion is contracted, but it comes back down $10.47$ radii further on rather than $2\pi = 6.28$.

The count is easy to check. One turn of the wheel takes a time $2\pi R/v$ by the axle’s clock. The ground sees that clock running slow by $\gamma$, so in the ground’s frame the turn takes $\gamma \cdot 2\pi R/v$, and in that time the axle moves $\gamma\cdot2\pi R$. Nothing is wrong with the arithmetic; what needs explaining is why the ground should get more rim than the axle measures.

The answer is that the rim has more length than the axle measures. The disc that cannot be spun found that rulers laid along the rim of a turning disc are contracted, in the axle’s frame, by the rim’s speed, so that more of them fit round the circumference than when the disc was at rest: the rim’s own length, measured by rulers moving with it, is $\gamma\cdot2\pi R$, larger than the circumference $2\pi R$ that rulers at rest beside it measure. Rolling transfers rim to ground at the contact point, and at the contact point the rim is momentarily at rest relative to the ground. So the ground receives the rim’s length as the rim itself measures it — $\gamma\cdot2\pi R$ per turn — just as a tape measure unrolled onto a floor gives the floor its own markings, regardless of how compressed it looked while it was spinning on the reel.

That is also why such a wheel could not be built. Nothing is allowed to be rigid found that no body in relativity can hold its shape against arbitrary changes of motion, and the disc essay found that a rim cannot be spun up from rest without being stretched: material that started at rest with a circumference of $2\pi R$ would have to stretch by the factor $\gamma$ to keep its radius while turning at speed $v$. A wheel rolling at $0.8c$ would have had to be stretched by two-thirds along its rim before it could turn at all. The kinematics here describe a wheel that is somehow turning; no material wheel could be brought to that state intact.

The same composition, every direction at once #

Read across the whole rim, the wheel is a sampler of the composition law. At each point a velocity of magnitude $v$, pointing along the rim, is composed with the axle’s velocity $v$ along the ground. The angle between them sweeps through every value from $180°$ at the bottom to $0°$ at the top.

When the two are opposite, the composition gives zero: a relativistic statement that is as simple as the everyday one. When they are parallel, it gives $2v/(1 + v^2/c^2)$, the classic result. When they are perpendicular — at the front and back of the wheel, level with the hub — the composition gives a velocity of magnitude $\sqrt{2 - v^2/c^2}\cdot v$, pointing up or down at an angle, the sideways component reduced by $\gamma$ because the ground sees the axle frame’s clocks running slow. That case, two perpendicular velocities composed, is the one in which the turn that two pushes leave behind found the Thomas–Wigner rotation: composing two boosts in different directions gives a boost plus a rotation. A wheel’s rim is a continuous sequence of such compositions, which is one way to see why its spokes bend.

The space of velocities itself, the space that speeds live in, is a hyperbolic one in which the composition of velocities is a kind of addition of points, and in that space the rim’s velocities — all of magnitude $v$ about the axle’s velocity — form a circle. In ordinary geometry a circle of radius $v$ centred at $v$ reaches from $0$ to $2v$. In the hyperbolic space of velocities, a circle of the same hyperbolic radius centred at the same point reaches from $0$ to $2v/(1 + v^2/c^2)$, and the plateau in the first figure is that circle’s far side crowded against the boundary of the space, which is the speed of light.

What the wheel leaves out #

It is not a real object. As above, no material wheel could be rolled at these speeds: the stresses needed to hold a rim together at even a thousandth of the speed of light exceed the strength of every known material, and spinning one up to relativistic speeds would require stretching it. The figures describe the kinematics of a wheel assumed to be turning steadily — what the parts’ motions must be, given that they are moving at all.

The spokes are not physical. A spoke in the axle’s frame is a straight line of material points, each moving on its own circle. Whether it stays straight there depends on its internal forces and its history, and no Born-rigid rotation exists for a disc, so the axle frame’s straight spokes are themselves an idealisation. What the snapshot shows is how a straight spoke in one frame appears in another, not what a real spoke would do.

Rolling is assumed. The contact point at rest on the ground is the definition of rolling without slipping. A real wheel deforms where it meets the road, as the push a millimetre ahead of the axle found, and slips slightly; at relativistic speeds the deformation of the contact patch would be dominated by the problem of a material’s response to forces arriving faster than sound in it can carry them.

Appearance is separate. Light-travel time, aberration and Doppler shifts all change what a camera would record, and none is included. Each would make an image of the wheel more distorted, not less.

Still open: whether anything natural rolls fast #

Nothing rolls at relativistic speeds, but things turn at them. The surfaces of the fastest-spinning neutron stars move at a fifth of the speed of light; matter in the inner edge of an accretion disc around a black hole orbits at half of it; and the plasma of the jets those systems launch moves at more than $0.99c$. In each, the parts of a rotating or moving object are composed with each other in exactly the way the wheel’s rim is composed with its axle, and the radiation they send out carries the imprint of it.

The open question is not about kinematics, which the composition law settles completely, but about how such matter holds together. A neutron star’s crust is the strongest material known, and how close to break-up its rotation can be pushed before it sheds mass, rings or deforms is set by its elasticity and its interior composition, both uncertain. The fastest known pulsar spins 716 times a second, well below the break-up rate for most models of its interior, and why none spin faster has not been settled — whether something in the stars’ own structure limits them, or the gravitational waves a deformed, fast-spinning star would emit carry away its spin before it can go faster.

The wheel asks a smaller version of the same question. Its kinematics say where every point would be; its dynamics, which no figure here contains, say whether any material could follow those motions without coming apart. The composition law, which made the top of the wheel slow down, is the part that is certain.