The rod that is level for only one observer

Now is a choice of slicing established that two events separated in space are simultaneous for some observers and not for others, and the pole that fits and does not used that to dissolve the best-known paradox of length contraction. A pole longer than a barn, carried through it at high speed, fits inside with both doors shut in the barn’s frame and never fits in the pole’s frame, and the two accounts agree because “both doors are shut at once” names two events whose order depends on the observer.

The barn and the pole are a story about two events. There is a version of the same paradox, invented by Wolfgang Rindler in 1961, that is about a whole object at once, and it shows something the barn does not: that the shape of a moving object, and in particular whether it is level, is a statement about simultaneity too. A rod is level if all its points are at the same height at the same time. Change what “the same time” means, and a level rod becomes a tilted one.

A rod over a hole #

Lay a thin flat plate on the ground with a hole cut in it one metre long, and slide a rod one metre long across the plate at a speed close to that of light. Arrange that, when the rod is over the hole, every part of it is given the same small downward speed at the same instant, so that it falls. Does it go through the hole?

In the plate’s frame the answer is plainly yes. At eight-tenths of the speed of light the rod is contracted to six-tenths of its rest length, so a rod as long as the hole at rest is shorter than the hole in motion, and when it is wholly over the hole it has room to spare. Every point of it starts falling at the same instant, it drops level, and by the time its leading end reaches the far edge of the hole it is below the plate. It goes through.

Now take the rod’s point of view. The rod is at rest and a metre long. The plate slides past it at eight-tenths of the speed of light, and it is the hole that is contracted, to sixty centimetres. A metre rod lying level over a sixty-centimetre hole cannot fall through it. And yet whether the rod ends up below the plate is not a matter of perspective: if it does in one frame it does in all, since being below the plate at a given place is a coincidence of events that every observer agrees on.

The fall does not start all at once #

The resolution is that in the rod’s frame the rod does not lie level while it falls. The instruction was that every point of the rod starts falling at the same instant, and the instant meant was the plate’s. Five points along the rod starting to fall are five events, simultaneous in the plate’s frame and spread along the length of the rod. In the rod’s frame they are not simultaneous.

The spacetime diagram shows it. The starting events lie on one horizontal line, one instant of the plate. The rod’s own instants are tilted lines, the slant that the scissoring of a moving observer’s axes gives, and each of them crosses the row of starting events at a single point. In the rod’s frame the starts happen one after another, and since the rod is moving forwards in the plate’s frame, the ordering puts the front end first: the front starts to fall eight-tenths of a light-metre-time before the rear does.

The starting events can be reordered because no signal connects them. Each is a separate push at a separate place, all at one moment of the plate’s frame, so any two of them are separated by more distance than light could cover in the time between them — they are spacelike separated, and the order of spacelike events is not something observers have to agree on. Had the push been delivered from the rear end and allowed to travel forwards along the rod, the starts would have been timelike separated and every observer would agree that the rear started first. The paradox depends on the push being arranged by agents spread along the rod, each acting on a clock synchronised in the plate’s frame.

What the rod does in its own frame #

Every snapshot in the figure is the plate frame’s own events, transformed. Nothing has been added. What appears is a rod that starts by lying level on the plate, then — as the hole sweeps under its front end — begins to droop at the front. The point at which the fall has started runs back along the rod, so for a while the rod is bent: level behind that point, sloping in front of it. Once every point is falling, the rod is straight again, tilted front down by 21.8°, and it is going through the hole the way a spear goes through a hoop, point first. Each part of it drops below the plate just before the plate’s far edge reaches it.

Two speeds conspire to make the tilt. In the rod’s frame each point falls faster than in the plate’s, at $\gamma u$ rather than $u$, because the plate frame’s clocks, which timed the fall, run slow in the rod’s frame. And the start of the fall runs along the rod with a time delay of $v/c^2$ per unit of length. The product is a slope:

$$\tan\theta' = \frac{\gamma\,u\,v}{c^2}.$$

At $v = 0.8c$ and $u = 0.3c$, with $\gamma = 5/3$, the slope is 0.4 and the tilt 21.8°. The rod’s own frame has no paradox to resolve: a long rod over a short hole can go through it, as long as it goes in at an angle.

A puzzle set for students, and argued over since #

Rindler posed the problem in 1961 in a teaching journal, as an exercise in the relativity of simultaneity: a rod of the same rest length as a hole in a table, sliding along the table fast enough to be contracted to a fraction of the hole. It has been rediscovered and re-argued in the same journals many times since, often by readers who were sure that one of the two accounts must be wrong and proposed a physical mechanism — stress waves, the stiffness of the rod, the force that pushes it down — to decide which. None is needed, and the figures here use none. The tilt in the rod’s frame is not caused by anything happening in the rod; it is what the plate frame’s level rod looks like when its events are sorted into the rod frame’s instants. A real rod would have its own stresses and its own delays, and they would modify the motion in both frames equally, without touching the conclusion that the two frames disagree about the rod’s shape and agree about whether it went through.

The same tilt, from the composition of velocities #

The tilt can be reached a second way, which connects it to a stranger fact. Once it is falling, each point of the rod moves in the plate’s frame with a velocity that has two components, $v$ along the plate and $u$ downwards. In the rod’s original frame, moving at $v$, the same point moves straight down at $\gamma u$ — the downward speed is increased, not reduced, because the plate frame’s time runs slow in the rod’s frame and the same distance is covered in less rod time.

A level rod moving at an angle to its own length is, in its own rest frame, not level, and the angle between the two descriptions is a combination of the two speeds that grows as their product. It is the static version of the rotation that two boosts in different directions leave behind: compose a boost along the rod with a boost across it, and the result is not a pure boost but a boost and a rotation. Carried round a closed path, that rotation is what makes an electron’s spin precess in an atom. Here it is a single tilt, the same algebra caught at one moment.

Level is a statement about one instant #

The general lesson is that orientation is not frame-independent, and the reason is not contraction but simultaneity. “The rod is level” means that at one instant every point of it is at the same height. A different observer’s instant contains different events along the rod — later events at the rear, earlier at the front — and if the rod is moving up or down, those events are at different heights. A level rod moving vertically is level only in the frame whose instants are the ones it was level in.

For a rod moving purely sideways, with no vertical motion, nothing tilts: every point has the same height at every time, so it does not matter which time is picked. For a rod moving purely vertically, with no sideways motion relative to the observer, nothing tilts either, because the observers agree about which events are simultaneous along a line at right angles to their relative motion. It takes both — motion along the rod’s length relative to the observer, and motion across it — to turn a level rod into a tilted one, which is why the tilt is proportional to the product $uv$.

When the fall is not quite instantaneous #

The bend in the figure deserves more than a glance. In the rod’s frame, for 0.80 L/c, the front half of the rod is falling while the rear half is still resting on the plate. A real rod would have to bend to do that, and the bend would require forces — but in the plate’s frame the forces were applied all at once and the rod never bent. Both are true: whether a rod is bent at a given moment depends on the moment, as whether it is level does.

What that means physically is that the instruction “start falling” cannot be obeyed by a rigid body. If each point of the rod is pushed separately by an agent stationed along it, at times that are simultaneous in the plate’s frame, then in the rod’s frame the pushes arrive one after another and the rod responds by bending. If the push is applied at one point and the rest of the rod is supposed to follow, it can follow only as fast as a signal travels through it, which is no faster than light and in practice the speed of sound in the material. Nothing is allowed to be rigid for exactly this reason, and the rod over the hole is the cleanest example of what non-rigidity looks like when the push is arranged to be simultaneous in a frame that is not the body’s own.

Where the tilt is large #

The tilt grows with both speeds, and with the sliding speed faster than linearly, because $\gamma$ grows. A rod dropped at a hundredth of the speed of light while sliding at eight-tenths tilts by less than a degree; dropped at half the speed of light, by a third of a right angle. As the sliding speed approaches $c$ the tilt approaches 90°, and a rod sliding fast enough over a hole goes into it, in its own frame, nearly end first.

For anything in ordinary experience it is far too small to notice. A cup dropped in a train going at thirty metres a second is tilted, in the frame of the ground, by a thousandth of a millionth of a millionth of a radian, which is why nobody has ever worried that “level” might depend on the observer. For particle beams it is not small. A bunch of electrons travelling at nine-tenths of the speed of light and given a sideways kick of a hundredth of the speed of light is, in its own frame, tilted by two hundredths of a radian if the kick was simultaneous in the laboratory.

Particle accelerators manipulate exactly this. In a collider, two bunches crossing at a small angle overlap less than they would head-on, and several colliders tilt their bunches deliberately with crab cavities, which give the front and the back of each bunch opposite sideways kicks so that the bunches meet broadside. The design is done in the laboratory frame, where the kicks are timed along the bunch; seen from the bunch’s own frame, the relation between the timing of kicks and the resulting tilt is the rod over the hole run in reverse.

The barn, the rod and the photograph #

Three related puzzles are worth keeping apart. The barn and pole is about whether two events — the doors closing — are simultaneous, and the answer that they are in one frame and not in another removes the contradiction. The rod over the hole is about whether a whole object has a shape — whether it is level — and the answer is that shape at an instant inherits all the frame-dependence of the instant. And the contraction no photograph shows is about what an object looks like, which depends on when light left each of its parts to reach the eye at once, and which for a sphere hides the contraction completely.

All three are the same fact — which events are simultaneous depends on the observer — applied to more and more of the object. None of them involves any force, any material property, or any physics beyond the Lorentz transformation of the events that make up the story. That is why the rod’s tilt can be computed exactly without knowing what the rod is made of: it is a property of a set of events, and of how two observers sort them into instants.

What the pictures cannot show #

The snapshots are drawn at a fixed instant of each frame, and a sequence of snapshots looks like a film. But a film is itself a choice of slicing. The plate frame’s film and the rod frame’s film show the same events sorted into different frames of film, and neither is a recording of what anyone would see; seeing adds the delay of light from each part of the rod to the eye, which would distort both further.

The rod is given a downward speed instantly at every point, with no acceleration phase, and falls at constant speed rather than accelerating under gravity. Real gravity, or a real push, would change the shape of the curves but not the conclusion, since the tilt comes from the timing of the start, not from the details of the fall. And the rod is drawn in two dimensions; a rod sliding past a round hole has the same tilt in the plane containing its motion and none across it.

The domain of the argument is special relativity: flat spacetime, inertial frames, and objects whose motion is specified as a set of events. Inside it, a level object moving across an observer’s line of sight is tilted by atan(γuv/c²) for any observer moving along it at v.

Still open: what shape means for an accelerating body #

For inertial frames the shape of a moving object is settled: it is whatever the object’s events look like on an observer’s instant, and the Lorentz transformation gives it exactly. For accelerating observers there is no unique family of instants, and the shape of an extended body seen by an accelerating observer depends on how the observer’s instants are extended away from the observer’s own position — the problem how big now is found has no unique answer for a rotating observer. How to define the shape of a body that is itself accelerating, in a way that does not depend on such a convention, is part of the long-standing question of what rigid motion can mean in relativity, and the definitions in use, Born’s among them, each work for some motions and fail for others.

The rod over the hole needs only the inertial case. A rod sliding at v and given a fall at u everywhere at one instant of the plate’s frame is level in that frame and, in its own, starts falling at the front v/c² per unit length earlier than at the rear, bends while the start runs back, and then is straight at a tilt of atan(γuv/c²) — 21.8° at 0.8c and 0.3c — threading a hole that is shorter than itself front first. Whether a moving object is level is a question about one instant, and the instant belongs to an observer.