The pole that fits and does not fit

A pole twenty metres long is carried at 0.866 of light speed through a barn ten metres long, and a farmer at each door shuts it as an end goes past. In the barn’s frame the pole is contracted to ten metres, so for an instant it is entirely inside and both doors are shut with it in there. In the pole’s frame the barn is contracted to five metres, the pole is four times its length, and no arrangement of doors could ever hold it.

Everything below is where those numbers come from, why the two accounts are not in conflict, and what the harder version costs — a bill that has nothing to do with relativity.

The paradox, stated so that it looks like one #

At $\beta = 0.866$ the Lorentz factor is $\gamma = 2.000$ — the same factor that slows a moving clock — which is why this speed is worth choosing: the arithmetic comes out in whole metres.

In the barn’s frame the pole’s rest length of 20 m is measured as $20/\gamma = 10.0$ m. The barn is 10 m. So the pole exactly fills it — the marginal case, in which the two doors shut on the two ends with nothing to spare and must reopen at once. A pole travelling slightly faster has room: at $\beta = 0.9$, where $\gamma = 2.294$, it measures 8.72 m and there is 1.28 m of clearance, which at 0.9c is a window of 4.75 ns during which both doors could stand shut. The marginal case is the sharper statement of the problem and the numbers below use it.

That factor is not an extra postulate. A light clock carried past at 0.866c has its pulse travel exactly twice as far between ticks as one standing still, because the pulse must cross the moving gap diagonally and it crosses it at $c$ like everything else — so $\gamma = 2.000$, and that single number sets both the ten metres the barn measures and the five metres the pole measures. One constant speed forces the ratio and nothing else is put in.

In the pole’s frame nothing about that reasoning survives. The pole is 20 m, because a length measured in an object’s own frame is its rest length by definition. The barn is rushing past at 0.866c and is therefore $10/\gamma = 5.0$ m long. A 20 m pole in a 5 m barn: three quarters of the pole is outside at every moment, and the claim that both doors were shut with the pole inside is not slightly wrong but absurd.

Both frames are inertial, neither has any privilege, and both apply the same formula correctly. What “the pole is shorter” means physically is worth settling before going further, because the trapped-pole version of the problem turns on it.

A contracted pole is not a squeezed pole. Take a lattice of atoms and measure its spacings from two frames moving at 0.866c relative to each other: the two sets of numbers differ, and nothing was done to the material in between — no force acted, no bond was strained, no energy was supplied. The contraction is a fact about what a measurement returns, not about what a rod has had done to it, which is why the contraction alone can never break anything. The same relabelling turns a magnetic force into an electric one with equally little happening to the wire.

The two door events, and the sentence that broke #

The resolution is not a compromise between the two accounts. Both are exactly right, and the fault is in one English sentence.

“The pole fits in the barn” means: both ends of the pole are inside the barn at the same time. That phrase is the one that does not survive a change of frame. Strip it out and the disagreement disappears, because what each frame actually asserts is a pair of events, and the two frames agree about every event.

There are two events, and they are the only ones that matter.

• Event N, at the near door: the pole’s rear end passes the near door, and that door shuts.
• Event F, at the far door: the pole’s front end passes the far door, and that door shuts.

Each is a coincidence at a single place — a door and an end of a pole in the same spot — so no frame can disagree about whether it happened or about what was where when it did. In the barn’s frame N and F occur at the same instant, 10 m apart. That simultaneity is the whole of the claim that the pole fitted.

Apply the Lorentz transformation. Put event N at the origin, $x = 0$, $t = 0$; event F is at $x = 10$ m, $t = 0$. In the pole’s frame,

$$t' = \gamma\left(t - \frac{\beta x}{c}\right),$$

so event N is at $t' = 0$ and event F is at $t' = -\gamma\beta(10\;\mathrm{m})/c = -57.8$ ns. The far door shuts 57.8 nanoseconds before the near one, and reopens in between; by the time the near door shuts, the pole’s front end has been out in the open for nearly sixty nanoseconds. That is the promised check against $\beta\gamma L/c$ with $L = 10$ m the barn’s rest length: $0.866 \times 2.000 \times 10\;\mathrm{m}/c = 57.8$ ns, the same number.

The pole’s frame tells a perfectly ordinary story. A short barn, 5 m long, sweeps backwards along a stationary 20 m pole at 0.866c. Its leading door reaches the pole’s front end and passes it; a moment later its trailing door arrives at the pole’s front end and shuts there, which is event F; the barn keeps going, and 15 m of travel later — 20 m of pole minus 5 m of barn — its leading door arrives at the pole’s rear end and shuts there, which is event N. Fifteen metres at 0.866c is 57.8 ns. The two frames agree.

What makes that an argument rather than an assertion is the classification of separations. A pair of events is timelike, null or spacelike according to the sign of $c^2\Delta t^2 - \Delta x^2$, and only the third kind can be reordered by any change of frame. The first two cannot: they lie inside or on each other’s light cones, something could have passed between them, and every observer without exception agrees which came first. Neither door event can have caused the other, so their order carries no physical claim to disagree about.

The classification is the point. In the barn’s frame the two door events have $\Delta t = 0$ and $\Delta x = 10$ m, so

$$s^2 = c^2\Delta t^2 - \Delta x^2 = -100\;\mathrm{m}^2,$$

which is spacelike. In the pole’s frame the same pair has $\Delta x' = \gamma \Delta x = 20.0$ m — precisely the pole’s rest length, since the events sit at its two ends — and $c\Delta t' = 17.32$ m. Then $17.32^2 - 20.0^2 = -100\;\mathrm{m}^2$, unchanged. Two frames, two entirely different pairs of coordinates, one number neither of them argues about.

The tilted axes on the diagram carry no scale of their own, which is the thing most often left off it. A unit of the pole’s time is not a unit of page length: it is fixed by the hyperbolae $c^2t^2 - x^2 = s^2$, and one unit of the pole’s time sits 2.000 page units from the origin at this speed. Without them the diagram shows the tilt and hides the stretch, and the 57.8 nanoseconds could not be read off it at all.

So the physics never disagreed with itself. Nothing was resolved by choosing a frame, because nothing needed resolving; a sentence containing “at the same time” was asked to mean something frame-independent, and it cannot.

What “shut and keep shut” costs #

The standard resolution stops there, and stopping there leaves the better question unasked. Suppose the doors are shut and kept shut. The pole is now genuinely trapped: it is inside a 10 m barn with 20 m of rest length, and the geometry admits no account in which nothing happens to it.

Something does happen to it. It is compressed — not by the contraction, which is a matter of measurement, but by the far door, which is a wall.

The mechanism is worth following in the pole’s frame, where the pole is 20 m and at rest. Its front end strikes the far door and stops. The rest of the pole does not know. Whatever signal carries the news backwards — a compression wave in the material — travels at the pole’s own speed of sound, and the one thing relativity insists on is that this speed cannot exceed $c$.

Take a generous value. If the news travelled at 0.5c, crossing 20 m of pole would take 133 ns. During those 133 ns the rear end is still moving at 0.866c and covers 34.6 m — over a length that started at 20 m. The far half of the pole is already deep into territory the front end has vacated before it has any information that the front end stopped at all. A real material is far worse: 0.5c is fantastically stiff, and the stiffest matter anyone models seriously, in neutron-star interiors, reaches sound speeds around 0.5c to 0.6c, with $c/\sqrt{3} = 0.577c$ the value for a conformal relativistic fluid.

No arrangement of internal signals gets round it either, and that is worth being explicit about because it is the obvious escape. Speeds do not add: a compression wave running forwards at 0.5c through a pole already moving at 0.866c is seen from the barn at 0.953c, not at 1.366c, and composing any two speeds below $c$ returns a speed below $c$. Chain any number of relays along the pole and the chain is still slower than light. There is no way to assemble a message from the far door to the near end that beats the light that is already trying.

And a signal that did beat it would not merely be fast; it would be a signal into somebody’s past.

That is why the ceiling on the pole’s internal signalling is a ceiling on causation rather than a limit on how stiff a material can be made. A rod that responded as a whole would be a rod carrying messages into its own past.

So “shut and keep shut” is a different problem with a different answer, and the answer is not a paradox but an accident: the pole crumples, or the barn breaks, or both. What relativity forbids is the one thing that would have made trapping impossible — a pole that responds as a whole.

Relativity permits no rigid body #

A rigid body is one whose parts hold fixed distances from each other whatever is done to any of them. Push one end and the other end moves — immediately, because otherwise the distance between them changed. “Immediately” is the problem. A push at one end that moves the far end with no delay is information travelling across the body in zero time, at infinite speed, and every argument against faster-than-light signalling applies to it in full.

So there is no rigid body in relativity. Not an unattainable one, not a good approximation with a small correction: the concept is inconsistent with the theory. Every object is elastic, every push travels through it at that material’s sound speed, and the far end of anything is always slightly behind the near end’s news.

The light cone is the whole of what a push can reach. Draw the future of the instant the near end is struck and it is a wedge opening at 45°; every event the push can affect lies inside it, and the far end’s worldline enters that wedge only after the crossing time has elapsed. A rigid body would require the far end to respond to something outside its own past cone — not a hard engineering target but a forbidden one, in the same sense that a triangle with two right angles is forbidden.

The size of the error has a clean form. Rigidity is a good approximation when the time a signal takes to cross the body is short compared with the timescale over which the motion changes. Write $\tau_{\mathrm{s}} = L/v_{\mathrm{s}}$ for the sound-crossing time and $T$ for the timescale of the applied motion; the fractional error in treating the body as rigid goes as $\tau_{\mathrm{s}}/T$.

For a steel rod, $v_{\mathrm{s}} = \sqrt{E/\rho} = \sqrt{2\times10^{11}\,/\,7850} = 5{,}050$ m/s, so $\tau_{\mathrm{s}}$ is 196 µs per metre — call it 200. A metre-long steel bar pushed by hand over a second has $\tau_{\mathrm{s}}/T \approx 2\times10^{-4}$, and rigidity is excellent. The same bar struck by a hammer whose contact lasts a millisecond has $\tau_{\mathrm{s}}/T \approx 0.2$, and rigidity is worthless.

Here is the surprise, and it is the reason this essay’s paradox is less exotic than its costume. The version of the argument that matters is not relativistic at all. A hammer striking a steel bar does not move the far end for 196 µs per metre, and the reason is that the bar’s sound speed is 5,100 m/s, which is $1.7\times10^{-5}$ of $c$. Relativity’s contribution is a ceiling 59,000 times higher than what steel actually manages. Put as a stiffness rather than a speed: causality requires $E \le \rho c^2$, which for steel’s density is $7\times10^{20}$ Pa, and steel’s Young’s modulus is $2\times10^{11}$ Pa — a factor of $3.5\times10^{9}$ of headroom. Every apparently rigid object in the world is a wave-mechanics problem being ignored on purpose, and it was already one before anybody knew about light cones.

Bell’s spaceships, and an instruction that cannot be obeyed #

One more instruction sounds harmless and is not: accelerate a rod rigidly.

Two spaceships, connected by a taut string, start at rest a fixed distance apart and fire identical engines on identical programmes, as timed in the launch frame. Their worldlines are congruent, so their separation in the launch frame never changes. The string, meanwhile, is a material object that is now moving, and its rest length must be contracting relative to what the launch frame measures. Held at a fixed launch-frame separation while its own natural length falls by $1/\gamma$, the string is stretched, and at some $\gamma$ it breaks.

Nothing in that requires a paradox to be dissolved. It says that “the two ends accelerate equally” is a frame-dependent instruction, and that keeping a rod’s own length constant during acceleration requires accelerating its parts by different amounts — the rear harder than the front, in the ratio of their distances from the accelerating frame’s horizon.

The correction is minute for anything terrestrial. A rod 20 m long accelerating at $1g$ sits $c^2/g = 9.2\times10^{15}$ m from that horizon, so front and rear need proper accelerations differing by two parts in $10^{15}$. No engineer will ever measure it. What matters is that the instruction was ambiguous, in exactly the way “the pole fits” was ambiguous, and for exactly the same reason: it presumed a shared now.

What it costs elsewhere #

The non-rigidity of matter is not a relativistic footnote; it is the daily business of several branches of engineering, and none of them needs relativity to meet it.

The strain gauge measures the delay directly. In a split-Hopkinson pressure bar, a striker hits a long steel bar and gauges bonded partway along it record the compression pulse arriving, reflecting from the far face and returning. The whole method depends on the pulse taking 196 µs per metre — long enough to resolve on an oscilloscope, which is how material behaviour at strain rates of $10^3\,\mathrm{s}^{-1}$ has been measured since the 1940s. If rods were rigid the technique would have nothing to observe.

Pile driving is the trapped pole, at 4,000 m/s. A hammer blow on a 20 m concrete pile takes about 5 ms to reach the toe and 5 ms to come back, and a hammer’s contact lasts a few milliseconds, so $\tau_{\mathrm{s}}/T$ is of order one and the pile is emphatically not a rigid body during the blow. Pile-driving analysers exploit precisely this: they read the reflected wave at the head and infer the resistance at the toe from its shape and timing. The whole diagnostic exists because what happens at a change of medium is written into the returning pulse.

A drill string is a rod with a half-second memory. Three kilometres of steel drill pipe carries a torsional signal at rather less than 5,100 m/s, so a change made at the surface is not felt at the bit for something like 0.59 s. That delay, coupled to a bit whose friction falls as it speeds up, is the loop that produces torsional stick-slip — the bit stalling and then spinning at several times the surface speed, which destroys cutters. The cure is a controller designed around the transit time, and it is a wave problem in a rod, not a mechanics problem in a shaft.

Where the model stops #

Two frames, both inertial, no gravity. Everything above is special relativity in flat spacetime, and the two doors and the pole are described by observers moving uniformly. The instant a door actually stops the pole, the pole’s own frame stops being inertial and the tilted axes of the diagrams no longer describe it.

The interval computation assumes point events. A door has width, a pole end has a face, and both were treated as single points in spacetime. Over 10 m and 57.8 ns that is harmless; over a barn whose doors are half a metre thick it introduces a correction of order the door’s thickness divided by the barn’s length, about 5%.

The paradox also has a threshold, which is easy to miss because 0.866c looks like a dramatic choice and is in fact an arithmetical one. A 20 m pole fits a 10 m barn in the barn’s frame only when $\gamma \ge 2$, and $\gamma$ reaches 2 at exactly $\beta = \sqrt{3}/2 = 0.866$. Below that speed the pole never fits in any frame and there is nothing to reconcile at all.

Below the threshold there is no paradox. At $\beta = 0.6$ the same pole measures 16 m, which does not fit a 10 m barn, and both frames agree that it does not. The apparent contradiction is not a general feature of moving objects; it requires the contraction to be large enough to reverse the comparison, which is a condition on $\gamma$ and therefore on $\beta$.

The rigidity bound is a bound, not a prediction. Relativity says $v_{\mathrm{s}} \le c$ and says nothing about what any actual material does. The sound speed of steel comes from its bonds, and the factor of 59,000 between the two is a fact about chemistry that relativity neither explains nor constrains.

Who argued about this, and when #

The barn is folklore and the argument underneath it is not.

Einstein noted in 1907 that relativity leaves no room for a rigid body. Max Born gave the surviving substitute in 1909 — a motion in which every neighbouring pair of points keeps a constant proper distance — and within a year Gustav Herglotz and Emmy Noether had shown how little of it there is: a Born-rigid motion has three degrees of freedom rather than six, so a body cannot be spun up rigidly at all. Wolfgang Rindler’s 1961 treatment of the rod-and-hole variant is the first careful published account of the two-door problem’s harder half, careful because it insists on the deformation; Dewan and Beran published the connected-spaceships argument in 1959, and John Bell revived it in 1976, reporting that a straw poll of theorists at CERN divided on the answer.

The dissolution — “at the same time is frame-dependent” — was available from Einstein’s 1905 paper and was never really in doubt. What took two decades of argument was the other half: that the rigid rod of elementary mechanics is not merely unattainable but incoherent.

What the picture cannot show #

Every figure here is two-dimensional, with one space axis. A barn has a width and a height, and a pole has a thickness; the transverse dimensions are unchanged by the motion, and a diagram with one spatial direction cannot make that visible or explain why the argument would break if they were not.

More seriously, nothing in these figures is deformable. Every worldline drawn is the worldline of a point, and every pole in them is a pair of parallel lines with no material between. The compression wave that decides the “shut and keep shut” version of the problem would appear as a third worldline slanting between the two ends, at a slope between the pole’s and the light line’s, and none of the diagrams have it. The figures are exactly good enough for the version of the paradox that dissolves, and blind to the version that has physics in it.

The interval hyperbolae show the invariance and not its meaning. They demonstrate that $s^2 = -100\;\mathrm{m}^2$ in both frames, which is what makes the door events reorderable, and they cannot show the more interesting negative fact: that no material process can join the two events, so their order is not a fact about the world.

And the diagrams have to choose whose axes are square. Drawing the barn’s frame with perpendicular axes makes it look like the real one. It is not; every statement here could be drawn the other way round with the roles swapped, and the asymmetry lives on the page rather than in the physics.

The ladder from here #

The next rungs on this anchor: the Lorentz transformation derived from clock synchronisation rather than quoted, with contraction and dilation read off it as two components of one object. Born rigidity worked out for a uniformly accelerating rod, including the horizon behind it. The rod-and-hole variant, in which a rod falls through a slot and is guillotined in one frame and bent in the other, which is harder than the barn because the deformation is unavoidable. Relativistic elasticity as a field theory, where the sound speed appears as a constraint on the equation of state. And the Penrose–Terrell rotation, which settles what a passing pole would photograph as, as opposed to what it measures.

Neighbouring ladders: the invariant interval and the diagram it is read from, which carry the geometry all of this rests on; the twin who comes back younger, which is the same kind of apparent contradiction settled by the same kind of care about which events are being compared; the invariant that survives a boost, where the quantity that replaces a frame-dependent one is a mass rather than a length; and a wave as a shape that travels, which is what a pole turns into once rigidity is given up, along with the medium deciding its speed.