The string that breaks between two rockets

Two identical rockets sit on a launch rail, half a light-second apart, with a delicate string tied between them. Both are given the same acceleration programme, started by the same signal. Their separation in the laboratory is fixed by construction — the two worldlines are the same curve shifted sideways, so any horizontal line on the diagram meets them the same distance apart, at every laboratory time, for ever.

Does the string break?

It breaks. Bell put the question to the CERN theory division in the 1970s and reported that a majority of a distinguished company got it wrong. That the disagreement was about a physical outcome rather than about a description is what makes the case worth a rung: two observers may legitimately disagree about which event came first, and they may not disagree about whether a string broke. That a distinguished company divided on it is a good indication that the intuition being tested is a load-bearing one.

Why the intuition says otherwise #

The argument for the string surviving runs: everything is moving at the same speed, so everything contracts equally, so the gap and the string contract together and nothing is stressed.

Every step of that is confused, and it is worth separating the confusions because they are the ordinary ones.

What “at the same time” picks out depends on who is asking, and the whole of this essay is a consequence of that one fact. A length is a distance between two events called simultaneous — the two ends of the object, now — so a length is a question with a frame attached to it. Two observers slicing the same spacetime differently are measuring between different pairs of events, and there is no reason their answers should agree.

“Everything contracts.” Contraction is not a process. Nothing shrinks; a different pair of events is being measured between. The rockets’ separation in the laboratory is set by their programmes and is not contracting, because nothing has been done to make it contract.

“The rockets’ frame.” There is no such frame. The rockets are accelerating, so no inertial frame keeps up with them for longer than an instant, and the phrase has to be replaced by the momentarily comoving inertial frame at each event — which is a different frame at each instant and slices spacetime differently each time.

“So the string contracts too.” The string is a physical object with a natural length — a length being a distance between two events called simultaneous, and the string’s natural length being the one measured in its own rest frame. If the distance it has to span exceeds that length, it stretches, and if it stretches enough it breaks. Whether it does is a question about the distance in the string’s own instantaneous rest frame, and that is exactly the quantity the figure measures.

What the measurement gives #

Take the leading rocket at some moment, take its momentarily comoving frame, and follow that frame’s slice of simultaneity back to the trailing worldline. The interval between the two events is the proper separation, and it is what a string spans.

At a gap of half the acceleration length, the drawn separations are 0.512, 0.557, 0.631 and 0.710 units at 0.3, 0.6, 0.8 and 0.9 c, against an initial 0.5. The stretch is two, eleven, twenty-six and forty-two per cent.

The textbook answer is $\gamma L$, and it is the limit of that measurement as the gap shrinks to nothing: at a gap a hundred times smaller the two agree to half a per cent, and at the drawn gap of half an acceleration length $\gamma L$ overstates the stretch by up to 38 per cent. That is the site’s standing obligation exercised on its own figure — the quoted result is a limit and the essay says which limit and how far the case in hand is from it.

What the stretch approaches, for closely spaced rockets, is $\gamma$ — and $\gamma$ grows without bound. So there is no material and no separation for which the string survives to arbitrary speed: at 0.99c it would have to stretch by a factor of seven, and at 0.999c by a factor of twenty-two. The question is never whether the string breaks but only when.

No material has that much strain available. Steel breaks at about a fifth of a per cent, so the string in the figure has failed before 0.06c.

What would have to be done instead #

The alternative is to insist that the proper separation stay constant, which is Born’s definition of a rigid body — and it is a programme rather than a property.

Keeping the proper separation fixed requires the trailing rocket to accelerate harder than the leading one, in the ratio of their distances from the horizon the acceleration creates, and to go on doing so for ever. For rockets half an acceleration length apart, the rear must push twice as hard as the front, permanently.

A steady push produces a hyperbola with a light ray as its asymptote, and the Born-rigid pair are two hyperbolas sharing that asymptote. That is the whole of what “accelerating rigidly” requires, and it is why the trailing rocket — being nearer the horizon, on a hyperbola of smaller radius — has to accelerate harder than the leading one. Equal programmes give congruent worldlines; rigidity needs nested ones.

A real rod does exactly this, and the mechanism is unremarkable. Push one end of a steel bar and the push travels along it at the speed of sound; the bar compresses slightly, the internal stresses redistribute, and after the transients have died away the bar is accelerating with a small permanent internal stress gradient — the front pulling the back, harder at the back than at the front. Born rigidity is what a real rod approximates, and it approximates it through elasticity rather than by magic.

The version with no string, which is the same statement #

The paradox can be stripped of the string altogether, and the stripped version is the one to keep.

Consider a single rigid rod, at rest, and ask what has to be done to it to get it moving at speed $v$ with no internal stress. Number the points along it. Each point has to end up moving at $v$, and each has to end up the right distance from its neighbours as measured in the rod’s own final frame. Those two requirements do not determine the histories: they determine only where each point ends and how fast, and a whole family of programmes gets there.

Two members of that family are the ones drawn above. Give every point the same acceleration in the laboratory, and the rod arrives at speed $v$ with its laboratory length unchanged, its proper length increased by $\gamma$, and therefore under a tension it did not start with. Give the trailing points more acceleration in the right proportion, and it arrives with its proper length unchanged and its laboratory length reduced by $\gamma$ — unstressed.

The same construction that slows a clock settles the rod, and it settles it in the same coordinate. What differs between the two acceleration programmes is the time at which each end is instructed to move, as judged by the rod — and time between two places is exactly what a moving clock disagrees about. Neither programme is wrong; they are different instructions, and only one of them is the instruction to hold a constant rest length.

So “the rod contracts” is shorthand for “the rod was accelerated in the way that leaves it unstressed”, and that way is a particular and non-obvious programme in which different parts of the rod do different things. Nothing about the rod enforces it except the rod’s own elasticity, which supplies the difference by pulling the laggards along and holding the leaders back.

That reframing also disposes of a question that otherwise looks deep. If contraction is real, what happens to the atoms? Nothing happens to them; the equilibrium spacing of a lattice is a property of its own rest frame and is unchanged, and what changes is which pairs of events an outside observer is comparing. The pole and the barn are the same statement with two observers instead of two programmes.

Length contraction, restated #

The rungs below this one established that a length is a distance between two simultaneous events and that simultaneity is frame-dependent.

The contraction is a real, measurable change in a charge density — it is what makes a current-carrying wire attract a moving charge — so it cannot be dismissed as a bookkeeping artefact. A lattice of charges seen from two frames genuinely has two different spacings, and an experiment reports the difference. It is nevertheless a statement about which events are being compared, which is why both halves of that sentence have to be held at once.

This rung adds the sharpest available test of that understanding. If contraction were something that happened to objects, the string and the gap would contract together and nothing would break. Because it is a statement about which pairs of events are being measured between, and the two rockets’ programmes fix a laboratory gap rather than a proper one, the two disagree and something has to give.

And a second warning, failing in the other direction: a photograph does not show the contraction, because light from the far side of a fast object left earlier than light from the near side. So contraction is neither a visual effect nor a squeezing. Treating it as the first makes it an illusion; treating it as the second makes it a stress in a material, and the string here breaks for neither reason.

What the two rockets see of each other #

The laboratory description is unambiguous and the on-board one is worth working out, because it is where the asymmetry the rockets themselves notice comes from.

Take the leading rocket. On its own slices of simultaneity, the trailing rocket is not merely further away than it was; it is also slower. The slices tilt as the speed rises, and a tilted slice cuts the trailing worldline at an earlier laboratory time, when that rocket had not yet reached the leader’s current speed. So the leader, using its own instruments and its own definition of now, reports the follower falling behind and losing ground — while the laboratory reports the two keeping perfect station.

Every disagreement in this essay is read off a diagram of one kind: two axes and one speed, with the horizontal slices belonging to the laboratory and the tilted ones to a moving observer. The two carve the same events into different presents, and the string’s fate is the one question on which those two presents give different answers about a material object.

Neither report is wrong and neither is more fundamental. What settles the physical question is that the string is made of matter with a rest length, so the measurement that matters to it is the one made in its own instantaneous rest frame — which is the leader’s, near the leading end, and the follower’s near the other. Both of those say the gap is growing.

Where the energy comes from #

A string that breaks has had work done on it, and it is worth asking by whom.

The rockets’ engines. Accelerating the pair with a taut string between them requires more fuel than accelerating them separately, and the excess goes into the string’s elastic energy until it fails. Once it fails, the two rockets fly on unaffected, which is the check that the accounting is closed: no energy has appeared and none has vanished.

Three cases the same analysis settles #

A spaceship’s crew never notices. Everything on board follows the Born-rigid programme automatically, because the ship is one elastic object and its internal forces enforce it. The crew measure the ship’s length as unchanged for ever, and they are right, because proper length is what they measure.

A charged particle beam does not. The particles in a bunch are not connected, and a linac accelerating them all with the same field is running Bell’s programme rather than Born’s: the bunch’s proper length grows as γ while its laboratory length is held. That is a real effect with a name — the bunch is said to be “frozen” longitudinally — and it is why space-charge forces in the bunch’s own frame weaken as the beam is accelerated even though the laboratory density is unchanged.

And two clocks on a rotating disc cannot be synchronised. The Herglotz–Noether theorem’s exclusion of rigid rotation is the same fact in a different geometry: going round the rim, each successive comoving frame’s simultaneity slice is tilted a little further, and after a full circuit the accumulated offset does not close. Every satellite navigation system has to deal with that offset, under the name of the Sagnac correction, and it is measured in hundreds of nanoseconds.

What actually holds a rod at its length #

Saying that a rod’s proper length is a property of the rod pushes the question down one level rather than answering it. A rod is a lattice of nuclei and electrons held apart at a spacing where the electromagnetic forces between them balance, and if that spacing is to come out contracted when the rod is moving, it must be the forces that change.

They do. The field of a charge in uniform motion is not the spherically symmetric one it has at rest. It still points away from the charge’s instantaneous position, but its strength depends on direction: it is weakened by $\gamma^2$ along the line of motion and strengthened by $\gamma$ across it, so the pattern of equipotentials is flattened into a disc. A lattice sitting in such a field finds its equilibrium at a different spacing along the direction of travel than across it, and working the equilibrium out gives exactly the factor $\gamma$.

This is Lorentz’s own route to the contraction, and it is a dynamical argument: the rod is short because the forces holding it together have changed shape, and one could in principle solve for the transient by which it settles into the new length. Bell did precisely that for the simplest possible rod — a single classical electron in orbit about a nucleus, with the nucleus pushed by an external force and the Maxwell–Lorentz equations integrated numerically — and found the orbit ending up contracted by $\gamma$ with no relativistic postulate used anywhere.

Two things follow, and they pull against each other. The first is that the contraction is not optional: any object whose internal forces are electromagnetic contracts, whatever it is made of, which is why a geometric statement can be made about matter in general. The second is that a string tied between two rockets is not such an object, because its length is being imposed from outside by the rockets rather than found by its own equilibrium. That is the whole difference between the two programmes, stated in terms of what the matter is doing rather than of which slice is being measured on.

Bell’s actual point #

The example is usually attributed to Bell and it is Dewan and Beran’s, published in 1959 and answered correctly there. What Bell added in 1976 was the poll — he put it to the theory division at CERN and got a clear consensus for the wrong answer — and an argument about teaching that the anecdote was in service of.

His claim was that the standard order of presentation, in which the transformations come first and matter is fitted to them afterwards, leaves a student able to manipulate $\gamma$ and unable to say what happens to a string. The remedy he proposed was to teach the constructive route above alongside the geometric one: work out how the fields of a moving charge deform, watch a bound system settle into a new equilibrium, and arrive at the contraction as a result about matter rather than as a rule about coordinates.

The two routes are not rival theories and Bell was careful to say so. They give identical predictions and they differ in which facts are taken as given — the geometry, or the dynamics that the geometry summarises. What the broken string demonstrates is that the second is not dispensable pedagogy: the distinguished company at CERN had the transformations perfectly and still could not say whether a piece of matter was under tension.

Where the model stops #

The string is a test object with no mass. A real string tied between two rockets pulls on them, changes their acceleration, and couples the two programmes together. The idealisation is fine for the paradox and wrong for any engineering.

Born rigidity is not fully available. A body can be moved Born-rigidly along a straight line, and — by a theorem of Herglotz and Noether — it cannot be Born-rigidly rotated and accelerated in general. There is no relativistic rigid body, only a set of motions along which a body can be moved without internal strain, and that set is much smaller than the Newtonian one.

The signal to start is idealised. The two rockets were started “by the same signal”, and simultaneity being frame-dependent, that phrase already picks the laboratory frame. Starting them simultaneously in some other frame gives a different configuration and a different answer, and stating which frame the programme is defined in is not a detail — it is the whole specification.

The elastic response has been idealised away. A real rod does not follow the Born programme exactly; it oscillates about it, ringing at its own longitudinal modes, and those oscillations are damped by internal friction and radiate. Everything above is the steady state after that has settled.

And nothing here is gravitational. The horizon that appears in the Born-rigid case is produced by acceleration alone, and although it behaves in many respects like a black hole’s, it can be removed by cutting the engines. The comparison is worth making and the identification is not.

What the pictures cannot show #

A spacetime diagram draws simultaneity as a set of lines, and the lines are the one thing in the picture with no physical existence. The worldlines are real, the light cone is real, and the slanted slices are a choice — a choice that the essay’s conclusion depends on entirely, which is an uncomfortable position for a drawing to be in.

Nor can the figure show the string. It is drawn as a separation between two curves, and what actually happens is a wave of stress propagating along a material object at its own speed of sound, arriving at each point at a different time, and eventually exceeding a breaking strain somewhere. The instant of failure is a fact about a material, and every quantity in this essay is a fact about geometry.

Where this ladder goes next #

Four rungs establish, in order: that a length is a pair of events; that a pole and a barn can both be right; that a photograph shows neither; and now that a rod not permitted to contract is a rod being torn apart. The last is the one that turns the first from a statement about measurement into a statement with a consequence — a broken string is not a matter of convention.

The habit worth carrying away is a discipline about specifications. When a relativistic problem says two things happen together, ask in which frame. Nearly every paradox in this subject is a specification that omits the frame and then imports a different one halfway through the argument. Here the omission is in the phrase “identical acceleration programmes”: identical as measured in the laboratory, and the rockets themselves do not agree that their programmes are identical at all. The leading one, on its own slices, sees the trailing one falling behind.

What is left on this ladder is what the string’s failure looks like from inside — how an accelerating observer describes the stress, what coordinates such an observer naturally uses, and why those coordinates cover only part of spacetime.