[Physics FAQ] - [Copyright]

Updated by Don Koks, 2014.
Original by Michael Weiss 1995.

Bell's Spaceship Paradox

John Bell described this special relativity paradox in the essay "How to teach special relativity" in his collection "Speakable and Unspeakable in Quantum Mechanics".  He did not invent the puzzle, but we'll call it Bell's Spaceship Paradox.

Bell considered two rocket ships connected by a string, each accelerating at the same constant rate in the "lab frame", with one ship trailing the other and both moving along one line.  The ships start out at rest in the lab.  Because they have the same acceleration, their speeds will be equal at all times in this frame, and so they'll remain a constant distance apart in the lab frame.  But after a time they will acquire a high speed, so we ask what has become of the idea of the distance between them being Lorentz contracted.  The actual paradox posed by Bell was that the string should eventually snap.  Why does it snap, if the ships are maintaining a constant separation?

What Happens to the String

Let's first ask what happens to the string.  Since the ships maintain a constant separation, some people say the string doesn't break.  But the string will eventually break, regardless of what it's made of.  We'll give three reasons in this article.

First we'll use no relativity at all, but will use a string that is held together by electromagnetism.  Later we'll add some relativity to show the same holds for strings of all makeup, and then we'll continue down the relativity path to investigate where the Lorentz contraction went.

The key to using electromagnetism to show the string breaks lies in determining what Maxwell's theory says about the electric field produced a by a fast moving charge.  The result is well known: Maxwell's theory says that as a charge moves faster and faster, its field line bunch up transversely to its velocity, and spread out before and after it.  In other words, the field it produces gets stronger alongside, but weaker in front and behind it.

Now, for the string to connect the spaceships, it will have to be fixed to the leading spaceship because we need some way to accelerate it.  If it's not fixed to the leading spaceship, it'll get left behind when the ships begin to accelerate.

Suppose that the ships have been accelerating for a while with the lead ship pulling on the string.  They are now moving very quickly.  Focus on the atoms that form the string and the electromagnetic forces that bind them together.  As the spaceships plus string zoom past us, we apply Maxwell's theory and conclude that the forces between the atoms, that normally hold those atoms together, are now very small.  Those forces will become arbitrarily small as the spaceships accelerate closer and closer to the speed of light.  But the front spaceship must always pull on the string to accelerate it—except that now, that pull can no longer by transmitted along the string's length, because there's really nothing holding the string together anymore; it has become just a loose chain of atoms.  So it can no longer be accelerated by the leading spaceship, and will instead begin to maintain a constant velocity.  That is, it will begin to be left behind, and so will separate from the leading spaceship.  In other words it will break, even though the spaceships are maintaining a constant separation.

What if the string is held together by forces that are not electromagnetic?  Relativity provides the answer, because what it says about the world is independent of the nature of the forces that hold objects together.  Of course, relativity respects Maxwell's electromagnetism, so it follows without any further analysis that relativity also says the string breaks, irrespective of the nature of the forces.  But how, exactly, does it do that?

As the string gets pulled along by the leading rocket and accelerated ever faster, it becomes more and more difficult to accelerate; its relativistic mass increases. (Some people don't like the term "relativistic mass", so they will just say that the string becomes more and more difficult to accelerate.  It's the same thing.)  The leading rocket is programmed to accelerate at a certain rate, but the force it applies to the string has to accelerate a string whose mass is growing without bound.  Do we require the rocket to be able to apply that force?  Even if it could apply an arbitrarily large force, it would only be applying that force to the piece of string that is attached to the rocket.  That force will have to be transmitted along the string to accelerate the rest of the string.  And we know of no material that is capable of withstanding arbitrarily large internal forces.  If the string could withstand such arbitrarily large internal forces, it wouldn't break.  But a real string will break.

What about the Rockets' Separation?

Let's return to the question of what happened to the notion of Lorentz contraction for the distance between the rockets.  By design, the rockets remain a constant distance apart in the lab frame.  Relativity says that the distance between two objects is contracted from the "proper distance", when measured by an observer whom they move past at some fraction v of the speed of light: it's contracted by the gamma factor γ = (1 − v2) −1/2.  The proper distance between the objects is the distance between them in the frame in which they are both at rest.  But there is actually no frame in which Bell's two rockets are always at rest, and so we can't define any "proper separation" for them.  That means we won't find that any simple expressions hold for length contraction.  But there is at least an approximate notion of the proper separation of the rockets, and we'll find that this is approximately γ times their lab separation.  So their approximate proper separation increases by roughly γ as the rockets move, and the result is then reduced by γ as measured by the lab observer, who thus does measure a Lorentz contraction back to the lab separation.  The rockets' proper separation—inasmuch as it can be defined—is set by γ; but γ is constantly increasing as the rockets speed up, so their proper separation must increase over time.  That increasing proper separation is what will break the string.

What is this approximate proper separation, and how can it be increasing as the rockets accelerate?  I'll sketch the solution without providing any of the maths.  Doing everything quantitatively is certainly enlightening, but because the rockets are accelerating, the details are not simple enough to put into this FAQ entry without obscuring the main points.

The picture at left is a schematic of the spacetime scenario, drawn in the lab frame.  We can treat the rockets as point particles because their extent is really extraneous to what's going on here.  Of course you can also talk about their extent, but the discussion then only gets back to what happened with the string, so let's not do that.

The blue curve is the world line of the trailing rocket, and the red curve is the world line of the leading rocket.  Their separation in the lab is a constant L.  Focus now on event A.  All events that the blue rocket measures as simultaneous with A lie along the dotted blue line passing through A.  (This dotted blue line is a reflection through an invisible (i.e. not drawn) 45° light ray passing through A of the tangent to the world line at A.)  In particular, the blue rocket says that event B is simultaneous with A.  The two rockets' clocks both read zero when they started out, so we see here that the blue (trailing) rocket measures the red (leading) rocket to be ageing quickly.  On the other hand, the red rocket doesn't say that A and B are simultaneous.  If it draws all the events that it measures as simultaneous with B, these events form the dotted red line, and this line crosses the blue world line at event C.  So the red rocket says that event C is simultaneous with B and, as a result, it measures the blue rocket to be ageing very slowly.  The factors by which the rockets each measure the other to be ageing relative to themselves are not inverses of each other.  For example, the blue rocket might say that the red rocket is ageing twice as fast as blue, while the red rocket says that the blue rocket is ageing far more slowly than one half as fast as red.  You can see the reason for these strange-sounding numbers by studying the diagram.

The distance that the blue rocket measures from A to B is approximately γL (in fact, it's somewhat more than γL).  Now the thing is that if we ask the blue rocket to re-measure the distance from itself to the red rocket at a later time marked by event P, then the set of events that the blue rocket measures as simultaneous with P is the upper dotted blue line (which is not parallel to the lower dotted blue line), and this line crosses the red world line at event Q.  The distance PQ will be larger than distance AB, and so the blue rocket will conclude that the red rocket is actually pulling away from it.  As the red rocket pulls away from the blue, any string connecting them will eventually break.  Also, given that blue measures the distance to red to be approximately γL, we can see that the separation L measured in the lab is indeed more or less a Lorentz contraction of γL back to L.  But I say "more or less" because this lack of shared agreement on which events are simultaneous means that the two rockets don't form a frame of their own, and so we can't talk about transforming one frame to another with a simple Lorentz contraction.

But is it possible to change the motion of one rocket so that the pair will form a frame?  Yes, it is.  The red and blue curves are hyperbolae, and they each approach (different) 45° lines as they accelerate ever closer to the speed of light.  (Note that they will always outrun light beams that approach from the left of those asymptotes.)  If we were to make the red rocket accelerate less than the blue in just the right amount to ensure that both hyperbolae shared the same asymptotes, then it turns out that the red rocket would say that event A was simultaneous with event B, and the rockets would not measure each other to be drifting apart.  This is remarkable, and in fact it forms the basis of Einstein's Equivalence Principle.  Both rockets would agree on the simultaneity of all events, and as such could construct one meaningful set of coordinates that both could use to describe the world in identical terms.  They would form a uniformly accelerated frame.  The key to constructing a uniformly accelerated frame is that its observers don't all accelerate at the same rate.  Even so, they do all measure themselves to be at rest relative to each other, and they all agree on the timing and simultaneity of events.

A study of how different observers accelerate to form a frame in which they might possible agree on simultaneity and might all be able to set their clocks to a common time is subtle and not mathematically simple.  In basic special relativity when we consider inertial frames, we always assume that these frames have been inertial forever; we don't ask how they got to be moving that way.  And that's fine.  But to see what happens when analysts get sidetracked into spending too much time thinking about how a frame "got to be that way", read the old FAQ entry The Rigid Rotating Disk in Relativity, and in particular read my note recently added to the end of it.

To see all the details of how different observers must accelerate in order to form a uniformly accelerated frame, see Chapter 7 of "Explorations in Mathematical Physics" by D. Koks (Springer, 2006).

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