Imagine taking a chain drive, of the sort used to power a bicycle, and accelerating it to relativistic speeds. For concreteness, let's say the two gears of the drive (call them $A$ and $B$) are fixed $L = 70\, \text{m}$ apart, and the speed of the chain is $v = \sqrt{3}/2$ (in $c=1$ units, so the relativistic distortion factor is $\gamma = 2$). Then there's an apparent length contraction paradox:

1. From the perspective of an observer watching the assembly from outside, although the distance between the chains stays fixed at $L$, each half of the chain—the half moving from $A$ and $B$ and the half moving from $B$ to $A$—contracts to $L/2 = 35\, \text{m}$. (There's some strange length distortion going on when the chain accelerates around the gears, but this shouldn't matter—if we lengthen the chain, then the length distortion of the straight segments increases proportionally, but the length distortion of the segments on the gears themselves remains constant. So the distortion due to the straight segment should be the determining factor, right?) There's $L = 70\, \text{m}$ of chain to cover a $2L = 140\, \text{m}$ round trip, so this suggests that the chain should break, similar to Bell's spaceship paradox.

2. From the perspective of an observer moving from $A$ to $B$ at $v=\sqrt{3}/2$, the length of chain moving from $A$ to $B$ stays at $L$, but the distance between the gears themselves contracts to $L/2 = 35\, \text{m}$. (I can't even visualize what this would look like.) The chain traveling from $B$ to $A$ has a relative velocity $\dfrac{2v}{1+v^2} = \dfrac{4\sqrt{3}}{7}$, with relativistic gamma $\gamma = 7$, and so would appear contracted to $L/7 = 10\, \text{m}$. In this case, we have an asymmetrically divided $8L/7 = 80\,\text{m}$ of chain for a round trip of $L=70\, \text{m}$.

So what happens here? Does the chain break? What would traveling alongside the chain look like? I know there must be something going on with relativity of simultaneity—points that appear to be on the same side of the chain in one reference frame are on opposite sides of the chain in another—but I can't say anything more precise than that.

• You didn't mention the Ehrenfest paradox in your question. Could you explain how much you already know about that? – Brian Moths Mar 24 '17 at 15:56
• I'd heard it described once or twice, but not by name. In any case, I don't think the gears themselves have much to do with the problem. – Connor Harris Mar 24 '17 at 16:01
• I answered this at physics.stackexchange.com/questions/309586/… . (That question was about electrons; this one is about chain links, but the questions are identical.) – WillO Mar 25 '17 at 1:20

1. When the the chain is accelerated, the tension of the chain increases, until the chain breaks.

2. The chain broke, so there's no moving chain that an observer could observe. Just before the breakage the moving observer saw slightly larger number of chain links on the side where the links were slightly more contracted according to him. Common sense says that if links are 1% shorter on one side, then there are 1% more of them on that side.

• This is a non-answer. The question is about the relativistic paradox, not material properties or engineering specifications. – Asher Mar 24 '17 at 16:49
• Then there is no answer. Unbreakable and unstretchable bike chain wrestles with equally rigid bike chassis, what happens? There is no answer to that. – stuffu Mar 24 '17 at 21:23
• @Asher The fact that there are no infinitely strong or rigid materials in special relativity plays an important role here. Maybe that should be explicitly stated in the answer to avoid the impression that the answer depends on all known materials "incidentally" having a finite strength. And notice the question does specifically ask, "Does the chain break", so saying that the tension increases and the chain breaks does not seem too out of line. – Brian Moths Mar 24 '17 at 22:40
• I apologize. My criticism was hasty and poorly founded. – Asher Mar 24 '17 at 22:44

The question becomes ill-defined at the moment when say "accelerate the chain". The chain has many parts. Which parts accelerate first? It's no use saying they all accelerate simultaneously, because an observer on the ground and an observer riding on the chain cannot agree about this.

Once you specify exactly the timing of the acceleration at various points along the chain (and specify whose point of view you are describing this from), the question becomes easy --- though the answer will be one thing or another depending on your specification. (See my answer here for some of the possibilities.) As long as you leave the timing vague, you can of course fool yourself into believing something is paradoxical, but that's always the way with relativity.

Just my two cents, not an "official" answer. First I want to consider what an observer moving along with the chain would see, so I will answer in reverse.

2) Let us simplify and assume that an observer at rest with the gears manages to get the chain to move at a constant speed, and thus "rigid" on his frame (let us say that it manages to do so using individual rockets for each link, or, that it is at least able to paint marks on the chain in such a way that these marks remain at the same distance). In such a case, each of the observers moving with the chain will see that the links on the other straight part of the chain become closer by length contraction, and this has to gradually happen during the circular (accelerated) part of the gear.

Thus we have an inertial observer (one moving with the straight part of the chain) that sees the chain either non-rigid (expanding and contracting) or breaking. To him the perpetrator is the accelerated part. But we have another inertial observer, the one at rest with the gears that do not see any non-rigidity (because the accelerated part does not change the speed). This last observer cannot explain why the chain should break.

My conclusion, perhaps wrong, is that the chain does not break. There is an inertial frame in which it does not, and there are other inertial frames in which it is seen non-rigid. The contradiction could perhaps be resolved if we conclude from this that any Lorentz invariant laws of force responsible for keeping the chain together will predict that the chain will behave as elastic on most reference frames.

1) Remember that I did not addressed yet if the chain breaks or not when accelerated from rest. In that case the chain will break, because the observer at rest will see the chain contracting as it accelerates (and changes speed) but the gears stay at the same distance.