Relativistic chain drive paradox 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:


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*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.

*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.
 A: *

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

*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.
A: 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.
A: 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.
A: This is a really excellent question which requires us to tie together the solutions of the Ehrenfest paradox and the train tunnel paradox.
The Ehrenfest paradox says that a relativistically spinning wheel should be length contracted on the edge, but a radial line isn't length contracted (because it's moving perpendicular to its length). So what is the radius of the circle? Is it 2pi*r? Ehrenfest resolved his own paradox by (correctly) asserting that there can be no rigid rotating objects in special relativity. Einstein later pointed out that one could imagine a wheel in relativistic rotation, not accelerated to that point, but constructed already in rotation. Its surface, as viewed by an observer riding the wheel, can only be described using non-Euclidean geometry.
But I think these answers doesn't give a good intuition for the question "what happens to a wheel (or chain/gear system) as I try to increase its rotation rate until its edge is relativistic?". I think a model which might give a better intuition is to imagine a wheel of particles (say spherical shaped particles) joined by springs - and somehow I apply an external force, or there's supports, such that from my perspective (the experimenter making the wheel rotate) the radius of the wheel remains the same. So from my perspective the distance between the particles remains the same. But something suspicious happens to the particles - they are no longer spheres, and rather ovals - obviously they're squished because of length contraction. And although the springs haven't changed length from my perspective, they are under strain. To better understand what's happening, enter the perspective of a particle. From that perspective, the length between particles has increased, and the springs are most definitely stretched. I look down at the other side of the wheel, and I see something similar to what the experimenter saw - particles that are actually closer together than the original length, but the particles are squished in their direction of travel by a bigger factor than the reduction in length. And to some extent, I infer that I need to be farther from my neighbors because the particles on the other side of the wheel are all bunched up from length contraction.
I think there are important takeaways here:
(1) the volume of a certain region of space does not need to be identical to different observers. From my perspective the length of the outer edge of the wheel never changes: its 2pi r, but I can see from the narrowing of the particles that from their perspective it's more than 2pi r.
(2) Length contraction is usually thought of as the length of a relativistic spaceship being smaller from my perspective. But if I somehow enforce that the spaceship doesn't get smaller from my perspective, then from it's perspective, it is stretched. As long as the ratio of the length perceptions is the lorentz factor, we're in accord with relativity. You need to be careful about what length, and from whose perspective, is enforced to remain constant.
So indeed, your relativistic chain which cannot stretch at all will instantly break the second it starts turning. Anything that rotates experiences a relativistic strain as bonds between atoms (or whatever holds the thing together) are stretched because from their perspective the length is increasing. This is usually a very small stretch in real life situations. But if we ever want to start spinning things relativistically, we will need to insert segments that can stretch (or extra chain links to span the increased distances from the chain's perspective).
But the introduction of the chain adds a kind of simultaneity layer of complexity here. Let's say an experimenter has two points in space (the gears) that are 1m apart - call them A and B. And let's say the experimenter takes a 1m long stick (when at rest) and moves it relativistically over the top of the two points. Since the stick (chain) is shortened, the back of the stick moves over point A before the front moves over point B (the chain is not long enough to span the two gears). But from the perspective of someone riding the stick, it is long enough, and the front of the stick passes over point B at the same time as the back passes over point A. But This is essentially the train tunnel paradox. The resolution of that paradox is that two events that are considered simultaneous in one perspective don't need to be simultaneous in another perspective. So when you ask your three observers "which chain link has just hit the top left gear/just left the top right gear," or equivalently "how many chain links are on top of the gears and how many are on the bottom" the three observers will disagree.
