That seems to be the crux of the issue. Uh, no, if that was the case then it would not be instantanenous, would it.
Can somebody please elaborate as to why the chain in the paradox in the opening post is required to move like a perfectly rigid body? It isn't a requirement that the chain is blocking the light, only the lead balls, therefore it isn't necessary that the molecular forces in the chain move faster than the speed of light. It's just a chain that connected between two balls and it is pulled tight. It isn't complicated and it's not impossible.
Zeno: you make the assumption that the chain is rigid for both observers. Think about how this might be a problem given both ends are being moved in the same direction and it's a chain made out of links, find out about rigid bodies and relativistic physics. Another error is your assumption that both machines can be made to move simultaneously; this isn't as easy to do in practice as it might seem, but go find out why, ok? I was just now thinking about what your setup is like trying to do. You can pull an ordinary chain, say a ship's chain at either end so it resembles a rigid body, if it's laying along the ground, which is flat; if this chain's length isn't too great that the curvature of the earth has to be factored in, then it's also "straight". But if you then lift either or both ends, i.e. apply forces transversally at the ends, it won't be straight. Or rigid, since the links at each end will be loosened. In fact a freely hanging chain is curved by gravity. If you do what you think can be done with a chain out in space or in the absence of gravity (i,e, in an SR setting), the same thing will happen and the chain will curve. This is because the forces applied at the ends by your machines are not "transmitted" to the rest of the chain immediately.
The clear implication of the OP was if the chain moved in any way not perfectly rigidly the chain would break: (emphasis added) Thus the implication is that the chain moves like a perfectly rigid rod in the first frame and the chain would fall apart if it assumed any other shape. That's saying the speed of sound in the chain is infinite in the rest frame. The "no slack whatsoever" says the material of the chain cannot support the slightest deformation so signals carrying momentum must travel instantly without differential deformation. But such a material does not exist in nature and cannot exist in a thought experiment exploring special relativity. The material of your chain has no physical properties -- it's just a magic set of claims, not all of which can coexist with special relativity.
The machines move simultaneously. There's no problem. Not it doesn't! It just means that it has no slack! It's not complicated! There is nothing preventing the movement of the chain along with the balls so it isn't going to break. If I have a steel bar connected to two weights, one on each end, is it impossible to move both weights simultaneously without the bar breaking? Of course not! Now then does the steel bar exhibit any kind of slack? No it doesn't. Maybe I shouldn't have used the word 'whatsoever'. Would that have made things more clear?
The problem is you are describing a material that cannot exist. It is like saying if you push one end of a steel rod then the other end will instantly move. That does not happen because the force propagates through the rod at only the speed of sound (for that material).
Thanks for the responses everyone. It is my opinion that the paradox posed in the opening post is an unresolvable paradox in special relativity. There is nothing in the paradox that requires movement along the chain to go any faster than what would be expected to occur normally. Four minutes is plenty of time for the chain to break for the moving observer whilst the chain remains intact for the stationary observer. Length contraction doesn't resolve the paradox. I'm not going to discuss this any further since nobody has resolved the issue and there appears to be no resolution.
rpenner's explanation resolves the paradox. In the "moving" observer's frame, the chain must stretch, because one ball moves after the other ball moves. Since this chain is incapable of supporting any more stress (i.e. it can't stretch to a longer length), it must break. This part is uncontroversial. The question is: why in the "stationary" frame should the chain break? There are forces exerted on the ends of the chain that pull the masses out of the way of the light sources. So, the balls pull on the ends of the chain (perpendicular to the chain's length), and the chain is supposed to move out of the way of the sources. But how does the middle point of the chain know to start moving perpendicular to the length of the chain? Answer: a signal must propagate from each end of the chain to the point in the middle. The propagation of that signal will take some time, so motion of the middle of the chain "lags" behind the motion of the ends. The result is that the chain bends in the middle as it is pulled out from between the light sources. But, the initial assumption was that this chain won't stand up to any more tension. Since the tension must increase as the chain is pulled out perpendicularly, the chain must snap in the "stationary" frame. This is consistent with the observation that it snaps in the "moving observer" frame, too. If you used a real, non-idealised chain for this, one of two things would happen. Either the chain could withstand the tension and not snap (in both frames), or it couldn't and it would snap (in both frames). Either way, this isn't a problem for relativity.
So you believe you have invalidated SR? I would rather believe that you are a rather stubborn ego inflated anti science troll. But that's only my opinion. Oh! Just noticed, you are in pseudoscience! Please Register or Log in to view the hidden image!
The same paradox would be true even without SR. Say we had a chain from here to the moon. I make a video showing me cutting the chain on earth. The moon reference will assume the chain is still intact, right after I cut it. The paradox is connected the speed of light setting on a limit on data transmission, so things do not appear to be synchronized. The instant the chain was cut; in universal time, it was cut everywhere, whether you see it now or later. A galaxy that is 100,000 light years across, rotates in synch, even though it should be all chaotic based on light speed, since each aspect of the perimeter does not know what the other does for 100,000 years. The way you get around the information speed limit of the speed of light is to separate space-time into separated time and separated space. This allows time to propagate without space limits; universal clock. The galaxy rotation is using the universal clock with time=K, space=0. That is the simplest explanation; two sentences.
It's hard to know where to start with that, wellwisher. Time dilation has nothing to with the speed of light limit on data transmission. It concerns when events actually happen, not when information is perceived about them at some distance away. There is no universal time. Galaxies rotate as they do not because of some kind of magical universal time, but because their contents react locally to the mass that is near them. We had separated time and space up until Einstein. Newton had separated time and space. Einstein showed that separated time and space doesn't cut the mustard. There are no universal clocks. Simplest explanation of the many errors in your post in four paragraphs.
Answer: The space ship wouldn't be able to determine the exact speed and position of the photons inside of the experiment. The experiment itself wouldn't be able to give a proper indication of that either. The chains would be just fine in both frames. Turning on the LHC and moving particles 99.9999999999999% the speed of light didn't crumble buildings across the world. It's safe to assume that is the experiment that proves it. This is why I think it is better to think of the affects of SR as being due to the light clock example instead of using relativity of simultaneity. Then the correct solution to the light clock example is: (ct')^2 + (vt)^2 = (ct)^2; which produces t'= t sqrt(1 - v^2/c^2)...
