In the pole-in-the-barn paradox, let's say the barn is 100 ft. wide and there is a rocket on top of each door and there is twine connecting the two rockets which is 100 ft. long or slightly longer. When each door is closed the rocket on top of that door is launched. When the events are simultaneous the rockets remain 100 ft. apart and the twine doesn't break. When the events are not simultaneous, one rocket is launched first and the twine breaks. So, the twine is both broken and not broken!

No, because the twine at one end can't instantly "know" that the twine at the other end is stretched. A signal has to travel along the twine.

Huh? Why is it necessary for one end to instantly 'know' that the other end has moved? If I hook up two rockets and launch one rocket, say 1 minute before the other, the twine will break. If they are launched at the same time the twine will not break.

You're right that it will either break or not break. But when it breaks, the question is: when did it break? In two different reference frames, the answer to that question may well be different.

There can never be a situation in which the twine breaks in one reference frame and does not break in a different frame. But there can be a situation in which the twine breaks in one frame and breaks at a later or earlier time in the other frame.

See?

If they are launched at the same time the twine will not break.
That's not necessarily true. If the rockets are launched at the same time, the twine still has to stretch a little if it is not to break.
The section of twine in the middle of the barn, for example, won't start moving immediately - the leading rocket will have moved some distance before that section of twine feels the pull and begins to move.

OK. Then let's replace the twine with a heavy-duty chain connecting Titan rockets. You guys are just beating around the bush without resolving the paradox. Obviously it is possible to have a connecting chain which can withstand acceleration without breaking but can't hold down a massive rocket. Either the paradox must be resolved or SR declared invalid because of logical inconsistencies. But, I for one will not be holding my breath for the latter.

Resolution:

From the pole's perspective:

The rocket at the front door (rocket 1) fires before the rocket at the second door(rocket 2). Assuming the chain reaches breaking tension before rocket 2 fires, the chain breaks.

From the barn's persepective: the rockets fire simultaneously and accelerate up to speed maintaining a constant distance between them. The chain accelerates with them. As the chain gains velocity with respect to the barn, it must undergo length contraction. If the chain is not strong enough to pull the rockets closer together (which it will not be if it can't tow rocket 2 behind rocket 1 from the pole's perspective.), it must break. So if the chain breaks from the perspective of the pole, it breaks fromt he perspective of the barn.

From the Rockets' perspective. The both rockets intially start at the same moment. As they build up speed, they must be considered from an accelerated frame. In an accelerated frame, clocks in the direction of the acceleration run fast, and clocks in the opposite direction run slow. Thus the clock in rocket 1 runs faster than the clock in rocket 2. As a result, the engines of rocket 1 run faster than rocket 2 and rocket 1 accelerates at a graeter rate than rocket 2. Rocket 2 can't maintain a constant distance from rocket 1 and begins to fall behind. This again puts strain on the chain and it breaks.

If the chain breaks in any frame, it breaks in all three.

OK. Then let's replace the twine with a heavy-duty chain connecting Titan rockets.
Hell just connect the rockets with an ultra-strong rod - the same thing applies.
When the rockets start moving, they will pull and push on the rod.. but they can't make the whole rod start moving all at once. The middle of the rod won't start moving until the signal (actually a compression or tension wave moving at the speed of sound in the rod) arrives from one end or the other.

Try this one, Zeno:
You have a really strong and really light rod one light-second long. You stand at one end, your friend stands at the other. Now you can communicate with each other instantaneously by pulling and pushing on the rod, right?
Obviously it is possible to have a connecting chain which can withstand acceleration without breaking but can't hold down a massive rocket.
The chain that is just strong enough to withstand acceleration is also strong enough to withstand being pulled by one rocket for long enough until the other rocket starts up as well.
This is no surprise - the difference in timing between the rockets starting is always going to be less than the time it takes the tension wave to travel through the chain.
In other words, the trailing rocket will always take off before the pull from the leading rocket arrives.

As the chain gains velocity with respect to the barn, it must undergo length contraction. If the chain is not strong enough to pull the rockets closer together (which it will not be if it can't tow rocket 2 behind rocket 1 from the pole's perspective.), it must break. So if the chain breaks from the perspective of the pole, it breaks fromt he perspective of the barn.

But doesn't the chain contract in the direction of travel?

This is no surprise - the difference in timing between the rockets starting is always going to be less than the time it takes the tension wave to travel through the chain.

Why? By varying the speed of the runner can't we make the difference in time between the launching of the rockets as great as we like? One rocket is launched weeks before the other one and has travelled less than 100ft?

But doesn't the chain contract in the direction of travel?
Yes. but that's the point. The chain contracts but the distance between the rockets stays the same. So either the chain pulls the rockets together and doesn't break, or it doesn't and it breaks. But which ever happens will be seen as happening from all frames.


Why? By varying the speed of the runner can't we make the difference in time between the launching of the rockets as great as we like? One rocket is launched weeks before the other one and has travelled less than 100ft?

But it will always take more time for the impulse to travel from one end of the barn to the other. Remember, the time delay in the launching of the rockets is as seen from the pole's frame of reference, and in the pole's frame of reference, the fasted the impulse can travel from end to end of the chain is c relative to the pole. From the Pole's reference frame the end of the chain attached to the second rocket is traveling at v in the same direction that the impulse is. IOW, the pole sees the impulse traveling at c chasing after the end of the chain traveling at v, which had a head start of X (the length of the chain. as measured by the pole. ) The time it will take for the impulse to catch the end will be equal to

t= \frac{X}{c-v}

For instance, assuming the chain (x') is 1 light sec long in its own frame, and the relative velocity between pole and barn is 0.866c, then the chain will be 0.5 light sec long as measured by the pole (X=0.5 ls), and the fastest the impulse can travel from front to back of the chain is:

t= \frac{0.5 ls}{c-0.866c} = 3.731 sec

if we increase the relative speed to 0.99 c, we get

t= \frac{0.141 ls}{c-0.99c} = 14.107 sec

At .9999c:

t= \frac{0.0141 ls}{c-0.9999c} = 141.4 sec

The time delay between rocket launches for these velocities respectively are:

1.732 sec
7.02 sec
and
70.7 sec

These are 0.4642, 0.498 and 0.5 times the time it takes for the impulse to travel from the front to back of the chain.

So yes, you could pick a relative velocity where the time delay between launches is weeks as measured in the pole's frame, but it still takes much longer for the impulse to traverse the length of the chain as measured from the same frame.

Einstein himself dealt with this paradox way back in the early days when he was still developing the Special Theory. The resolution is simple, as I believe the post above explains in more technical terms. In relativity, there is no such thing as a truly rigid object. Any force applied to one end of the rod cannot have an effect any quicker than a beam of light would on the other end. A wave of compression/expansion/stretching/contracting would travel along the rod no quicker than the speed of light.

So if from a central reference frame, where both rockets lift off simultaneously, the rod doesn't snap, then in any other inertial frame, when you watch the wave travelling across the rod, it will not at any point produce enough tension to snap the rod. Any tensile force in the rod disobeying this principle would be thrown out as nonrelativistic.