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My professor recently gave us this paradox to solve as extra credit. The problem involves a runner moving at 4/5c holding a pole that has a proper length of 15 feet. They are moving towards a barn with a proper length of 10 feet. There is another observer standing off to the side somewhere who is not moving relative to the barn.

So, the observer sees the barn with its proper length, but the pole held by the runner is seen as 9 feet due to length contraction, so to him the pole fits in the barn. However, the runner sees the barn undergo length contraction to where its 6 feet wide, but his pole is still 15 feet long.

We identify two events, the pole hitting the back of the barn, and the back of the pole coming into the front of the barn. In each case the events are flipped relative to the other, and by calculating delta s^2 we find that they are space-like separated.

The question is of course, does the pole fit in the barn?

I'm not asking for the answer, just want pointed in the right direction here, so if this is the wrong place to ask this let me know.. and I didn't check other questions on the same topic because I'm afraid of coming across the right answer by accident.

My greatest difficulty with this is telling who is wrong, if either of them are. I did some reading on the relativity of simultaneity, and from that it seems like they would both be right! I remember reading that if two events are space-like separated, it's not possible to say that they absolutely occur at the same time. I'm not sure how the timing of the events play into this though. The delta t for the observer is positive while the delta t for the runner is negative, which I think is what indicates that they don't observe the events at the same time?

The main question though, 'does it fit?' seems to imply that it does or doesn't fit, so I feel like I'm supposed to give a yes or no answer here, but I don't think I can, because I think this situation entirely depends on which observers frame you're in.

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    $\begingroup$ Hint: "does it fit?" really means "are the front and back of the pole inside the barn at the same time?". That last bit is the key. $\endgroup$ – Emilio Pisanty Apr 17 at 20:48
  • $\begingroup$ Can you convert your intuition about what it means for a pole to fit in a barn, into a precise condition that should be satisfied by spacetime events? To start off with, don't even worry about the pole moving near the speed of light. For ordinary velocities where you can ignore length contraction, etc, what does it mean precisely for a pole to fit in a barn? $\endgroup$ – Andrew Apr 17 at 20:49
  • $\begingroup$ Well, in terms of the events I mentioned, it would be that the back of the pole enters the barn before or at the same time the front of the pole hits the back of the barn, right? This happens in the observers case, but in the runners those events are flipped, which I interpret as it not fitting from his perspective. $\endgroup$ – sushi Apr 17 at 20:56
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    $\begingroup$ Draw a position vs time graph (a Spacetime diagram). Your drawing can support and provide intuition to your calculations. (A Spacetime diagram is worth a thousand words.) $\endgroup$ – robphy Apr 17 at 21:53
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    $\begingroup$ This may help. $\endgroup$ – mmesser314 Apr 17 at 23:50
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Three things I learned from studying these Relativity paradoxes are:

  • The events always happen. If one observer sees "pole hit the back of the barn", so does the other.
  • Most of the "paradox" stems from our intuition of absolute time and / or simultaneity. Two things that are simultaneous in one reference frame need not be simultaneous in the other reference frame. It's also why Emillio Pisanty in the comments highlighted at the same time - the "same time" is not the same time for both observers.
  • The English wording of the question can be quite loose, because of #2. Be very careful whenever "at the same time" - is involved.

In this case, one observer sees "pole hit the back of the barn" and "back of the pole coming into the front of the barn" at the same time. The other doesn't. You can calculate the time interval between which the other observer sees both events using the Lorentz transforms, as you have done.

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The answer is that the pole can either fit or not fit depending on who is observing it.

The observer standing to one side of the barn will consider that it fits if the trailing end of the pole enters the barn before the leading end leaves it. But that assumes he can view both ends of the barn simultaneously. Crucially, two simultaneous events for that observer do not appear simultaneous for the runner with the pole.

The apparent contradiction results from the fact that there are two 'real' events, namely the moment when the leading end of the pole pops out of the Barn and the moment when the trailing edge disappears into it. What the observers disagree upon is the order of those two events. If your clocks say the former event happened first, you will say the pole is too long for the barn as both ends were sticking out at some time. If your clocks say that the latter event happened first, you will consider that the pole fitted because its rear had gone inside before its tip had emerged at the other end.

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