The Pole-Barn Paradox

Herbert is a pole vaulter who uses a pole that is exactly 10 m long.  He can't easily store this in his small apartment, so his friend Arthur, a local farmer, offers to let him keep it in his barn.  But when Herbert stops by the farm, the two discover that the barn is only 8 m long, so the pole won't fit.

Or will it?  Arthur knows some physics, so he tells Herbert to walk back to the end of the field, hold the pole horizontally over his shoulder, and run as fast as he can towards the barn.  If he can run through the barn's front door moving at 80% the speed of light, the pole's length will have contracted from 10 m to 6 m -- a 40% reduction -- and it will fit inside the barn!  Just to prove the point, Arthur and his daughter Gertrude will stand by the front and back barn doors and will slam them shut just when Herbert reaches the middle of the barn, with 1 m of open space in front and behind.

(Yes, we're using the term "friend" loosely here.)

Herbert proceeds to carry out this mission.  However, as he speeds across the field towards the barn, he notices that things aren't as he and Arthur had discussed.  Suddenly he realizes that in his own reference frame, he and the pole are sitting still while the barn, Arthur, and Gertrude race towards him at 0.8 c, so his pole remains 10 m long but the barn's length contracts by 40% from 8 m to 4.8 m.  There's no way that the full-length pole will be able to fit within the shrunken barn!

We seem to have a conflict, depending on which reference frame we adopt: Will the pole fit inside the barn or won't it?

The special relativity principle tells us that so long as neither Arthur nor Herbert is accelerating, we can use either one of their two reference frames to analyze the situation and obtain correct answers.  "Yeah, that's easy for you to say," shouts Herbert as he zooms past us, "but how can this pole both fit inside and not fit inside?!?"

The key to resolving this paradox is to consider just what it means to say that the pole "fits inside the barn."  Here's a precise, "legalese" version of that statement: The two ends of the pole are inside the barn at the same time.

The crucial words in that precise statement are "at the same time."  Arthur and Gertrude have agreed to close the two doors at the same time, and in their frame of reference, that's just what happens, and both ends of the pole are simultaneously inside the barn at the moment when those doors slam shut.  But the relativity of simultaneity ensures that in Herbert's frame of reference, the two doors close at two different times, and the two ends of the pole are never inside the barn at the same time.

In order to analyze this situation using special relativity, we have to assume that Herbert (as viewed by Arthur) zooms right through the back wall rather than splatting to a halt against the wall -- that is, we must assume that Herbert remains in a state of uniform motion.  So in Herbert's frame, he and the pole sit still while a squashed barn moving leftward at 0.8 c plows into them and then past them.

Without getting into the calculations, here's the sequence of events as seen by Herbert:

• 1. The open front door reaches the right end of the pole.

  2. Gertrude shuts the back door.

  3. The closed back door slams into (and past) the right end of the pole.

  4. The closed back door, which now has a hole in it, slams into (and past) Herbert.

  5. The open front door reaches the left end of the pole.

  6. Arthur shuts the front door.

The point, once again, is that in Herbert's frame, the right end of the pole is back outside the barn (event 3) well before the left end is ever inside the barn (event 5).  The pole didn't fit.

Of course, Arthur sees these same events in the order 1, then 5, then 2 and 6 simultaneously, then 3, and lastly (ouch) 4.  Here's how Arthur would describe things -- and remember that in his frame, the (full-size) barn is sitting still and it's Herbert and his (contracted) pole that are moving:

• 1. The right end of the pole reaches the open front door.

  2. The left end of the pole reaches the open front door.

  3. Gertrude and Arthur shut both doors simultaneously.

  4. The right end of the pole slams into (and past) the closed back door.

  5. Herbert slams into (and past) the closed back door, turning a small hole into a large hole.

Thus in Arthur's frame the pole did fit, at least for a moment.

The two observers can't agree on which events are simultaneous; they can't agree on the order of events; and they can't agree on whether or not the pole fit in the barn.  And it's crucial to remember that they're both right.

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The Pole-Barn Paradox

Written by Chris Magri

Last modified on February 7, 2017

URL: https://sites.google.com/a/maine.edu/magri/phy110e/polebarn