Last evening, I suddenly realised that I hadn’t yet written down the solution to the pole-barn paradox that I wrote about a while ago. Gawd, it’s been more than a month!
Just to put the problem in perspective, let me refresh our memories with this image.

Like most other paradoxes of relativity, this one too stems from the fact that simultaneity has no meaning at speeds relativistic enough. Now what do I mean by that? As Lorentz’s transformation goes,
\begin{aligned} \Delta x' = \Delta x \cosh \theta + \Delta t \sinh \theta\\ \Delta t' = \Delta t \cosh \theta - \Delta x \sinh \theta\\ \end{aligned}
theta being the rapidity, the primed and the unprimed coordinates representing those in the rocket frame and laboratory frame respectively.
So suppose there are two events which happen simultaneously in one frame, but have a spatial separation, i.e. if $\Delta t'=0$, then $\Delta t$ will not be $0$, thus those two events are not simultaneous in the second frame.
Coming back to the problem at hand, in our case there are two events, the pole head merging with the barn back door, and the pole tail merging with the barn front door. In the barn frame, it seems that the two events are simultaneous. And hence, someone standing by the barn would say that the pole was (momentarily) fully contained inside the barn. Now, to you (in the pole’s frame), these two events happen at different instants of time, which basically means that to you, the pole is never completely contained inside the barn.
The next logical question would be, what is really happening? How could two things so different physically, i.e. being or not being contained by the barn, be happening in the two different frames? There must be only one of these that is actually happening, right?
Well, no.
What you’re implicitly assuming when you say this is that there must be a grand universal frame of reference which shows the actual truth. However, one Mr Einstein showed us how every frame, no matter how big or grand it may be, has an equal footing in terms of the validity of its observations. So to the observer standing by the barn, the pole is contained fully by the barn (momentarily), while, to you, carrying the pole, it seems that the pole is never completely contained within the barn.
If you’re keen enough on taking this further, you might say, what if the observer in the barn frame closes its back door at exactly the moment he sees the pole head merging with the door?
Well, in that case the pole obviously won’t be able to move any further ahead after its head reaches the back door. So in the pole frame, it seems that the pole should crash with part of it’s tail sticking out of the front door of the barn. All we’ll be left with then is a pole at rest inside a barn with part of it sticking out of the front door. But, nothing can stop instantaneously, not even this pole. So what that amounts to is the fact that once the pole’s head is stopped, a shockwave is sent out along its length, carrying the information of the crash. Think of it as a kind of a vaporising effect being carried along the length of the pole, so that as soon as the wave crosses a point, it gets vaporised.
However, the speed of that shockwave is also limited by the speed of light. In the barn’s frame, the pole was already inside, so when the shockwave reaches the tail of the pole, it’ll be well inside the barn. It turns out, even in the pole frame, the shockwave can never reach the tail of the pole before it’s well inside the barn’s front door. A more graphical explanation of this last statement can be found here.
As always, if you’re reading this, I’d love to know your opinion, and suggestions on the write-up. Thanks for visiting. :simple_smile: