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The Pole and Barn Paradox

Five seconds into the future

The pole and barn paradox is a classic demonstration of the breakdown of universal simultaneity in special relativity. Barney stands beside a barn with doors open at both ends. Polly carries a pole; when it and the barn are at rest, the pole is longer than the barn. Polly runs toward the barn so quickly (a significant fraction of the speed of light) that Barney measures the pole she carries as being shorter than the depth of the barn due to Lorentzian length contraction. Yet to Polly, it's the barn that's contracted. We ask both “will there be a moment when the pole fits entirely inside the barn?”. Barney says yes; to him, the moving pole is shorter than the barn's depth. Polly says no; to her, the pole is still longer than the now-shallower barn. Who's right?

The standard answer is "both", and it's usually followed by spacetime diagrams, calculations, and talk of simultaneity and multiple frames of reference. What this page attempts is to make the lack of simultaneity obvious. Here's the standard spacetime diagram from Barney's reference frame; the blue region is the barn and the red region is the pole, which is moving left to right. The dashed lines are the lines of simultaneity: to Barney, everything on the blue dashed line happens at the same moment, while to Polly everything on the red dashed line happens at the same moment.

This diagram holds all the information needed to resolve the apparent paradox, but is still prone to fostering a misconception about Barney's view of the pole. What is it that he sees, exactly, at the moment indicated by the dashed blue line? Is Polly's pole completely within the barn?

One minor tweak to the problem makes things clearer. Imagine that in Polly's reference frame the pole changes color smoothly over time, continuously cycling through the visible spectrum. At any moment Polly sees the entire pole as one hue, but which hue she sees depends on which moment she looks.

How will the pole look to Barney and Polly? To figure this out, look at each of the lines of simultaneity in the second diagram. Along Polly's, the pole is a uniform dark blue hue:

What does the pole look like to Barney? For one thing, it's significantly shorter than the pole Polly sees. Much more interesting, though, is the fact that the pole isn't a uniform hue. Its color varies continuously along its depth.

Since each hue represents a different moment in Polly's reference frame, the pole Barney sees at a given moment is not simply a shortened version of one of “Polly's” uniform-hue poles, but is instead a smoothly varying collection of infinitesimally thin slices from different moments in Polly's reference frame. In essence, he's seeing both past and future moments in the life of the pole as seen from Polly's frame. If Polly had mounted synchronized clocks on the ends of the pole before starting her run, in her frame they'd still be in synch. In Barney's frame, the clock at the leading end of the pole would show an earlier time than the clock at the trailing end.

The argument can be reversed for the two frames. If the barn were changing color so that Barney always saw a uniform hue, to Polly its hues would vary along its length.

The hidden assumption in the apparent paradox is that when we envision “the pole”, we imagine an actual pole. In our daily existence poles don't change from moment to moment, so our mental model of a pole is of something uniform, something always the same at every moment for all observers. Once we recognize and break that assumption, it becomes easier to comprehend how observers in different frames can disagree about an object's appearance.

Associating proper time with color can help explain other apparent paradoxes of special relativity, particularly ones involving rigidity (e.g., the manhole paradox). In general, using color to indicate a scalar quantity is a useful tool, particularly for visualizing higher dimensions.

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Thanks to Prof. Tevian Dray for stimulating discussions, and to Christopher for making me redo the diagrams correctly.

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