According to
the special theory of relativity an object that is in motion relative to a
givenframe of reference contracts – becomes shorter in length – along the
direction in which it is moving. But
this seems to lead to a paradox. Consider
a pole that is 15 meters long when at rest, and consider a barn that is 10
meters long when at rest. Suppose that
the barn has a door at each end. Clearly
the pole should not fit inside the barn with both doors closed.

Now suppose
you can get a friend to run through the barn carrying the pole at (4/5)c – i.e.
at 4/5 the speed of light. Then, from
the barns rest frame, the barn is still 10 meters long, but the pole is only 9
meters long (i.e. the length of the pole in the barns rest frame = 10m x
sqrt(1-(v/c)^2) = 15 x (3/5) = 9m). So, for a brief moment, as the pole passes
through the barn, the pole fits completely inside the barn with both doors
closed.

However,
from your friend’s frame of reference, where the pole is at rest and barn is
moving toward him at (4/5)c, the pole remains 15m long. But the barn becomes
shorter. Its length becomes 6m (i.e. the length of the barn in the poles
rest frame = 10m x sqrt(1-(v/c)^2) = 10m x (3/5) = 6m). So, as it passes
through there is no way that the pole should fit completely inside the barn
with both doors closed.

But the pole
either fits completely inside the barn (for a brief moment as it passes
through), or it doesn’t. This is the
paradox.

So from the
point of view of the pole, its length never changes, it is always 15 meters
long, and the length of the barn is ever decreasing so that from the poles
frame the pole could never fit within the barn all at once. And, from the point of view of the barn, the
pole is ever decreasing so at some point it should be able to fit within both
doors of the barn. So the diagram shows
that in the barn frame the pole will fit and in the pole frame it is quite to
long to fit. Which one of these is the
correct assumption?

The solution to the paradox is in the
relativity of simultaneity. That is that
what one observer sees as simultaneous is not the same as what another observer
sees as simultaneous. “At the same time”
does not apply to those events which are separated by space in special
relativity theory. This thought
experiment shows that there are circumstances when there is no correct
answer. Neither observer has privileged
status or rightness. And even that all
included observers can claim to be correct, at least in their own independent
reference frames, even if their ordering of events differs from another’s.

*This article originally written February 25th, 2008 for OU PHIL 3623 - Physics and Cosmology.*

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