The Barn-Pole Paradox - Maple Help

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The Barn-Pole Paradox${}$

Main Concept

The barn-pole paradox is a thought experiment that asks what appears to be a simple yes or no question, namely, whether or not a certain pole can fit into a certain barn; but apparently this paradox does not have a simple answer. In particular, no matter what the answer may be, if the pole is at rest relative to the barn, that answer can change if the pole is moving fast enough relative to the barn, due to relativistic length contraction and an appropriate choice of reference frame.

Suppose there is a pole of length 20 m and a barn of width 10 m. According to Einstein's theory of special relativity, if the pole is moving fast enough from the barn's perspective, it will be length-contracted and thus fit inside the barn. In contrast, from the pole's perspective the barn is moving and is length-contracted so there is no way the pole will fit!

The paradox results from the mistaken notion of absolute simultaneity. If both ends of the pole are in the barn simultaneously, then the pole fits in the barn. In special relativity, having two events be simultaneous in one reference frame does not imply that they are simultaneous in another frame.

Therefore, since simultaneity is relative to each observer, both observers were correct! From the barns perspective the pole fit in the barn, and from the pole's perspective, it did not fit in the barn.

Special relativity

Einstein's theory of special relativity describes the relationship between space and time; in particular, they are not independent quantities, but rather two sides of the same coin, namely spacetime. There are two postulates in special relativity:

 • The laws of physics remain the same in all inertial (non-accelerating) frames of reference.
 • The speed of light is constant and does not depend on the speed of the light source relative to the observer.

These postulates are in direct contradiction with the traditional notions of absolute space and time that provided the background for pre-20th century physics, but experiments have proven them to be correct.

The Lorentz transformation, time dilation and length contraction

The coordinates defining the position of an event in spacetime depend on the choice of inertial reference frame. The two postulates mentioned earlier imply that these coordinates must transform in a certain way known as the Lorentz transformation when switching reference frames. Suppose that $S$ and $S'$ are two reference frames with coordinates $\left(x,y,z,t\right)$ and , such that the origin of both frames corresponds to the same event, and $S'$ is moving at speed $v$ in the $x$-direction relative to $S$. The coordinates are related via:

 , , , , where .

In particular, if an object accelerates to a speed $v$ close to the speed of light, relative to an observer, then that observer will see that the object has shrunk by a factor of $\mathrm{γ}$  relative to its rest length. Similarly, it will appear to be aging more slowly.

 Worked example As mentioned earlier, suppose there is a pole of length 20 m and a barn of width 10 m. Exactly how fast would the pole have to be going for it to fit inside the barn from the barn's perspective? Let the common origin  in both frames be the event when the leading end of the pole gets to the back door of the barn (assuming the pole is moving through the barn from front to back). We hypothesize that at the same time in the barn's frame of reference, the trailing edge of the pole coincides with (has the same $x$-coordinate as) the front door of the barn. The coordinates for the second event in the barn's frame are , and in the pole's frame of reference they are . Plugging these values into the second equation of the Lorentz transformation yields:   , which requires $\mathrm{γ}=2$, and thus . From the pole's point of view, what time is it when the end of the pole finally makes it into the barn?   The answer is just the value of $t'$ stated earlier. To find it, use the first equation in the Lorentz transformation:    .   Admittedly, this is a really short time after $t'=0$, when the front of the pole was already hitting the end of the barn, but given the speed involved it is still an important difference, and it proves that the pole did not quite fit into the barn (in the pole's reference frame). From the pole's perspective, how much of the pole had still not fit into the barn when the front of the pole hit the back of the barn? We need to find the $x'$ coordinate at time $t'=0$, so we use the second equation of the inverse Lorentz transformation, noting that the $x$ coordinate for this event is -10 m:   . Plugging in the numbers gives:   , .   In other words, from the pole's perspective, only one quarter of the pole can fit into the barn!

Adjust the speed of the pole relative to the barn and manually move the pole to see if it fits into the barn. Then switch between reference frames to see how much difference it makes! Also, try starting with a pole with rest length less than 10 m and try adjusting the parameters so that the pole no longer fits.

 Reference Frame Rest length of pole l (m) Speed $\frac{v}{c}$ Distance $x$ (m)

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