Relativity in brief... or in detail..

The pole and barn paradox (ladder and garage paradox).

The pole and barn paradox (or the ladder and garage paradox) poses the question: Is the symmetry of length contraction paradoxical? Einstein's theory of Special Relativity makes several predictions, some of which seem counter-intuitive. These predictions relate to different inertial frames of reference with a constant relative velocity v. The effects are only noticeable when v is a substantial fraction of c, the speed of light. Length contraction refers to the observation that measurements of length of an object made within a frame in which the object is stationary (its 'proper length') are larger than those made by an observer in another frame moving at v with respect to the first. In other words, moving objects appear to contract in the length of their relative motion. Time dilation refers to the observation that measurements of elapsed time between events occuring at the same position as measured in one frame (the 'proper time' between those events) are less than those made by an observer in another frame moving at v with respect to the first. In other words, moving clocks seem to run slow. Despite its internal logic, some aspects of Special Relativity are thus counter-intuitive, because we are (nearly all) unaccustomed to measuring objects moving at speeds approaching c. Here is an example.

Pole vaulter Emma travels at speed v and holds a pole. If Emma measures the length of the pole, which is stationary in her frame, she measures length Lo, which is called the proper length. ('proper' here means characteristic of its own frame, not correct or superior.) According to spectator Eric, Emma is moving so fast that he sees her pole to have a relativistic length contraction. He measures its length as L < Lo. In fact, Emma's back-up pole is lying on the ground next to Eric, and he sees it as longer than the one she is carrying. He explains this to Emma. Emma disagrees: she measures her pole as Lo and on the contrary she reckons that Eric must have shrunk her back-up pole.

Size of poles is an important issue to pole vaulters and Eric organises this experiment: he builds a shed which is just as long as Emma's moving pole, as he measures it. The shed has a door at either end. Emma, with her contracted pole, will run into the shed and he will shut both doors just when her pole fits inside. For an instant at least, Emma's pole will be entirely inside the shed and he will have proved that her pole has shrunk. (Emma is holding the pole in an unusal position, which will be useful further on.)

The experiment is run and Eric thinks that it is conclusive (first diagram).

Emma differs, however. Here is how she saw it:
"You cheated," she says "you closed the back door when my pole had already poked through the front door! My pole was always longer than your shed." Their disagreement is now about the timing of the closing of the doors. Did two distant events happen simultaneously or not?

So they try again, and this time Eric sets off a flash bulb simultaneously when the doors are closed. Here is his view of events, with the flash bulbs represented by white circles. It just so happens that, according to Eric, Emma was at the midpoint of the shed when the flash bulbs went off. (Hence the strange and athletically unhelpful way of holding the pole.)

Now Eric expects that Emma will receive the flash from the front before the one from behind, because he sees her moving to the left. No problem for him. Emma also receives the light from the front before she gets the light from behind — and this is the basis for her charge of cheating. For her, light travels at the speed c, she receives the light from the front first. Hence her claim that Eric fired this flash first. Simultaneity is relative: one consequence of the theory of relativity is that two observers may disagree on whether or not two events are simultaneous. (Indeed, if the two events are a long distance D apart, but close together Δt in time (if D/Δt >c), they can even disagree upon which happened first.)

Their respective points of view are shown below in what are called space-time diagrams. This means that we plot time on the vertical axis as a function of the position of events. Because the two characters disagree over time and length, we must give them two separate diagrams (below) and, to convert between the (x,t) and the (x',t') frames, we need to use the Lorentz transform equations. In this figure, the bold lines are the ends of Emma's pole, the dashed bold lines are the ends of Eric's shed, and the fine lines are the flashes of light.

Note that the two flashes are simultaneous to Eric, but not to Emma. This paradox is set up by the incautious use of the word 'when', which is italicised above for that reason. In its normal use in speech, absolute simultaneity is assumed: English grammar does not require a clause specifying the frame of reference in which the simultaneity implied by 'when' is observed*. In discussing relativity, such a qualification is required. In fact, this whole story is analogous to the animation shown in our section on Simultaneity.

* Has anyone ever invented a language that included relativistic requirements in its grammar? It might be more useful than Vogon poetry.

Let's forget about light!

A reader posed this question: "But what if Eric decides not to use flash bulbs and light? What if he mechanically closes the doors using long, rigid rods, one connected to each door? He pulls simultaneously on each rod and, he reasons, the doors will close simultaneously. Simultaneously for everyone!"

The problem lies in the word "rigid". When he pulls on the end of a rod, the other end does not move instantaneously. It takes a finite time for a mechanical pulse to travel along a solid (and in fact the mechanism involves electromagnetic interactions among atoms, so it cannot be faster than light and in practice is vastly less). The speed of such a wave is not infinite (just a few km/s if the rods are steel). So there is no such thing as an infinitely rigid rod. Eric would see a mechanical wave travel along each rod at the same speed. But Emma would see the waves to have different speeds, so the two events would not be simultaneous.

So, who is right? From Eric's point of view, he is: the pole's two ends were simultaneously (in his frame) inside the barn. From Emma's point of view, she is: the pole's two ends were never (in her frame) simultaneously inside the barn: the pole is longer than the barn. Both are right — according to their own point of view. And in relativity, there is no absolute point of view.

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