Ladder paradox

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The ladder paradox or (barn-pole paradox) is a thought experiment in special relativity. If a ladder travels horizontally it will undergo a length contraction and will therefore fit into a garage that is shorter than the ladder's length at rest. On the other hand, from the point of view of an observer moving with the ladder, it is the garage that is moving and the garage will be contracted. The garage will therefore need to be larger than the length at rest of the ladder in order to contain it.

Figure 1a: The problem, part 1 - A length contracted ladder being contained within a garage

Figure 1b: The problem, part 2 - A length contracted garage being too small to contain the same long ladder

[edit] Relative simultaneity

Figure 1c: The solution from the garage's point of view: A length contracted ladder entering and exiting a two-door garage

The solution to this dilemma lies in the fact that what one observer (e.g. the garage) considers as simultaneous does not correspond to what the other observer (e.g. the ladder) considers as simultaneous (relative simultaneity). A clear way of seeing this is to consider a garage with two doors that swing shut to contain the ladder and then open again to let the ladder out the other side.

Figure 1d: The solution from the ladder's point of view: A length contracted two-door garage passing over a ladder

From the perspective of the ladder what happens is that first one door closes and opens and then, after the garage passes over the ladder, the second door closes and opens.

The situation is illustrated in the Minkowski diagram below. The diagram is in the rest frame of the garage. The vertical blue band shows the garage in space-time, the red band shows the ladder in space-time. The x and t axes are the garage space and time axes, respectively, and x′ and t′ are the ladder space and time axes, respectively. The ladder is moving at a velocity of in the positive x direction, therefore . Since light travels at very close to one foot per nanosecond, let’s work in these units, so that c = 1ft/ns. The garage is a small one, G=10 feet long, while the ladder is L=12 feet long. In the garage frame, the front of the ladder will hit the back of the garage at time . This is shown as event A on the diagram. All lines parallel to the garage x axis will be simultaneous according to the garage observer, so the blue line AB will be what the garage observer sees as the ladder at the time of event A. The ladder is contained inside the garage. However, from the point of view of the observer on the ladder, the red line AC is what the ladder observer sees as the ladder. The back of the ladder is outside the garage.

Figure 1e: A Minkowski diagram of ladder paradox No. 1. The garage is shown in blue, the ladder in red. The diagram is in the rest frame of the garage, with x and t being the garage space and time axes, respectively. The ladder frame is for a person sitting on the front of the ladder, with x′ and t′ being the ladder space and time axes respectively. The x and x′ axes are each 5 feet long, and the t and t′ axes are each 5 ns in duration.

Using the one-dimensional Lorentz transformation (see special relativity) we can get:

Event	Garage Frame	Ladder Frame

[edit] Relativistic trains passing

Another dramatisation of the same effect is to consider two relativistic trains approaching each other along a single track, with only a short length of double track for the trains to pass, at a station platform.

The platform is shorter than the rest length of the trains (just as the garage is shorter than the ladder); but the trains can just pass (from the perspective of an observer on the platform), thanks to the relativistic length contraction: the head of each train reaches the end of the platform just as the tail of the other train clears it, going in the other direction.

From the train perspective, it is now the platform which is contracted. As before, the nose of the train reaches the end of the platform just as the tail of the other train clears it. However, in this perspective, the train still has its full rest-length. So the tail of the first train is still in the section of single-track - the front of the platform has not yet reached it. But there is no collision, because the oncoming second train is even more length contracted than the platform, and so its nose has not yet reached the front of the platform. There is just time for tail of the first train to clear the front of the platform, as the nose of the second train reaches it.

So a crash is again averted, in this frame because of the relativity of simultaneity: what is simultaneous to an observer standing on the platform is not simultaneous for an observer on one of the trains.

[edit] Resolution of the paradox if the ladder stops

A ladder contracting under acceleration to fit into a length contracted garage

When the stationary garage traps the moving ladder, what happens after the event is either 1)the ladder continues out the other side of the garage (the above two-door garage example) or 2)the ladder comes to a complete halt within the garage. Considering just the latter, we can say that every point of the ladder is simultaneously at rest from the perspective of the garage. From the perspective of the ladder this cannot be true. What the ladder experiences is one end decelerating in order to catch up to the garage, then the next point of the ladder decelerating, followed by the next point, and so on until finally the entire ladder decelerates. In other words, the ladder contracts under its own deceleration as it suddenly decelerates to catch up to the garage (enter the inertial reference frame of the garage). And so from both perspectives the garage manages to contain the ladder. (After that, since it is no longer moving in relation to the garage, the ladder may revert back to its original length, possibly causing damage to itself and the garage.)

Figure 1 - A Minkowski diagram of the case where the ladder is stopped all along its length, simultaneously in the garage frame. When this occurs, the garage frame sees the ladder as AB, but the ladder frame sees the ladder as AC. When back of the ladder enters the garage at point D, it has not yet felt the effects of the deceleration. At this time, according to someone at rest with respect to the back of the ladder, the front of the ladder will be at point E and will see the ladder as DE. It is seen that this length in the ladder frame is not the same as CA, the rest length of the ladder before the deceleration.

[edit] Man falling into grate variation

A man (represented by a segmented rod) falling into a grate

This paradox was originally proposed and solved by Wolfgang Rindler ("Length Contraction Paradox": Am. J. Phys., 29(6) June 1961) and involved a fast walking man, represented by a rod, falling into a grate.

From the perspective of the grate the man undergoes a length contraction and falls into the grate. However from the perspective of the man it is the grate undergoing a length contraction and seems too small for the man to fall through.

Using the solution of relative simultaneity, we can see that from the perspective of the grate every part of the man falls into the grate at the same time while from the perspective of the man his front falls into the grate before his rear, as can be easily seen if the man is represented by a segmented rod. This causes him to bend into the grate, which is similar to the ladder contracting to fit into the garage. In fact, if the man hits a bar of the grate as he falls in, since he is travelling forwards as well as falling and hitting the bar brings him to a halt, he undergoes the same contraction that the ladder experiences as well as bending into the grate.

[edit] Recent criticism

However, the above "man falling into grate" solution has recently been criticised in an article by Van Lintel and Gruber in the Eur.J.Phys.26 (Jan. 2005), p.19: the above solution implied that proper stiffness could be affected by relative speed. The new solution is that the rod's (unaffected) proper stiffness is related to the thickness of the rod, which thickness implies a time delay before any part of the upper surface of the rod can start falling. In this particular case, the rod will even arrive at the other side of the hole before any part of the upper surface "feels" the effect of the hole.

In other words, the above pictures are misleading: the correct solution is that the rod remains as straight (stiff) as one would expect, and that therefore the rod doesn't fall into the hole.

[edit] See also

• Bell's spaceship paradox
• Ehrenfest paradox
• Physical paradox
• Supplee's paradox
• Relativity of simultaneity

[edit] References

• Rindler, Wolfgang (2001). Relativity: Special, General and Cosmological. Oxford University Press. ISBN 0-19-850836-0. 

• H. van Lintel et al, The rod and hole paradox re-examined (abstract only)

[edit] Further reading

• Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics (2nd ed) (Freeman, NY, 1992)

- discusses various apparent SR paradoxes and their solutions