Ladder paradox
From Freepedia
The ladder paradox or (barn-pole paradox) is a thought experiment in special relativity. If a long ladder travels horizontally at almost the speed of light, it will undergo a length contraction and will therefore fit into a garage which is shorter than the ladder's rest length. On the other hand, from the point of view of the ladder, it is the garage which is moving, and is therefore length contracted, and will need to be larger than the rest length of the ladder in order to contain it.
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Relative simultaneity
Image:Ladder paradox garage irf2.png
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.
Image:Ladder paradox ladder irf2.png
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 <math>v=c\sqrt{1/2}</math> in the positive x direction, therefore <math>\gamma=\sqrt{2}</math>. Since light travels at very close to one foot per nanosecond, lets work in these units, so that <math>c=1</math>ft/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 <math>t_A=G/v\approx14.14</math> ns. 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.
Using the one-dimensional Lorentz transformation (see special relativity) we can get:
| Event | Garage Frame | Ladder Frame |
|---|---|---|
| <math>A\,</math> | <math>t_A=\frac{G}{v}\approx 14.14\,</math> ns
<math>x_A=L=10\,</math> ft | <math>t^'_A=\frac{G}{\gamma v}=10\,</math> ns
<math>x^'_A=0\,</math> ft |
| <math>B\,</math> | <math>t_B=t_A\approx14.14\,</math> ns
<math>x_B=G-\frac{L}{\gamma}\approx 1.514\,</math> ft | <math>t^'_B=18.48\,</math> ns
<math>x^'_B=-L=-12\,</math> ft |
| <math>C\,</math> | <math>t_C\approx2.142\,</math> ns
<math>x_C\approx-6.971\,</math> ft | <math>t^'_C=10\,</math> ns
<math>x^'_C=-12\,</math> ft |
| <math>D\,</math> | <math>t_D=12\,</math> ns
<math>x_D=0\,</math> ft | <math>t^'_D=16.97\,</math> ns
<math>x^'_D=-12\,</math> ft |
Resolution of the paradox
Image:Ladder paradox contraction.png
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 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.)
Man falling into grate variation
Image:Ladder paradox grate variation.PNG
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.
Reference
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Further reading
- Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics (2nd ed) (Freeman, NY, 1992)
- - discusses various apparent SR paradoxes and their solutions
