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Ladder ParadoxThe ladder paradox is a thought experiment in special relativity. A long ladder travelling horizontally at almost the speed of light undergoes a length contraction and is made to fit into a smaller garage. Length Contracted Ladder Entering a Garage However, because of symmetry, it can be argued that from the perspective of the ladder it is the garage undergoing a length contraction and therefore cannot contain the ladder at all. Length Contracted Garage Too Small for Ladder 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 opens again to let the ladder out the other side.Length Contracted Ladder Entering a Two-Door Garage 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. Length Contracted Two-Door Garage Passing Over a Ladder This simple analysis can be applied to each part of the ladder itself and the problem of how a long ladder can fit inside a small garage can be solved. Ladder Contracting to Fit into the 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 accelerating in order catch up to the garage, then the next point of the ladder accelerating, followed by the next point, and so on until finally the entire ladder accelerates. In other words, the ladder contracts under its own acceleration as it suddenly accelerates 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 reverts back to its original length, possibly causing damage to itself and the garage.) 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.
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