69. Rashomon:

The Ladder Paradox

The Question:

A carpenter contemplates how to store away a ladder. His workshop would be the ideal place but, unfortunately, the ladder is slightly too long to fit inside. What to do? One way would be to cut off the ladder’s top (or bottom) rung. But that would be a waste. Thinking out of the box, he decides to run towards the workshop, carrying the ladder horizontally on his shoulder. It’s no ordinary run, but a quick dash close to the speed of light. According to the Special Theory of Relativity, at such high speeds the ladder contracts and, lo and behold, it will fit into the workshop.

Correct?

The Paradox:

Yes and No.

Yes, when observed by someone standing next to the workshop: by the Theory of Special Relativity, the ladder does indeed contract to a length shorter than the workshop.

No, from the point of view of the carpenter: by the Theory of Special Relativity, it is the workshop that approaches the carpenter with a speed close to the speed of light. Hence, it is the workshop that contracts! Now the ladder is even less likely to fit inside.

The situation reminds of Akiro Kurosawa’s masterpiece from 1950, the black and white film Rashomon: the same event can be interpreted in contradictory ways when viewed from different perspectives.

So, what is going on?

Background:

Albert Einstein’s Theory of Relativity says that all movement is relative: one may consider, for example, a policeman standing at an intersction may spot a car speeding towards him; or the driver in the car may observe the policeman getting closer and closer. In our scenario above, we first assume the ladder approaching the stationary workshop. But by the same token, one may assume the ladder as being stationary, and the workshop approching it.

Another key concept of the theory, postulated by Hendrik Lorenz in 1892, is that moving objects contract. The contraction is so minute, however, that it is only noticeable at speeds close to the speed of light.

The upshot of all this is that what’s correct for the carpenter, is wrong for the observer, and vice versa. The paradox was dreamt up by the physicist Wolfgang Rindler in 1961.[1] His paper was about a man walking on the street, albeit at close to the speed of light, towards a manhole that has been left uncovered. A man standing next to the manhole fully expects the fast man to fall in. “Yet to the fast man, the manhole is much narrower even than to the stationary man, and he certainly does not expect to fall in.”[2] So, Rindler asks “which is correct?”

Dénouement:

What does ‘to fit inside’ actually mean? The notion implies that at some brief moment in time, the doors at both ends of the workshop can be shut simultaneously, while the ladder is entirely ocntained inside. But in the context of the Theory of Special Relativity, simultaneity itself is a relative concept. Two occurrences that seem simultaneous to one observer, may not seem to happen at the same time according to another observer. What happens at the front end of the ladder and what happens at the back end may seem simultaneous to an observer near the workshop, but not simultaneous to the carpenter running with the ladder.

Let us say, you are the observer, standing next to the workshop, and look on, as the caprenter with the ladder approaches the back end of the workshop. Both doors are open. The carpenter enters the workshop through the back door and the front end of the ladder approaches the workshop’s front door. As soon as the ladder, now contracted, fits inside the garage, both doors shut simultaneously for the briefest of moments. Then the front door opens to let the ladder out again.

According to the carpente, this is not at all what what occurred. He claims that the doors did not close shut simultaneously. Running with the ladder on his shoulder, he saw the following: as the front end of the ladder approached the front end of the garage – while the back end of the ladder was still outside – the front door closed. Then, the front door opened, the front end of the ladder moved outside, and only then – when the ladder’s back end has cleared the workshop’s back threshold – did the back door close. So, at no point in time did the ladder fit entirely inside the workshop.

What the observer standing next to the workshop saw as simultaneous, was not at all simultaneous according to the carpenter. It was Rashomon all over again.

Technical supplement:

In Rashomon, five people tell of a horrifying story. In a forest, a woman was raped by a bandit, and her husband, a samurai, was murdered. A woodcutter witnessed the crime. At court, they all tell their stories, as they perceived them, the dead samurai through a medium. Shaped by the different viewpoints, they all give different interpretations to the event. The audience is left, at the end, wondering which version is the correct one. It’s the Theory of Relativity all over again.

[1] Wolfgang Rindler, "Length Contraction Paradox": Am. J. Phys., 29(6) June 1961

[2] I have replaced ‘grid’ in Rindler’s paper with ‘manhole’.

© George Szpiro, 2019

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