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Interesting Paradoxes: The Ladder Paradox

Mohammad Yasir is a physics graduate from the University of Delhi and currently enrolled in the master's programme at IIT.

The Ladder Paradox

The Ladder Paradox

Setting Up the Problem

If you compare it to the other paradoxes I have discussed, the Ladder Paradox, also known as the barn-pole paradox, is a relatively modern problem, especially as far as Aristotle's Wheel or Achilles and the Tortoise are concerned. Regardless, it is still a remarkably interesting discussion. If you enjoyed the topic of special relativity in your studies, keep reading. You will feel totally at home.

On the other hand, if physics bores you (which shouldn't happen), worry not. This article will have you hooked on the problem at hand in no time.

Relativistic Velocities?

For those of you who skipped out on special relativity, let us take a refresher. This will also serve to set up the problem for any reader who might not be from a scientific background.

We start by recalling what happens to the length of an object when it moves so fast that its speed becomes comparable to the speed of light (300,000 km/s). This is what is known as the relativistic scenario. At such high speeds, a plethora of strange effects start to occur. One of them is known as length contraction. Here's what it entails:

Imagine that a rigid object, say a cube, is moving parallel to the ground at an enormously high speed. According to the concepts of special theory of relativity, the dimension of such an object parallel to the direction of motion will contract. That is, it will appear shorter than it is to someone who happened to be sitting nearby.

Note that this contraction only occurs along the direction of motion. The other dimensions won't be affected. Thus, a cube would become a cuboid, not a smaller cube. Here's a little diagram.

The black line represents the ground. A cube moving along the ground will suffer length contraction only along its direction of motion.

The black line represents the ground. A cube moving along the ground will suffer length contraction only along its direction of motion.

Back to the Topic at Hand

Now that we know what length contraction is, let's see the effect it will have. Imagine that you are sitting near a garage. You see the ladder approach the garage at a relativistic velocity. You already know that the ladder cannot fit inside.

But here's where the fun begins! Due to length contraction, the ladder is shorter now. If you are sitting near the garage, you will be able to witness a moment where both the front and back end of the ladder are simultaneously inside the garage. In other words, the ladder fits inside the garage.

Due to length contraction, when the ladder is moving, it is able to "fit" inside the garage.

Due to length contraction, when the ladder is moving, it is able to "fit" inside the garage.

Let's Change Perspectives

Trouble begins to brew when you change your perspective. Initially, you were sitting near the garage as a silent observer. Suppose that was not the case. Suppose you were somehow superhuman enough to carry the ladder at relativistic velocities towards the garage. What then?

In such a case, the length of the ladder would remain the same per your perspective. For you, the ladder is at rest, but the garage isn't. Instead, it is coming towards you with relativistic velocities. Now, can you guess what would happen to the garage? It would suffer length contraction and thus, it would become shorter than it already was when everything and everyone was at rest.

That being the case, you would say that the garage is too short for the ladder to fit inside. In fact, it is shorter than it already was and now, an even larger portion of the ladder protrudes out. And therein, our paradox starts.

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When you are moving with the ladder, the garage suffers length contraction and now, is even shorter than it was when everything was normal.

When you are moving with the ladder, the garage suffers length contraction and now, is even shorter than it was when everything was normal.

The Paradox

We just discussed two different scenarios as follows:

  1. You are sitting near the garage and the ladder is moving at relativistic velocities compared to you. The ladder can fit inside the garage due to length contraction.
  2. You are the one moving the ladder and thus, the garage suffers length contraction from your perspective. Therefore, the garage is even shorter than it was, and the ladder cannot fit inside.

Thus, in the second case, your observation contradicts your observation from the first one. Naturally, it is not possible for the garage to be both shorter and longer than the ladder. So, which one is it?

A Small Digression

Before we can understand why this discrepancy arises, we need to know one more thing. Riddle me this:

You already have an intuitive answer to this question. What I wish for you to grasp is the scientific definition of one object being "completely inside" another object.

In the language of physics, the ladder is said to fit inside the garage when the positions of its front and back ends are inside the garage a specific instant in time. And that last portion is important. Both ends must be inside the garage simultaneously.

Resolving the Paradox (Finally!)

Now how does this knowledge help us resolve the paradox? Well, as it happens, another postulate of special relativity is that simultaneity is relative. That is, depending on the observer or reference frame, two events may or may not be simultaneous.

For example, suppose you were sitting in a room, and you heard two firecrackers go off simultaneously. For an observer moving at relativistic velocities compared to you, the two firecrackers wouldn't go off simultaneously. Instead, there would be a delay. Thus, one observer's perception of two events being simultaneous does not necessarily guarantee the same for all other observers. It's all relative.

Now, as I said, for the ladder to fit inside the garage, its ends must be inside the garage simultaneously. Since simultaneity is relative, the definition of when the ladder fits inside the garage is also relative. Therefore, you will observe different results in the two cases we discussed in this article. That does not mean your observations are contradictory. The ladder fitting inside the garage depends on the simultaneity of its front and back ends being inside. And thus, the paradox is resolved.

Other Paradoxes

If the world of Paradoxes interests you, here are a few other fun problems you can look at:

  1. Achilles and the Tortoise
  2. Aristotle's Wheel
  3. The Monty Hall Problem

That's all for now. Hope you have a fun time racking and riddling yourself. Good luck.

© 2022 Mohammad Yasir

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