Ladder paradox

The ladder paradox (or barn-pole paradox) is a thought experiment in special relativity. It involves a ladder, parallel to the ground, travelling horizontally at relativistic speed (near the speed of light) and therefore undergoing a Lorentz length contraction. The ladder is imagined passing through the open front and rear doors of a garage or barn which is shorter than its rest length, so if the ladder was not moving it would not be able to fit inside.

To a stationary observer, due to the contraction, the moving ladder is able to fit entirely inside the building as it passes through. On the other hand, from the point of view of an observer moving with the ladder, the ladder will not be contracted, and it is the building which will be Lorentz contracted to an even smaller length. Therefore, the ladder will not be able to fit inside the building as it passes through. This poses an apparent discrepancy between the realities of both observers.

This apparent paradox results from the mistaken assumption of absolute simultaneity. The ladder is said to fit into the garage if both of its ends can be made to be simultaneously inside the garage. The paradox is resolved when it is considered that in relativity, simultaneity is relative to each observer, making the answer to whether the ladder fits inside the garage also relative to each of them.

Paradox

Imagine a garage with open front and back doors, and a ladder that is too long to fit inside when both are at rest. If we move the ladder very fast through the garage, something interesting happens. Because it is moving so quickly, the ladder appears to become shorter due to a rule in physics called length contraction. For a moment, the ladder can fit completely inside the garage, and we could even close both doors at the same time to trap it inside.

But here’s where things get puzzling. If we look from the ladder’s point of view, it seems like the ladder is still its normal length, and instead, the garage is moving and appears shorter. From this angle, the garage looks too small to ever hold the whole ladder. This clash of viewpoints — where each observer sees the other as shorter — creates what we call the ladder paradox.

Resolution

The solution to the ladder paradox comes from the idea of the relativity of simultaneity. This means that what one person thinks happens at the same time might not be the same time to someone else moving very fast.

Imagine you have a garage and a ladder. If the ladder is moving very fast, it appears shorter to someone watching from the garage. When the ladder is moving, the doors of the garage close for a short time when the ladder seems to fit inside. But if you were moving with the ladder, you would see that the doors did not close at the same time, so the ladder never really fit inside the garage all at once.

A Minkowski diagram can help show this. In the garage’s view, the ladder looks like it fits at one moment. But in the ladder’s view, it never fully fits inside the garage at the same time. This shows how motion changes what we see as happening at the same time.

Shutting the ladder in the garage

In a more complex version of the ladder paradox, we can trap the ladder inside a garage by closing both doors while it is moving inside. From the garage's view, once the ladder is fully inside and the doors are closed, the ladder stops and becomes longer than the garage, which would seem to cause a problem.

However, from the ladder's viewpoint, it was always longer than the garage. The key to understanding this is that the parts of the ladder stop moving one after another, starting from the front and moving to the back. This explains how the ladder ends up trapped inside the garage even though it appears longer from its own perspective.

This situation is similar to the twin paradox, where one twin travels at high speed and returns younger than the other. In both cases, the difference comes from the acceleration and deceleration involved in the journey.

Ladder paradox and transmission of force

If the back door of the garage is closed and will not open, the ladder will stop when it hits the door. This creates a puzzle: in the garage’s view, the ladder fits inside before hitting the door, but the ladder thinks it is too long to fit. The problem comes from thinking of the ladder as perfectly solid and unchanging shape. But special relativity says nothing can send information faster than light. So when the front of the ladder hits the door, the back of the ladder does not know yet and keeps moving forward. Only later, when the news of the collision catches up, does the back of the ladder slow down. Both ways of looking at it agree on what happens.

After the news reaches the back of the ladder, different things could happen — the ladder might bend or, at very high speeds, break apart. light cone

Man falling into grate variation

This version of the paradox was suggested by Wolfgang Rindler. It imagines a fast-walking person, represented by a rod, falling into a grate. From the grate’s view, the rod gets shorter and fits into the grate. But from the rod’s view, the grate looks shorter, making it seem like the rod is too long to fit.

The key idea is that the acceleration happening at the same time for the grate does not happen at the same time for the rod. In the rod’s view, the front starts moving down first, then the rest of the rod follows. This can make the rod bend in its own view. For this effect to be noticed, both the rod and the grate need to be big enough for the time it takes to matter.

Main article: paradox

Bar and ring paradox

The bar and ring paradox is a simple version of a similar problem called the rod and grate paradox. In this scenario, imagine a bar that is slightly longer than the diameter of a ring. The bar moves upward and to the right while the ring stays still. Because of special relativity, the bar appears shorter when it moves very fast—this is called Lorentz contraction. When the bar’s center lines up with the ring’s center, the bar looks short enough to fit through the ring.

But here’s the puzzle: if we look from the bar’s perspective, the ring is moving instead. Now the ring looks squashed, but the bar does not. How can the bar go through the ring? The answer lies in how we think about “at the same time.” Because what’s happening “at the same time” can change depending on how you’re moving, the bar and ring aren’t always lined up perfectly. This lets the bar pass through the ring even when it looks like it shouldn’t.