Theoretical Physics, Relativity, and Ladders

The Problem

Meet you.

You’re an aspiring young roof tiler. Your boss is horrible, your coworkers are stuck up veterans, and you make below the minimum wage, despite working overtime. But you need to pay off that college debt from your master’s degree in theoretical physics, so you put up with it all anyway.

This is You (representation)

When you arrive home one night, you realize that the new ladder you bought didn’t fit into your garage; it was simply too long, and the folding mechanism was broken. Since you’re already behind on your electrical bills, you can’t afford to have it stolen in the middle of the night. Getting it in the garage is your top priority, and you’d do anything to do it.

But then you have an idea.

You tap into the grueling 8 years of your life you spent studying theoretical physics and formulate a solution. You understand that according to the principles of special relativity, when an object is moving relative to another observer, that object will be horizontally compressed in the direction that the object is moving. This is known as Lorentz contraction, and it can be calculated exactly. The closer the object moves to the speed of light, the more compressed the object will become.

Operating under this principle, you call two of your best buddies, Bob and Mark, to come and manage the doors on either side of the garage. The plan is that you will accelerate the ladder close to the speed of light towards the garage. You know that relative to Bob and Mark, you and the ladder will contract in length to the point where the ladder could fit inside the garage. Once you and the ladder enter the garage, the two of them will slam the two doors shut, trapping the ladder inside.

But no! Mark, who also has a master’s degree in theoretical physics, says. If you do that, sure, the ladder’s length will contract relative to us. But relative to you with the ladder, the garage will be the one that contracts in length! How do you expect us to close the doors when the garage will be even smaller than before!

What you think Mark will see

What Mark thinks you will see

But you know that there can only be one possible outcome: either the ladder will fit, or the ladder won’t fit. You can’t have two conflicting outcomes in the same timeline. So that being said, will your method work in getting the ladder inside the garage?

The Solution

The solution to the apparent paradox lies in the relativity of simultaneity. This means that what one observer (Bob or Mark) considers to be two simultaneous events may not, in fact, be simultaneous to another observer (you). When you want the ladder to “fit” inside the garage, you essentially want, at one specific time, for the position of the back of the ladder and the position of the front of the ladder to be both inside the garage simultaneously. But since simultaneity is relative, both you and Mark can disagree whether or not the ladder fits without contradiction.

To imagine this, let’s say that at one point, Mark and Bob both close the doors and open them immediately again as the ladder passes through the garage. In the frame of reference of Mark and Bob, the ladder would fit, as they could close the doors. From your perspective however, it would be rather different. You would first see the front end of the ladder approach the exit door of the garage. The exit door closes, and opens again allowing for the ladder to pass through. At the same time, the back of the ladder passes through the entrance door, which closes and then opens. In this case, the two doors closing were not necessarily simultaneous relative to you, and the ladder didn’t fit inside the garage.

From your frame of reference, the doors
will not necessarily close at the same time

From Mark and Bob's frame of reference, the doors
will close simultaneously around the ladder

So in the end, will you be able to succeed in locking the ladder into the garage? Well to do that, it means that the exit door will have to remain closed and the front door will also have to remain closed as well. Let’s take a look at this from the frame of the garage. If we assume that the exit door is immovable, it means that once the ladder hits the exit door, it will instantaneously stop. By this time, the entrance door has closed, and the ladder is stuck inside the garage. Since its relative velocity is now zero, and is not length contracted, it is now longer than the garage. The ladder must now bend, snap, or explode. So the question here, this time, is how did we ever get to close the doors to trap it inside in the first place?

The answer again lies with the relativity of simultaneous events. In the garage’s frame, all parts of the ladder decelerated at the same time. However, in your frame of reference, the ladder accelerates sequentially. First the front decelerates, and then the middle, and then the back end of the ladder. By the time the back end of the ladder has decelerated, the entire ladder is already in the garage, after which you can close the back door. Only issue is, after this, that ladder will have to bend or snap to accommodate its new restrictive garage.

So is it a viable method?

Well assuming you can accelerate to about 95% the speed of light, have an immovable garage, extremely fast and hyper coordinated people, and too much time on your hands…

Probably.