The ladder paradox

My friend and I, neither of whom are physicists, got into a discussion about special relativity recently. Essentially, special relativity leads to many paradoxes that result from time dilation, Lorentz contraction, etc.

The famous example is that of a garage that is 9 feet long, a ladder that is 10 feet long when “stationary” (but remember, it is only “stationary” relative to some frame of reference), and a mover that is holding the ladder and moving at a relativistic speed. If you are in anyway familiar with relativity, Lorentz contraction says that if an object is moving at a speed on the order of the speed of light (relative to your frame of reference), it will begin to look shorter to you. So let’s say that this ladder is moving at a speed of 3/5*c relative to the observer, such that its length shrinks to 8 feet. The garage has doors located in the front and back that are initially open, but when the ladder enters the garage, the doors immediately close. Then they open before the ladder exits the garage. From the observer’s point of view, there is no contradiction, because the ladder 8 feet long can fit within the garage.

But what if you were viewing it from the mover’s point of view? In his world, the garage is moving at 3/5*c, so its length shrinks to 7.2 feet. However, the ladder is 10 feet long! Can he actually fit inside the garage then? Well, thanks to the relativity of simultaneity, there is no contradiction based on what he sees moving at such high speeds. The mover will actually see the front door open, then the back door open. After he slides through, the front door closes, and then the back door closes. Pretty weird.

Now my friend went ahead and proposed another example, to which I might have an answer but will need to verify. Suppose that from the observer’s point of view, the back door is closed. The front garage door open, and the ladder goes in. Then the doors close, and the mover immediately stops moving after the doors close. What will happen inside the garage when the ladder suddenly expands?

I will leave this to the reader as an exercise. 😉