On the Boundary
When I first started learning about physics, I thought it was amazing how we had these equations and patterns that emerge in nature to the point that we could actually predict what would happen if we threw an object or slid it on a certain surface. The classes were interesting to me because they allowed us to describe things we actually saw.
At this point, I wasn’t familiar with the notion of classical physics, or that there was anything other about physics that I was learning. I didn’t know there were inherent limitations into what we were learning. I imagined that physics was as intuitively simple as the principles I was learning.
However, the trouble came when taking those intuitive principles and extending them in my imagination. For example, imagine moving a book across your desk. When you push on it with your hand on one side, the whole book starts to slide across the desk. Have you ever wondered why this is?
It might seem like a stupid question, but hang in there with me for a moment. It seems obvious that the whole book will move despite only touch a small portion of it, but let’s extend this experiment. Imagine having a long broomstick that is one kilometre long. When you push it, what happens?
Intuition will give us the same answer: the broomstick moves as a complete “unit”. However, if we extend this already-crazy scenario even further, what happens when the broomstick is 300,000,000 m long?
If we were to continue with our intuition, we would be able to push one end of the stick and move the whole thing. But wait a moment. If pushing the broomstick means effectively moving some sort of “signal” throughout the atoms of the broomstick from one end to the other, you’ve just made a signal move faster than the speed of light! Since this goes against Einstein’s special relativity, something in our scenario is wrong.
As it turns out, the rod would not actually move together in one moment. Instead, it would take time for the signal to be passed from one atom to the other along the entire broomstick. After all, pushing the broomstick only means moving the atoms on one side of the broomstick, and so the atoms have to “communicate” to the adjacent ones that they too should move, all the way through the broomstick. It is much like what you can see if you film somebody letting go of a slinky with a high-speed camera. The top of the slinky falls first, followed by the next slowest ring and so on.
Personally, I find this extremely cool. It’s unexpected: we’re taught that an object in free-fall will, you know, fall. But here the situation seems to defy the usual assumptions (though a closer inspection would show that indeed, we could describe the slinky with tension as it falls).
What I wish I had more of in my classes of classical physics were boundary conditions: situations that showed the limitations of the wonderful equations we learnt. It’s great and all to know the ideal cases, but I think it would have been equally instructive to show where they don’t work. I feel like we often gloss over what doesn’t work in favour of what does, but when learning, the cases where something doesn’t work can be of great help.
When learning an idea, investigate where it’s domain of applicability exists, and what happens when one leaves it.