Special Relativity

The plan for the next few weeks is to start with Einstein’s special theory of relativity. (It’s “special” because it only deals with things that aren’t accelerating; if you want acceleration, you’re looking for the general theory of relativity, but I would advise you not to go looking for it anytime soon.) Special relativity is a good place to start because you already have all the math you need, namely algebra and trig. Plus it was figured out before all the other stuff I’ll talk about in this blog, and some of the other stuff is based on it.

The whole theory comes from just two postulates:

1. Principle of relativity: The laws of physics are the same in all inertial (non-accelerating) reference frames.
2. Universal speed of light c: The speed of light in vacuum c (about 300 000 m/s) is the same for all inertial observers, regardless of the motion of the source.

The first postulate has actually been around for hundreds of years, but it probably deserves a little explanation. As far as we’re concerned, it means this: if you’re moving along without speeding up or slowing down, you can never really prove that you’re moving at all. After all, how would you prove it? You’d have to do some sort of physics experiment, and according to postulate 1 you’ll get the same result as if you weren’t moving. If you see me walking down the street you could say I’m moving while the ground stays still, but it’s just as fair to say I’m staying in one place while the ground moves underneath me. Like what happens on a treadmill, for example.

The upshot is that you can pick any (non-accelerating) reference frame you like and decide it’s stationary. If someone else picks a reference frame that’s moving relative to yours and claims that’s the one that’s stationary, that’s cool too. Neither one of you is wrong; you’re just describing the same thing two different ways. If I walk into a wall, it doesn’t matter whether you think I’m moving forward or the wall is moving backward, the physics when I hit it stays the same.

Okay, so hopefully you believe me about the principle of relativity. It’s the universal speed of light which ought to really upset you. You certainly wouldn’t believe me if I’d said that the speed of a bowling ball is the same for all observers, regardless of their relative motion. If I’m on a train going 60 mph and I roll a bowling ball at 5 mph down the corridor, then the ball’s total speed relative to the ground is “obviously” 65 mph—the speed of the bowling ball (A) with respect to the ground (C) is equal to the speed of the ball (A) relative to the train (B) plus the speed of the train (B) relative to the ground (C). Our common sense tells us that this should work for any three objects A, B, and C. But according to postulate 2, if instead I send a light signal down the corridor, its speed is c relative to the train and c relative to the ground!

Impossible? When I was a kid I sure thought so. In fact, I was absolutely convinced there was a logical paradox there, and I’d spend days coming up with thought experiments that would give contradictory results. But, infuriatingly, each time there was some clever way out so that you could just barely avoid a contradiction. As it turns out, there really isn’t any paradox at all, but special relativity does require you to change dramatically how you look at space and time.

There are three major consequences to special relativity. We’ll work out the first one this week and I’ll give you a little problem, and we’ll save the last two for next week. If I can direct your attention here, we’ll get started: https://mywebspace.wisc.edu/mweinberg/web/Simultaneity.pdf