Wednesday, Nov 01, 2023

Lately I've been getting a lot of questions from my children about Special Relativity and to be honest my answers haven't been as detailed as I'd like. I took a course in modern physics and enjoyed it and did well, but that was decades ago.

So, given that writing is one of the best ways to think and learn, I'd like to write about Special Relativity every day until I feel like I mostly understand it.

Backing out, let's start with the speed of light. When you put Maxwell's equations together and get a wave out of them, that wave has a speed. Specifically, that speed is given by c^2 = 1/ (epsilon_0 mu_0).

The thing that blew me away when I first learned this is that epsilon_0 and mu_0, the permittivity and permeability of free space were old friends by this point, having shown up in equations describing the strength of forces from electric and magnetic fields on charged particles.

So here are these fundamental constants that show up together to give a speed. We're used to speeds changing. But fundamental constants don't change. Or, if they did, that would be a big deal. It would change the nature of interactions between every particle in your body and every other partner. It would change the size of atoms and molecules. It was change energies and binding strengths.

So then the question becomes, what if you take this relationship seriously and say that since epsilon_0 and mu_0 are fundamental constants that don't change if your velocity changes, and since the speed of light in a vacuum is determined by them, the speed of light in a vacuum must also be constant.

A constant speed still isn't that strange. The strange thing is that it goes farther than that. The speed of light doesn't depend on your speed. A ray of light may be traveling northward at c, and you may also be traveling northward but at v1 and I may be traveling southward at v2. In our non-reletivistic vector math, I would perceive the light to be moving faster than you would. Compared to me, the light would be moving at c + v2. Compared to you, the light would be moving at c + v1. But neither of these are allowed by the prinicple that the speed of light you measure does not depend on your own velocity. It just depends on epsilon_0 and mu_0, which as we have said, can't change without many other physical consequences, which are not observed.

So suddenly we find that our most basic velocity math fails. How then do we recover something that works? As a hint for next time, let's note that velocity is always a ratio of distance to time. The only way for velocity to change is for the measured ratio of distance and time to change. In other words, saying that the measured speed of light will be the same for two observers who themselves have different velocities from each other means that somehow either the measured distance traveled by the light between two events or the measured time that has passed between two events or both must be altered somehow.

That's the eye-popping thing about special relativity: it's not too difficult to accept that epsilon_0 and mu_0 are fundamental constants and constant even with a change in relative velocity. But then suddenly by holding them still you end up with the measured velocity of light itself being constant. And the only way that can happen is if space and time measurements change from one reference frame to another.