## On the invariant measure in special relativity

I’m writing this for my string theory class. We are basing our lectures on Zwiebach – A First Course in String Theory, and starting off with special relativity. Not everybody in the class has a physics background (pure and applied mathematics students), and so there are likely to be questions which come up which show where I have to fill in some knowledge. We had a question about the invariant measure in special relativity (SR) and why there was a different sign in front of the time term compared with the space terms. I’ll do my best to explain here. Note that I am not explaining it in the precise chronological order of discoveries.

We start the picture off with relativity before SR – that is, Galilean Relativity. This simply states that the laws of motion are the same in all inertial (non-accelerating frames). That may sound straightaway like SR, but there’s a crucial ingredient missing which we will see in a bit.…