I’m reading Spacetime and Geometry by Sean Carroll and working all the proofs as I go. I recorded this video after struggling to understand where a seemingly magic transform came from and why it simplified a complicated equation. If you’re working on his equation 5.139, this will help you understand how you might have thought of the transform and why the thing called m really is a mass.
Unfortunately, I discovered only after I recorded this that the whiteboard had been switched off for the introduction and, for reasons to tedious to mention, was unfixable with the tools at hand. So I provide here the equations I refer to. (Once I get launched on the explanation proper, you can see everything I’m writing.)
Here is the original time component of Einstein’s Equation, Carroll’s 5.139:
Here is the transform he conjuries out of thin air, which I try to explain (Equation 5.142):
And here is the resulting simplified equation: