Terence Tao on Nostr: I had the idea to try to perform a change of variables to rotate these tangent planes ...

I had the idea to try to perform a change of variables to rotate these tangent planes to line up and become parallel, in the hope of simplifying the equation. At the time, I did not have the required background in differential geometry to recognize that I was actually trying to perform a gauge transformation; but in fact this project finally set me on the path to learn differential geometry properly (I had not fared well with it as an undergraduate, and was slightly intimidated by it as a graduate student), which has definitely helped my mathematical career since.

The problem was that the coordinate changes I wanted to implement were not unique. If I wanted to rotate a tangent plane on the sphere to be tangent to the north pole, say, I could easily visualize rotating the sphere so that the point of tangency was on the pole, but then I could also twist the plane around that pole arbitrarily. So, for each point in space and time, there was an additional twist I could perform, and this transformed the equation that I was trying to solve in a way that could either make the equation more like the free wave equation (and thus easier to solve), or less like the wave equation.

If I focused only on how these twists evolved in a single spatial direction - such as the x-axis, it was like trying to smooth out a chain of disks to all be parallel, and I could see how to perform a change of variables to do that. (Now that I know differential geometry, I know that I was integrating a connection along this direction.) But then this operation made things curve in a funny way in the y-axis. I couldn't visualize all this in my head.
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