# Covariant

Penultimate lecture today in the graduate electromagnetism class. These last four lectures are a lightning tour through some important concepts – showing how many of the things we’ve been doing all semester fit with Special Relativity. (For example, amusingly, showing that the Lorenz gauge condition is in fact Lorentz invariant…) It is fun to show a powerful example of how an important guiding principle (such as writing equations in a Lorentz-covariant way) can end up making several features of the theory seem much more natural, while also leading to new physics. This is fun to do, although it does mean that I end up writing whole new notes for this since I am not a fan of the way some of these electromagnetism books (Jackson included) decide to treat time in an odd way, such as treating it as imaginary (which must be so confusing to some students) just so as to write Lorentz transformations like a rotation, or using mostly negative signatures for spacetime, and so forth. And, inexplicably, using different units of measurement from the choices made in the rest of the book… Anyway, so the last two lecture-writing sessions have mostly been one of fiddling with minus signs and factors of c, 4 Pi, minus one, and so forth. Joy. Well, the group seems excited since they’re beginning to see things that they’d seen in other classes and it is all making some sort of sense now (Klein-Gordon equation, duality, etc., etc…) I think the last class will show how many of these things flow from variational principles. Maybe I’ll find a little time to do the Born-Infeld model? I’m excited too, although I’ll be sad to end the class and stop working with this fun group of students.

Today I managed to grab a few sketches on the train. This afternoon coming home on the Expo line these two snoozing gentlemen were kind enough to sit still for a few minutes each for me to get down a few impressions of their features. This was all helped a bit by the train sitting still for a while as we waited for a truck to get off the line. Apparently it was parked or stuck there.

Perhaps not helpful was this young guy who watched me drawing and then decided to yell over to me “Did you take a class?” to which he responded “Wow!” with a shake of his head when I said no. I was worried he’d draw attention to me and the subjects of my drawing would notice that I was drawing them. They seemed not to notice. Later he came and sat next to me and we struck up a short conversation about sketching. I explained that my “class” was simply to draw when I could in situations like the one we were in – that one really learns to draw by just drawing a lot. He wondered about how I managed to capture faces since people kept moving, and in particular why I prefer a pen. I said that with practice you rely less and less on exactly drawing a fixed pose and learn to internalize more the features so that you’re just doing glances at the subject, not staring, and that drawing is a combination of drawing what you see and what you know and finding the balance between the two is what I strive for, and then someone moving a bit while drawing becomes not so much of an issue, and so on and so forth. I explained that I don’t necessarily prefer a pen, and use pencil too, although for short sketches a pen is sometimes the better way for me since it forces me to focus on doing the drawing all the way through and not worrying about making mistakes and trying to correct them. You’re more committed to the marks you make on the page with a pen, and you learn to use them and/or live with them as you go along. I explained that it is a bit like with playing a musical instrument – it is good to try to play a piece all the way through even if you stumble a few (or several) times along the way. The alternative – stopping and starting again every time you make a mistake – robs you of seeing and feeling the overall structure of what you’re doing, of the sense of completion, of the real point of the exercise. (You know, I really should listen to all this splendid advice!) I encouraged him to take up drawing again and just practice when he can (he’d said he can’t find the time, and how he likes to do cartoons…), and told him where I got my little notebook (since he said did not know where to get little notebooks)…

Pretty sure he was happy the train arrived at the station so that he could say goodbye. Sometimes I talk too much.

-cvj