So it is that time. A new semester is upon me, and a totally new course to deliver. Today was my first day back on campus after the break (at least during a regular working day).
Sunday saw me sitting down (in the newly completed study) thinking about how I was going to structure the course. This usually has me sitting with the textbook, a pen, a hand drawn calendar on a big sheet of paper, and a frown on my face trying to figure out roughly what topics I will cover, how many lectures I will devote to each, how many class worksheets (see earlier post) I might have, when the midterms will be, and so forth.
What I am teaching? Why, only one of my most favourite topics to teach in the entire physics world – General Relativity! It is just as fun to teach as electromagnetism (see here), and it is particularly fun to teach it to undergraduates. I first did this back when I was a professor at the University of Kentucky, when I convinced them that they needed an undergraduate GR course on the books, and I designed a new course and delivered it to a cohort of a wide range of ability. It was well received. I owe it all to UCSB’s Jim Hartle, who many years ago circulated some excellent notes on introductory GR, taking an approach that was (then) rather untraditional – rather than teach it the usual way, introducing lots of differential geometry and tensor analysis, then presenting Einstein’s field equations and only then studying the physical consequences, why not teach it the way we teach almost every other field of physics? In other words, build up the students’ intuition for what is going on and the physical consequences by studying particular solutions, adding more of the mathematical techniques as you need them.
There’s an analogy with electromagnetism, for example. We tend not to learn electromagnetism by starting with the whole formalism of differential vector calculus and writing down Maxwell’s field equations, only to later learn about simple solutions like the electric field due to a point charge, or the magnetic field due to a steady line of current. We learn about the latter first, and the physics and mathematics intuition associated with them, and then we build up until we are ready for the full content of Maxwell, appreciating it more once we’ve got there since we deduce it from the ground up. So this is what Jim Hartle encouraged people to do with GR as well, and in this way you can teach General Relativity to a mixed audience of undergraduates without hugely front loading it with seemingly intense mathematical formalism. Most the mathematics you need to go quite far is already known by juniors and seniors, and so within relatively short number of lectures, students are analyzing fully relativistically the properties of the gravity of spherical objects and deriving the classic experimental consequences of GR such as the precession of the perihelion of Mercury and the deflection of light by the sun and so forth – without ever having seen a tensor or worried about general covariance, things they’ll have plenty of time to appreciate later on in the course.
I had such fun teaching it back then, using Jim’s approach, and now I’m going to do a version of it here, going from studies of the sun, to black holes, cosmology, and more. Some things have changed since then, and some things have not. Among the changes? Jim Hartle has now converted those notes he’d circulated to the community over many years into a full textbook, which is simply excellent. (“Gravity: An Introduction to Einstein’s General Relativity”, James B. Hartle, Addison Wesley).
Among the things that have stayed the same? Well, I went to my office that night to do a bit of archeology. I tend to be unable to throw things away, and somewhere in among the shelves of papers and notebooks is a slice of stuff from the Kentucky days. I was hoping to lay my hands on my old notes from the course, to use those as a basis for the lectures I would give this semester. Not only did I find them pretty quickly (sometimes being a junk collector pays off), but this morning, a few minutes after starting to write lecture one for today, I noticed something amusing. It was 11 years ago to the day (click image above right for larger view) that I gave lecture one. Another thing that has not changed? The group I gave the course to back then (eleven years ago – makes me sound so old) was really excellent – just such fun to teach: bright, enthusiastic and willing to put in hard work to learn more about a fascinating subject. Same as now. I’m so lucky to have a huge number of that excellent group I taught last electromagnetism to last semester (see here and here). It’ll be like old times again. I think we’re going to have fun with this…
-cvj
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I liked Bernard F Schutz’s A first course in general relativity, from which I was able to learn both special and general relativity with what seemed to be an absolute minimum of fuss.
I don’t see many people talking about that book, however.
It is the lot of the old to envy the young. I first encountered General Relativity when I came upon a copy of Weyl’s ‘Space Time Matter’, owned and autographed by A.S. Ramsey, and defaced by his son (Frank Plumpton): the blighter injected a ‘badly’ in front of “translated from the German by Henry L. Bose”. But, hey, it was nonetheless a blast, as we used to say. Frank’s marginalia could yet inspire a Ph.D, or two (the defaced translation still lies in the library of Magdalene College, Cambridge.) Nonetheless: your route to the essentials of GR does seem to rock. Put the course notes on-line, so that an old fart can read them.
See the class schedules online.
Best,
-cvj
Hi Clifford! When and where are you teaching GR? I’d love to sit in on a few lectures.
Sounds like a great class indeed. In my course, relativity only comes in the third year of bachelor. Elektromagnetism is coming up in my second year.
Why is all the great stuff so far off?
Right, I’ll be acquiring that book, then, and adding it to the growing stack. Sounds like a lovely class — wish I could take it!
Carl, yes. Have a look at Jim’s book. Your applets sound like fun. Will look. Thanks!
-cvj
Clifford, this is the way it should always be taught. I’m guessing that you introduce the Schwarzschild metric and how to make calculations with it, before you show that the metric satisfies Einstein’s equations for a point mass.
One normally takes the metric and derives the equations of motion for orbits using an affine parameter so you have 4 differential equations, three for spatial coordinates and one for the time coordinate.
An alternative method to calculate the orbits that is painfully obvious and requires less fancy math (and so is very suited for undergraduates) is to write down the elapsed proper time for a path, i.e. integral of (ds/dt) dt, and use the calculus of variations to determine the orbit (since the orbits extremize proper time s). The result is a much more difficult calculation than usual, but it’s just basic undergraduate crank turning rather than tensor magic. And instead of ending up with differential equations in the affine parameter, you end up with differential equations with the derivatives taken with respect to coordinate time.
I did this for Schwarazchild and Painleve coordinates and turned it into a java applet that a professor at MIT intends on using in his class the next time he teaches GR. The applet is set up so that it automatically sequences through a series of gravity demonstrations. But you can stop it and modify the initial conditions and start it up again. My favorite demonstration is the circular orbits near the accretion radius, where they start spiralling into the hole.
A demonstration that I don’t have is one that shows frame dragging. I think it’s possible for the applet to do this. If you or one of your students figures out a set of initial conditions that will do this, take a “print screen” of your display (which will show the initial conditions) and email it to me and I’ll add it to the demonstrations.
Hey, fly over and sit in. It will be good to see you – it’s been a while!
You’ll love Jim’s book by the way…
-cvj
You make me want to take your course, Clifford! But I put Hartle’s book on my Amazon wish list, instead. 😉