As I type, the students in my Spacetime and Black Holes class are putting the finishing touches on their final exams. Unlike Clifford, I prefer to give take-home finals rather than in-class ones. Not a strong conviction, really; it’s just easier to think of interesting problems that can be worked out over a couple of hours than ones that can be done in half an hour or so. Here’s the final (pdf), if you’d like to take a whack at it. The colorful problem 4 was suggested by Ishai Ben-Dov, the TA; the terse calculational ones were mine.
This is one of my favorite classes to teach, and this quarter the group was especially lively and fun. It’s an undergraduate introduction to general relativity, using Jim Hartle’s book. (It’s okay, Jim uses my book when he teaches the graduate course.) GR is not a part of the undergrad curriculum at most places in the U.S., believe it or not. (There are plenty of grad schools that don’t offer it, and almost none where it is a requirement.) Here in the World Year of Physics, it’s astonishing that the huge majority of physics majors will get their bachelor’s degrees without knowing what a black hole is.
We didn’t have an undergrad GR course at Chicago until a few years ago, when I started it. To nobody’s surprise, it’s become quite popular. Each of the three times I’ve taught it, we’ve had over 40 students; this in a department with maybe 20-30 physics majors graduating each year. At one point I proposed an undergraduate course in classical field theory, which would have been a nice complement to the GR course. It would have covered Lagrangian field theory, symmetries and Noether’s theorem, four-vector fields, gauge invariance, elementary Lie groups, nonabelian symmetries, spontaneous symmetry breaking and the Higgs mechanism, topological defects. If we were ambitious, perhaps fermions and the Dirac equation. But this was judged to be excessively vulgar (you shouldn’t teach classical field theory without teaching quantum field theory), so it was never offered.
The real trick with GR, of course, is covering the necessary mathematical background without completely losing the physical applications. Jim’s book does this by covering the geodesic equation (motion of free particles) and the Schwarzschild solution (the gravitational field around a spherical body) without worrying about tensors, covariant derivatives, the curvature tensor, or Einstein’s equation. It’s like doing Coulomb’s law for electrostatics before doing Maxwell’s equations — in other words, completely respectable. Personally, after studing Schwarzschild orbits and black holes, I zoom through the Riemann tensor and Einstein’s equation, just so they don’t think they’re missing anything.
And when the students pick up the final to spend the next 24 hours thinking about general relativity, I try to remind them: “Three months ago, you didn’t even know what any of these words meant.”
Update: replaced a nearly-unreadable pdf file for the exam with a much cleaner one.
Yes, Jim’s book is wonderful. I had the pleasure of being one of the testers of the notes he had circulating before it got turned into a book. Taught a course at the University of Kentucky with those notes, twice.
Oh, I have nothing against take-homes. As I said in the post, “those have their place”. I sometimes set them……it realy depends upon the course content, and what kind of material you want to test from the content….. and also how much grading you want to do! I also happen to love the art of setting good fixed time exams. It is becoming a lost artform, (take homes require different approaches) and I want to preserve it as part of our heritage…..
Cheers,
-cvj
It really is a pity that GR is not given a more important place in Physics curricula. The situation is all the more ironic as many of the students who choose to major in Physics developed an interest in the field by reading popular articles on quantum mech and GR.
Any opinion of the undergraduate text by Taylor & Wheeler (I think its called “Exploring Black Holes”)?
Plenty of real exams here in the Motherland. 😉
Citrine, suffice it to say that “what the students are interested in” is not always a major motivating factor in designing the curriculum. Not as much as it should be, certainly.
When I was an undergraduate we had to do a slightly more difficult examination in class. We got three hours to do the problems but I wasn’t able to finish them in time.
Levi, I have somewhat mixed feelings about Taylor and Wheeler’s book. It has some very nice stuff in there, and it has the considerable virtue of being short. On the other hand, while Taylor is a great explainer, he’s not an expert in general relativity. Wheeler is, of course, but he had nothing to do with the book. (Is he the only author ever who is extensively quoted in his own book?)
Another good undergrad GR book is the one by Bernard Schutz, who does things in a more traditional tensors-curvature-Einstein’s equation way.
The situation is even more silly considering that Riemann geometry has managed to poke its way into most everything thse days, and if people learned about covariant and lie derivatives and the like early in their lives, it would make learning the ‘standard core material’ all the much easier. And all of that is the main reason that people have issues learning GR in the first place.
After all, knowing how to do covariant derivatives enables one to not have to remember those damn experssions for the gradient and divergence from E&M texts.
Will Sean check the work of the students or will he let his poor Ph.D. students do that 🙂
Yep. That’s the trick. And I’m not sure that I’ve mastered it. We don’t have an undergraduate course, but the graduate course always contains about 25% undergrads.
My approach has been to write down the Einstein equations the first day together with a “road map” for understaning them. As I cover the math in the first part of the course, I constantly refer to the road map to give students a sense of progress toward our goal. Only after they’ve mastered the math, do we delve into the Kerr-Newman solution, trajectories, and the rest of GR.
The way I painlessly introduce quantum field theory for undergraduates who’ve been exposed to both classical and quantum mechanics is through the use of “wave functionals”. They know how convert to an equation featuring a 1-particle Hamiltonian into the corresponding Schroedinger equation. So I show them how the same approach can be applied to the Hamiltonian density of a field to get a Schroedinger-like wave functional equation. Of course, it useless for calculations, but it is nevertheless conceptually powerful. At the end of the GR course, I introduce quantum gravity this way. That is, I show them the Wheeler-DeWitt equation and describe its problems.
Oooh! Problem 4, me likes!
Sean, while I imagine you would recommend Hartle’s book over Schutz’s, since I already have the latter, is Schutz’s good enough? Meaning, do I have to shell out $$…again? 🙂
From my point of view Schutz is certainly good enough, but Hartle is done from a different perspective, allowing one to jump into the implications before mastering all the mathematics. There are advantages and drawbacks to this. I really like Jim’s book, but have always liked Schutz also.
Schutz’s book is great, I used it the first time I taught the course (and a great deal when I was learning it myself). But if you already own it, you should go through it and see what you think!
Last spring Occidental College offered an intro GR course, the only specialty course that semester, using Hartle’s book. About 8 people enrolled but there were only 3 come the second week (myself included). Even though I struggled through each and every homework, I gained a solid understanding of special relativity which also nicely complemented my e/m course and the GR material made for an excellent comps talk. Moreover, I discovered that I definitely want to learn more GR in grad school. Due to the low enrollment in that course (amongst other things), the department (not surprisingly) hasn’t offered any specialty courses this entire year, which sucks. I’m glad to hear that’s not the case in Chicago.
Will you be posting solutions to that final??
Twaters, considering that you pay ten times more tuition fees than here in Europe, I don’t think you get value for your money.
Solutions won’t be posted. But if you want to do it and send me your answers, I’ll let you know if you’re right.
I took GR because I thought Neil Turok was dreeeamy.
Total disconnect between the lectures and the assignments, though, so not much was retained (which I regret to this day).
Well I dunno? 🙂 He certainly got me thinking about brane world collisions, along with steinhardt, that’s for sure.
As to “online resources” for General Relativity, is there one preference if you do not have access to the Hartle book or the other?
Or a link to this one for a historical look?
Relativity
The Special and General Theory
….since you are moderating you might as well take out your lectures notes blockquotes, as I see them here
Interesting Question Paper, especially the third. I wonder if there’s something special about GR which makes you want to set interesting Question papers? Coincidentially, the exam for the GR course at my institute was today as well. And of all the courses I’m doing it was the only one with an interesting exam which made me think rather than carry out quite routine calculations.
Sean, as for verifying the answers, I’ll take you up on that. Should I post answers to some of the questions here, on the blog or elsewhere?
Just curious, are they allowed to use GRtensor (in homework and/or exam)?
Wow, your lecture notes are very well written for lecture notes, and with a nice friendly style! Something like this could have steered me into studying more physics. However, while skimming through it I noticed on page 34 the statement “The notion of continuity of a map between topological spaces (and thus manifolds) is actually a very subtle one, the precise formulation of which we won’t really need.” Can you see me with my arms tightly crossed, a frown on my face, and tapping one of my toes? I hope that hand waving over fundamental math concepts such as this one is not consistent practice in physics courses. It’s precisely the subtle nature of such concepts that can land you into hot water if you’re not playing with them rigorously.
Cygnus, if you email me the answers, I’ll let you know how you did. Once there are too many questions with answers floating around the internet, it becomes impossible to come up with good problem sets or take-home exams; students just spend all their time looking for the answers on the web.
Moshe, no, they’re not allowed to use computer manipulation software. You should be able to calculate the Riemann tensor yourself, I think.
Richard, waving over fundamental math concepts is absolutely standard operating procedure in physics courses. And there is simply no other way to do it: if you took all of the math at a serious and rigorous level, you’d be bogged down in point-set topology and functional analysis and never get any physics done. For some purposes it’s important to dig into the math, and for others it’s not; different students should be able to pursue material beyond the course as they choose.