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.
Good, for undergraduate class I tend to agree, though I can see arguments going both ways.
Oh, Moshe, you’ve let the cat out of the bag now! 🙂 We’ll never get another student to compute these things by hand ever again….. I usually wait until they’ve done them by hand a few time and then mention in passing (when I have a clear escape route to run along) that there’s computer algebra….
-cvj
Clifford, I waited patiently with my question till the exams were handed in…
But, it is not uncommon for graduate students and even advanced undergraduates to do some of their calculations (integrations and such) using Mathematica or Maple. Generally, my feeling is that if they get the answer right consistently, this is fine. However, it is difficult to get the answer right consistently if you don’t know how to calculate it by hand.
True…but all to often students hand in nonsense they got from the computer because they never took the time to learn the meaning of what they were calculating. It is frustrating. In most of the integrals we do at undergraduate level, I would claim that the art of solving the integral by hand also helps your understanding of the physics more often than not. There is often a substitution or change of variables that renders the integral very easily doable…. in those cases, the change of variables tells you something useful about the physics too (the new variable is telling you about a conserved quantity, etc….) I know you know that…but it is worth mentioning…. This is the same good reason that it does not pay to get young children doing arithmetic on calculators too early. They end up knowing nothing about numbers, and the patterns they encode, etc.
-cvj
Good points Clifford, there is an interesting discussion there, I think computation should be a part of physics education, but maybe not too soon as you say. Way off-topic here I fear.
Off-topic? It’s only a matter of time before this thread degenerates into a rant about how string theory is the biggest swindle since swindling was invented….. like most other threads…….. just wait.
-cvj
I meant off-that-topic, of course, nevermind…enjoy the movie…
“This is the same good reason that it does not pay to get young children doing arithmetic on calculators too early. They end up knowing nothing about numbers, and the patterns they encode, etc.”
Good point Clifford! When I was in early grade school I had a terrible memory (and it’s still not impressive), so I never really totally memorized the multiplication tables. It was easier for me to make rapid calculations like 7 x 8 = 7(7 + 1) = 7^{2} + 7 = 49 + 7 = 56 in my head, and so I started thinking in terms of algebra very early, even though I had never heard the word “algebra”. And later on it was very instructive to do interpolations with log and trig tables, and … ah … didn’t we also do some interpolation with slide rules? When you push keys on the calculator and out pops a number, you don’t really get a good feel for where it came from.
Actually, when working with Mathematica, I often need to do a lot of calculations by hand and program that into Mathematica. E.g. I am now busy calculating some integrals that I have to evaluate numerically from zero to infinity. If you use
NIntegrate[f[x],{x,0,infty}] then Mathematica will extrapolate to infinity but that’s not very accurate (in my case it is complaining about oscillatory behavior).
So, I did the numerical integration till some not too small R and performed the integral from R to infinity using an asymmptotic expansion. Unfortunately Mathematica isn’t good at performng asymptotic expansions. It will complain about some (essential) singularity and refuse to do the expansion. So, I had to do that (largely) by hand.
Jo Anne
So “six weeks” has long past in terms of Eric Adelberger experiments at Eotvos.
Modifications to General Relativity
Any News?
I will be teaching the undergraduate GR course here at UCSB in the Spring. I plan to use Hartle, with Carroll as supplementary reading for the more ambitious students. This is actually the first time I have ever taught an undergraduate course at UCSB after more than 20 years here! I was talked into it by my colleague Lars Bildsten here at KITP, who claimed that some undergrads are actually smarter than some grad students (— kind of an empty statement!) I stumbled into this discussion; I should probably read it in its entirety and pick up a few pointers. I am just about at the stage of starting to think about what I should cover.
A. Zee — in my experience as both an undergrad and a grad student, *most* undergrads are smarter than most grad students.
Well, obviously most UGs are smarter than most PGs – the “smart” ones are those who leave university to get jobs which pay, and don’t hang around to be insulted by those who are presumeably supposed to be encouraging and supervising them…
Tony, I’m sure you’ll enjoy it, it’s a fun class to teach. And Jim’s book is great, but has far too much good stuff in it — you have to be disciplined about sticking to the important bits (especially in a one-quarter course).
Have a blast Tony. I have an undergraduate relativity and cosmology class in the Spring, in which I’ll be drawing heavily on Hartle, and a graduate class next Fall, in which I’ll be turning to Carroll. Both excellent, and both requiring careful choices of the most important topics, since so much fun stuff is covered.
For me the best undergraduate material for GR remains the collection of Sean’s notes. I think it helps to have Sean around to explain what is going on. But the notes are very good on their own as well.
I started with Schutz but the notation and the vagueness of some chapters led me to migrate to Sean’s notes in the end. In college, I used Wald, but while I liked the rigour, I hated the lack of insight onto the physical implications of all the calculations. For another undergraduate class I used Kip’s book, but frankly, this one is nothing more than a great reference and should be never mistaken for a textbook.
I am surprised that GR is not an undergraduate requirement. It is not nearly as complicated and insightful as QM, plus the two were developed around the same period. Why teach the one without the other, especially when the clash between the two is such a hot modern topic!!! The concepts in QM are more shocking and sutble than the ones in GR. When GR starts to send electric shocks then one can say that a merge is cooking!
Sean and Mark, thanks for the encouragement. I am sure that it will be fun; that’s why I volunteered for an undergrad course, the first I will teach here at UCSB. GR is by far the most beautiful subject in physics! Teodora, thanks also for the useful remarks. I agree completely that GR is easier to understand than QM; that was certainly true for me as an undergrad. I intend to go to the physics department and ask Hartle and Marolf some questions about their teaching experience. Two questions are on my mind. One is how fast I can zoom through special relativity in order to get to the “good stuff”. The other is that most of the remarks posted by people (for example regarding GR texts on Amazon) is that they find the math difficult. To me that is somewhat puzzling because I think that any student with a future in theoretical physics should have lots of problem understanding the physics, but not any difficulty with Riemannian geometry, which is after all totally logical and algorithmic. But perhaps I have simply forgotten what physics undergrads are capable of. Any thoughts about these two issues? My general tendency is wanting to cover too much, for example Penrose diagrams.
Teodora, it’s even worse than that. Among the more bizarre academic experiences of the Dissident is a “big wig” theoretician (at least by his own reckoning) getting upset upon finding out that a grad student was taking a GR class: he considered it a waste of time. And this was a guy with a professed interest in astroparticle physics. Oh, the humanity…
A. Zee,
I can recommend Gerard ‘t Hooft’s lecture notes:
http://www.phys.uu.nl/~thooft/lectures/genrel.pdf
The book ”A short Course in GR” by J Foster and J.D Nightingale:
http://www.springer.com/sgw/cda/frontpage/0,11855,1-40197-22-81654042-0,00.html
Is an excellent introduction to GR for physics students. The book seems to have changed a bit from the first edition which I own.