Spacetime and black holes

By Sean Carroll | December 2, 2005 10:27 am

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.

CATEGORIZED UNDER: Academia, Science

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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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