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.
December 2nd, 2005 at 10:46 am
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
December 2nd, 2005 at 11:04 am
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.
December 2nd, 2005 at 11:37 am
Any opinion of the undergraduate text by Taylor & Wheeler (I think its called “Exploring Black Holes”)?
December 2nd, 2005 at 11:41 am
Plenty of real exams here in the Motherland.
December 2nd, 2005 at 12:36 pm
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.
December 2nd, 2005 at 12:51 pm
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.
December 2nd, 2005 at 12:52 pm
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.
December 2nd, 2005 at 12:53 pm
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.
December 2nd, 2005 at 12:54 pm
Will Sean check the work of the students or will he let his poor Ph.D. students do that
December 2nd, 2005 at 1:08 pm
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.
December 2nd, 2005 at 1:26 pm
Oooh! Problem 4, me likes!
December 2nd, 2005 at 1:50 pm
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?
December 2nd, 2005 at 2:05 pm
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.
December 2nd, 2005 at 2:14 pm
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!
December 2nd, 2005 at 4:32 pm
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??
December 2nd, 2005 at 4:51 pm
Twaters, considering that you pay ten times more tuition fees than here in Europe, I don’t think you get value for your money.
December 2nd, 2005 at 4:56 pm
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.
December 3rd, 2005 at 2:46 am
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).
December 3rd, 2005 at 12:44 pm
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?
Preface
These lectures represent an introductory graduate course in general relativity, both its foundations and applications. They are a lightly edited version of notes I handed out while teaching Physics 8.962, the graduate course in GR at MIT, during the Spring of 1996. Although they are appropriately called \lecture notes”, the level of detail is fairly high, either including all necessary steps or leaving gaps that can readily be filled in by the reader. Nevertheless, there are various ways in which these notes differ from a textbook; most importantly, they are not organized into short sections that can be approached in various orders, but are meant to be gone through from start to finish. A special effort has been made to maintain a conversational tone, in an attempt to go slightly beyond the bare results themselves and into the context in which they belong
Or a link to this one for a historical look?
Relativity
The Special and General Theory
December 3rd, 2005 at 12:58 pm
….since you are moderating you might as well take out your lectures notes blockquotes, as I see them here
December 3rd, 2005 at 3:17 pm
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?
December 3rd, 2005 at 4:50 pm
Just curious, are they allowed to use GRtensor (in homework and/or exam)?
December 3rd, 2005 at 5:21 pm
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.
December 3rd, 2005 at 6:00 pm
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.
December 3rd, 2005 at 6:02 pm
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.
December 3rd, 2005 at 6:30 pm
Good, for undergraduate class I tend to agree, though I can see arguments going both ways.
December 3rd, 2005 at 8:43 pm
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
December 3rd, 2005 at 8:49 pm
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.
December 3rd, 2005 at 8:59 pm
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
December 3rd, 2005 at 9:02 pm
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.
December 3rd, 2005 at 9:35 pm
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
December 3rd, 2005 at 9:39 pm
I meant off-that-topic, of course, nevermind…enjoy the movie…
December 3rd, 2005 at 10:40 pm
“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.
December 4th, 2005 at 8:19 am
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.
December 4th, 2005 at 8:45 am
Jo Anne
So “six weeks” has long past in terms of Eric Adelberger experiments at Eotvos.
Modifications to General Relativity
Any News?
December 5th, 2005 at 5:10 pm
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.
December 5th, 2005 at 10:01 pm
A. Zee — in my experience as both an undergrad and a grad student, *most* undergrads are smarter than most grad students.
December 5th, 2005 at 10:08 pm
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…
December 5th, 2005 at 11:47 pm
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).
December 6th, 2005 at 1:56 am
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.
December 7th, 2005 at 12:58 pm
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!
December 11th, 2005 at 2:20 pm
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.
December 11th, 2005 at 2:50 pm
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…
December 11th, 2005 at 7:26 pm
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.