# How Much Mathematical Background Should We Acquire When Studying General Relativity?

1. Nov 16, 2006

### andytoh

I think most students develop their mathematical background up to classical tensors before digging into the physics of general relativity.

I think this approach is good, but once this "Introduction to General Relativity" is finished, then the student should refine his mathematical background further and study tensors (with the modern coordinate-free approach), differential manifolds, point-set topology, differential topology, group theory, fiber bundles, etc... and then study general relativity again with a more mathematical general relavity book. This way, he gets a strong taste of general relativity from the both the physicist's and mathematician's point of view, with which he can then decide upon how to specialize. What do you think?

Last edited: Nov 16, 2006
2. Nov 16, 2006

Staff Emeritus
I think that the three tier format of MTW Gravitation more or less implements your idea. In any case, what books would you recommend for the three stages (prliminary tensorss, physical introduction, mature restudy) that you suggest.

3. Nov 16, 2006

### Daverz

I like the approach in Ohanian. He introduces linearized GR from the beginning, which allows him to cover the bending and retardation of light, gravitational lenses and gravitational waves, before getting into tensor analysis. If I were teaching a course, I think I'd use this approach so that the class isn't so frontloaded with mathematics.

For the math itself, I like the approach in Carroll, which mixes modern ideas with the "tensors are objects that transform as..." approach. I do think differentiable manifolds should be introduced at the beginning (as Carroll does); it's just too useful an organizing principle. But I don't think it's useful to inject a lot of topology into a first course, even a graduate course. Topology, Lie groups, fiber bundles, etc. would be topics for math methods course (e.g. from Choquet-Bruhat et al.)

4. Nov 16, 2006

### robphy

One question to ask yourself is
"At what level and for what purpose are you studying General Relativity?"
(Certainly a similar question can be asked of any subject.)

5. Nov 17, 2006

### notknowing

I think it is better to start with mathematics and not with physics and then accidentally move into the field of general relativity. Then, if you are really good, and work very hard, you could probably understand most of GR

6. Nov 17, 2006

### Daverz

7. Nov 17, 2006

### robphy

I really like parts of Sachs & Wu because they get straight to the point, connecting the mathematics with the physics it is trying to model.

8. Nov 19, 2006

### Chris Hillman

Some general advice for would-be autodidacts of gtr

Hi, Andy,

I dare say that every teacher discovers with experience that how a given student learns best varies greatly (and often, unpredictably!) between individuals. This is one of the major reasons why classroom teaching, especially of technical topics, is so challenging. And why any advice offered here might be good for some students but very bad for others, so caution is advised. Ultimately, students should probably weigh any advice they may be offered according to how well they guess it corresponds to their individual learning style and how well they guess the advisor knows the subject. (Incidently, are you a teacher mulling how to advise a student, or a graduate student seeking advice for yourself?--- I am guessing the former.)

Students who are enrolled in university have the benefit of an established curriculum, which has generally been carefully thought out to ensure that all dependencies are satisfied and that the intellectual demands made upon the students in each course are not unreasonable. Even better, graduate or undergraduate students generally can seek guidance from advisors who have some notion of their individual background and abilities, and who can help the student tailor coursework to best meet individual needs. Therefore, it seems reasonable to assume that in this thread we are tacitly discussing SELF-education in general relativity, rather than university coursework. (Of course, at the second year graduate level or above, the distinction rapidly becomes blurred.)

Teaching oneself gtr from a standing start poses a nontrivial challenge, no doubt about it! Fortunately, autodidacts are blessed by an extraordinary number of truly superb textbooks on general relativity--- this is certainly NOT true for many other subjects of equally compelling beauty and interest. My own recommendations as of 2005 are at http://www.math.ucr.edu/home/baez/RelWWW/reading.html and they do conform to the "helical method" you espouse, in which the student revisits material once he has acquired greater sophistication.

From the posts I've just looked over in this forum (unfortunately I have noted quite a bit of misinformation offered by well-intentioned posters, as well as some good contributions from a few posters who obviously DO know enough about this beautiful but subtle subject to offer guidance to newbies), it seems safe to say that many readers here will greatly enjoy and benefit from the two popular books by Robert Geroch and Robert Wald listed at the beginning of my reading list (see link above). Popular books on physics in general and gtr in particular are in my experience generally more misleading than helpful, but these two are truly exceptional--- no doubt because the two authors happen to be leading experts on gtr who work at the University of Chicago (one of the great centers for gtr research). Wald is of course also the author of a standard graduate level gtr textbook.

Regarding textbooks, two of the texts mentioned so far, by MTW and by Ohanian and Ruffini, are on my list and I think very valuable. Sachs and Wu is not, because I happen not to consider this as valuable as the books I do list. (Certain techniques pioneered by Sachs in re Bondi radiation theory and optical scalars are essential topics for serious students to master, but I feel that other books, such as the textbooks by D'Inverno and Carroll, or the book by Poisson, do a better job of introducing these topics--- at least, for most students.) In general, I urge serious students to pay top dollar, as it were, for a recent book such as Carroll or Poisson, rather than paying one tenth as much for a book which was published twenty or thirty years ago. In my view, with a few exceptions such as the wonderful textbooks by MTW and Weinberg, the problem book by Lightman et al., books first published before 1975 or so are generally too outdated in notation and topics covered to efficiently bring the student to the point where he (or she) can easily read recent papers or arXiv preprints. Unfortunately, Dover books are often old books, and in some cases were not very useful even when new. There are exceptions, such as the wonderful book by Flanders on differential forms.

Regarding specific topics which are invaluable to serious students of gtr, there are some fairly obvious prerequisites which should not be omitted even by autodidacts, including:

1. abstract algebra (linear operators, quadratic forms, trace and ideally a bit of invariant theory, matrices as representations of linear operators, vector space bases etc. are all essential),

2. differential equations (the geodesic equations are ordinary differential equations; the field equations are partial differential equations; up to and including some local versus global existence; "local versus global" is a constantly recurring theme in gtr, and failure to recognize this critical distinction is one of the most frequent causes of student confusion),

3. mathematical methods generally (e.g. basic notions of real and complex analysis including integral theorems, the standard theory of the basic equations of mathematical physics such as the heat equation and the wave equation, e.g. solution by separation of variables or by power series, special functions including Bessel functions, Legendre polynomials, spherical harmonics, and hypergeometric functions, Sturm-Liouville theory, potential theory for the Laplace equation, multinomial Taylor series, asymptotic expansions, perturbation theory),

4. differential manifolds (including vector fields as first order linear partial differential operators, tensor fields, exterior forms, Lie algebras and Lie groups) and ideally some prior exposure to surface theory or even abstract Riemannian geometry.

But in addition to these, some essential prerequisites often overlooked (I think) by autodidacts include:

1. mathematical models (especially the notion of a "theory" in mathematics viz. physics, plus the interplay of experiment and theory in physics and the other "hard" sciences),

2. electromagnetism at the level of Landau and Lifschitz (because some essential topics in gtr are best motivated, at least initially, via presumably familiar topics in EM, and also because some of the most important applications of gtr involve EM on curved spacetimes in some way),

3. undergraduate physics generally (e.g. some interesting topics in gtr, which is a classical theory, turn out to be closely connected mathematically to the beautiful theory of the Schroedinger equation!)

My own particular interests in this area seem to center around exact solutions. Here, a graduate student wishing to learn the theory of the Ernst equation or colliding plane waves will want to be familiar with the lovely theory of point symmetries and variational symmetries (at least) of (systems of) ordinary and partial differential equations, to have encountered the notion of Baecklund automorphisms, and to know something about theta functions.

In addition to these, one should mention that acquiring considerable familiarity with standard software which is extremely useful at all levels for mathematical physics generally, such as Maple and Mathematica, should not be neglected. In particular, I highly recommend GRTensorII, which is ideally suited for student explorations of gtr at a variety of levels. Note that the book by Poisson is best studied with a working installation of GRTensorII at hand. GRTensorII does not always conveniently support passing information to some other valuable Maple packages, but it works very well with powerful commands like "casesplit" (e.g. for solving the vacuum field equations or the Killing equations.)

Speaking of Bondi, I have important news both good and bad. The good news is that GRTensorII is free; see http://grtensor.phy.queensu.ca/. The bad news is that it runs under Maple, which is certainly not free. To be sure, Maple is invaluable for all kinds of computations in mathematical physics (far more than can be covered in a single course, in fact, although many universities do now offer courses in using symbolic computational engines like Maple or Mathematica.) Some might suspect that anyone who is unwilling to purchase software costing a few thousand dollars in order to study mathematical physics probably isn't sufficiently determined to succeed in their studies anyway, but I do regret that this poses a serious obstacle for some enthusiasts. (Registered university students can obtain Maple at something like one twentieth the list price, incidently, and I urge them not to pass up the opportunity!)

GRTensorII is ideally suited to making the kind of computations with specific spacetimes which students will find most valuable for gaining physical intuition into the theory, but is not suitable for symbolic computation in the style of "tensor gymnastics". (I should note that GRTensorII is also useful for working with approximations such as in the above-mentioned Bondi radiation theory, or in perturbations of black holes, not to mention working with large families of solutions such as the Ernst vacuums, so I don't wish to give the impression that it is only useful for working with specific exact solutions.) However, other packages are available which can handle that kind of thing to greater or lesser degree. As always, a wise student will beware of possible bugs.

This is off the top of my head, so no doubt I have forgotten some essential topics and will feel very silly once I realize what I omitted!

One last thought: I tend to feel that any student who assumes (not that I really think you were doing this!) that he must specialize in either mathematical or physical approaches to gtr will have limited success. In such a mature subject, any contributor worthly of note will be, I think, reasonably familiar with all relevant mathematical techniques and theorems (e.g. existence and uniqueness results), as well as well as being constantly mindful of the difficult and all too often neglected issue of physical interpretation of all our impressive mathematical techniques. I have recently been studying the collected papers of Chandrasekhar (you can picture me winding my way up my own helix here!) and enthusiastically echo those who would urge serious students to try to take him as something of a role model. Anyone who assumes from casual perusal of his monographs that Chandra exhibited much mathematical insight but comparatively little physical insight has greatly underestimated his genius.

Chris Hillman

(The same "Chris Hillman" cited in wolram's sticky above; this is my very first PF post, so please note that I hope to avoid discussing psychoceramics here; the "debunking" page cited by wolram was only a tiny part of a larger website offering guidance to on-line resources for serious students of gtr.)

9. Nov 19, 2006

### Daverz

Someone who was really serious would write their own package.

10. Nov 19, 2006

### robphy

As a follow up to Chris Hillman's post, these resources [directed to teachers of relativity] are worth mentioning:

http://arxiv.org/abs/gr-qc/0511073 Wald, "Teaching General Relativity"
http://link.aip.org/link/?AJPIAS/74/471/1 Wald, "Teaching the mathematics of general relativity"

http://arxiv.org/abs/gr-qc/0506075 Hartle, "General Relativity in the Undergraduate Physics Curriculum"

http://www.aapt-doorway.org/TGRU/ "AAPT Topical Workshop: Teaching General Relativity to Undergraduates" (see the articles, slides-from-talks, and the posters)

By the way, I would add Thomas Moore's "A Traveler's Guide to Spacetime:
An Introduction to the Special Theory of Relativity"
http://www.physics.pomona.edu/faculty/prof/tmoore/travguide.html
to the list of resources. (Moore made some insightful comments at the workshop above.)

Also worthy of reference:

http://arxiv.org/abs/gr-qc/0506065
"Classical General Relativity" by Malament
(who was also at UChicago; discussion of deeper foundations)

"General Relativity" by Ludvigsen

"Relativity: The Special Theory" and "Relativity: The General Theory" by J.L. Synge [for being one of the first "geometrized" (emphasizing invariance) treatments of relativistic physics]

Finally, here's my old list of references:
http://www.phy.syr.edu/courses/modules/LIGHTCONE/references.html

Last edited: Nov 19, 2006
11. Nov 19, 2006

### Daverz

It looks like the Moore book is now part of his "Six Ideas That Shaped Physics" series, which looks like an interesting way to divide up the freshman physics curriculum.

12. Nov 21, 2006

### andytoh

On a personal note, I first studied General Relatvity at the undergraduate level with only the minimum required mathematical background. Getting an A+, I thought I was the guru of GR. Then I started perusing Hawking's "Large Scale Structure of Spactime" and was overwhelmed by the level of math and realized that I didn't know so much GR after all.

To overcome my new lack of confidence, I then studied differential geometry (Wasserman's "Tensors and Manifolds" being my favourite) and topology, only to realize that I found topology to be my favourite subject and studied that in great depth (too many topology books to mention here). But I did not forget that I initially had my interest in GR and so currently, my specific interest is to do earn a PhD by doing research solely in the topological aspects of GR, notably proofs, just as did Zeeman, King, McCarthy, Penrose, and Hawking himself have done. I am open to other mathematical aspects of GR as my mathematical interests continue to evolve but point-set topology and differential topology are by far my greatest interest.

I don't think I am walking down a mathematical dead-end street because I do have a good physical background on GR and so am prepared to give physical interpretations of my topological findings. Thanks for the advice guys, especially Dr. Hillman.

Last edited: Nov 21, 2006
13. Nov 21, 2006

Staff Emeritus
Andy, that's a great personal history! More power to you!

14. Nov 21, 2006

### robphy

andytoh,
Are you studying the arguably more-physical topologies (e.g., the Zeeman, Alexandrov, and fine topologies)? Have you looked at the Fullwood-McCarthy topology?

15. Nov 21, 2006

### Daverz

Hmm, I got a copy of Wasserman (1st ed; I didn't realize there was a 2nd edition) for \$10, but wasn't too impressed with it, finding it too mathematical for my taste. Guess I'll have to give it another try.

16. Nov 21, 2006

### andytoh

At one time I wanted to specialize only in point-set topology and nothing else. But my interest in switching to researching topology in GR began when I studied Zeeman's "fine" topology as his alternative to the Euclidean topology for Minkowski Spacetime. I realized that I was hooked when I then studied Hawking, King, and McCarthy's "path topology" for Minkowski spacetime which had the same homeomorphism group as Zeeman's fine topology, and which amazingly required in its proof nothing more than my beloved point-set topology. I have not looked at the Fullwood-McCarthy topology in great detail yet, as I am still only a graduate student (and thus still trying to discover how I would like to approach GR/topology/differential geometry), but I am not biased to either physical or theoretical topology in GR.

In my opinion, if history's greatest topologists did some research in general relativity, we would certainly understand general relativity better today.

Last edited: Nov 21, 2006
17. Nov 22, 2006

### Chris Hillman

How I got to be such a knowitall

Hi, Andy,

I've told this story before, but you (or others) might not have seen those long-ago posts (in sci.physics.*):

I encountered MTW before I knew any math other than high school trig and algebra, or any electromagnetism or even a solid grounding in Newtonian physics, with one key addition: by a happy accident, I had recently studied (as an extracurricular activity) Taylor & Wheeler, so I was familiar with hyperbolic trigonometry and Minkowski geometry. When I saw MTW, I was immediately entranced by the pictures of hyperslices in the Schwarzschild geometry and so on. I thought: "hey, if the point is to be able to draw a picture, I can do this stuff!" So I started studying MTW and bye and bye was able to come up with some interesting results which were not mentioned in the book.

For example, I remember embedding various hyperslices other than those they discuss. I also embedded the FRW dust and radiation fluid and discovered they look like an American football and a lemon, respectively. Since I didn't know any calculus at that time, I simply used a table! :-/ I later found the FRW dust picture in the book The Shape of Space, by Jefferey Weeks Later still I was able to ask him if he had carried out the computation or just guessed, and he said he had just guessed--- rather impressive because the exact shape very closely resembles what he drew. I had a similar conversation with Teukolsky in which he was able to draw this shape correctly without doing any computations. This might be of interest here because in my sample of three individuals who -can- make computations, it is obvious that all three have enough intuitive understanding to know what the answer is without actually computing anything.

I also remember that, inspired by the discussion of the Eddington chart in MTW, I had the idea of trying to transform the Schwarzschild chart to "pull down" hyperslices orthogonal to the world lines of the LeMaitre observers (who fall in freely and radially from rest "at infinity") so that they appeared as coordinate planes. Again, I used a table of integrals (I did know enough differential calculus to verify by differentiation that I was using the correct indefinite integral),
and noticed that the slices are locally isometric to $$E^3$$, a fact which many might find surprising (and which amplies the discussion in MTW about the Coulomb field showing up in gtr with suprisingly little change from Newtonian gravitation). Much much later I learned that this apparently first done by Painleve in 1921! But his chart has only recently become popular; see http://arxiv.org/find/gr-qc/1/ti:+Painleve/0/1/0/all/0/1

Although I didn't know about the semipopular books by Geroch and Wald at the time, when I did come across them I saw that they would have been perfect for me at this stage, incidently, which is one reason why I recommend them so enthusiastically to anyone curious about gtr who has little mathematical background.

While I wouldn't recommend anyone to try to duplicate my own path (to state the obvious, MTW is a graduate level textbook intended for students with a strong background in math and physics), my experience does show that what counts in learning gtr is a strong visual imagination. My experience probably also shows that a certain amount of ability and a pragmatic approach (e.g. using a table of integrals under the assumption that I would soon pick up integral calculus) can be helpful. It also shows that it is quite possible to learn gtr before learning Newtonian gravitation (although I wouldn't recommend this route).

Continuing my story: I soon discovered Feynman's Lectures on Physics and started filling in my almost complete lack of background in undergraduate physics. I remember being very impressed with his introduction to complex numbers and his remark about the same differential equations playing the central role in the theoretical analysis of quite different physics phenomena. (A kind of force multiplying effect, as they would say in the Pentagon.) I soon decided I was more interested in differential geometry, differential equations, group theory, and information theory than physics (probably not quite the effect which Feynman and MTW had intended!), so I suddenly (and quite "illegally") declared myself a math major and insisted on enrolling in the abstract algebra and real analysis courses. I think my undergraduate advisor, John Hubbard of Hubbard-Mandlebrot set fame, expected a train wreck, but fortunately I did very well in those courses and eventually went on write a Ph.D. thesis on generalized Penrose tilings. Thus, ironically enough, I might be the only entity other than Roger Penrose which is familiar with both Penrose tilings and Newman-Penrose tetrads :-/ I once pointed out to Penrose that you can endow a rhomboidal Penrose tiling (with the vertices deleted) with a Lorentzian metric, so that both "thin" and "fat" rhombs correspond to
$$ds^2 = -dt^2 + dx^2, \; 0 < t, \, x < 1$$
Then the "vertices" correspond to deficits and excesses as in the Regge calculus, and the null geodesics resemble "curves" which had been introduced by J. H. Conway. This proved amusing!

Chris Hillman

Last edited: Nov 22, 2006
18. Nov 22, 2006

### andytoh

This reflects my general researching motif. If I study in depth the level of topology that topologists would understand but general relativists would not, and study in depth the level of general relativity that general relativists would understand but topologists would not, then something new is just bound to be found.

Thanks for your anecdote and time in writing it.

19. Nov 22, 2006

### robphy

I would add philosopher's and historian's point of view, if you have a philosopher of physics or historian of physics who teaches relativity in your institution.

I have been lucky to have taken and sat-in on numerous undergraduate and graduate relativity courses, ranging from line-element-based classical differential-geometric to more-abstract tensor-geometrical to philosophical-foundations-of-physics. Each course taught me different aspects of the subject (which are often not mentioned in the other courses). Taking these courses from various professors, of course, gives you a wide range of viewpoints and skills. (Unfortunately, I've never taken a relativity course taught by an experimental relativist, a cosmologist, or a historian of physics.)

When it becomes your turn to teach the subject [including courses for nonscience majors], you can often learn a lot from some of the good introductory books, old textbooks, lecture notes, and pedagogical articles that you may not have encountered [or overlooked] in the classes you took.

(Some of these recollections reminds me of Synge's "My Relativistic Milestones". I'm curious to read of others.)

Last edited: Nov 22, 2006
20. Nov 25, 2006

### najia

i thought alot about it but it is very hard i have no idea sorry.
bye i hope you find a good idea about this.