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.)
