Tom1992 said:
i was told that riemannian geometry is too restricted for general relativity, that it needs to be generalized to pseudo-riemannian geometry to accommodate gr by removing the positive-definiteness of the riemannian metric. and much of riemannian geometry is lost in the transition to pseudo-riemannian geometry, right? like the hopf-rinow theorem no longer holds.
Correct, if you replace "much of Riemannian geometry" by "many of the most useful theorems in Riemannian geometry which assume a compact manifold". More generally, many of the crucial distinctions between Lorentzian and Riemannian manifolds involve nice behavior lost because the isotropy group (acting on suitable bundles, e.g. providing "gauge freedom" in the frame bundle I keep yakking about) is no longer compact, which is closely related to the fact that the inner product on tangent spaces is no longer positive definite; see
http://www.arxiv.org/abs/gr-qc/9512007 (if this wasn't clear, I am referring to passing from SO(n) to SO(1,n).)
mathwonk said:
i recommend one of einsteins expository essays for the general public.
I don't agree at all, in fact I think that would be the worst imaginable starting point for someone who has some mathematical aptitude and wishes to quickly learn enough gtr to appreciate the central ideas.
One reason for this is that gtr was not very well understood either geometrically, mathematically, or
physically until the Golden Age of Relativity (circa 1960-1970; I give different dates every time I mention this Age, but at least the "circa" is stable!). Another is that Einstein's intent in those essays was completely different from offering an introduction to gtr for either physics or math students!
Those who want a quick overview can try
http://www.arxiv.org/abs/gr-qc/0103044
With this said, I think I pretty much agree with most of what mathwonk said in his next post in this thread:
mathwonk said:
simple explanations... physics is not math... the main point is to try to understand gravitation...I think we give th wrong impression very often that physics can be understood just by learning the mathematical language in which its concepts are expressed...i also recommend taylor wheeler's spacetime physics from 1963, a period when good books on math and science by outstanding figures were being produced for students.
I'd just add a bit of clarification to a few points:
mathwonk said:
Tensors are just notation for writing down and quantifying curvature.
Among other things of interest to physicists!
I'd add a broad qualification,
specifically regarding the literature on classical gravitation: this literature is huge, but one cannot deny that--- exaggerating greatly to make a point--- much of it might well appear both "math-heavy" and "physics-light" (no pun intended) to a physically senstive mathematician. IMO, much of the literature which presents this appearance tends to divide up fairly cleanly into two groups: inconsequential and serious-minded.
Among the former group I might mention papers which (due to the notorious difficulty of finding "honest solutions" of the EFE, of the well-motivated kind with clearly understood and well justified boundary conditions, which Einstein had in mind) make highly artificial assumptions to concoct the ten zillionth "exotic" "solution" (if this is even the proper term) of the EFE, but in whose limitations the authors fail to trouble to emphasize. To be frank, many of these papers read like student exercises, except that the authors are often not students, and there seems to be a venerable but bad tradition of tolerating such publications which might have been justifiable in the 1930s but hardly seems so today. (Strong stuff, perhaps, but W. B. Bonnor, among others, have frequently expressed similar views; one of Bonnor's contrarian essays is
http://www.arxiv.org/abs/gr-qc/0211051)
Among the latter group I would mention papers which are intensely mathematical for reasons which might not be explained but which are in fact well justified (in particular, papers on difficult existence problems, nonlinear stability, etc.)
The point here is that many questions in gtr are very difficult, mathematically speaking, and overall the responses in the literature to such difficulties seem to present an hourglass profile, with many "cheap tricks" and much "hard work", but surprisingly little "middle ground".
mathwonk said:
As a mathematician I have been very disappointed in many lectures supposedly on physics which contained no physics, just math i already knew.
I am going to go out on a limb here and guess that you are attending lectures at your local university on theoretical physics for physics students. If so, since these students often cannot be assumed to be familiar with the necessary mathematical concepts/techniques, it is reasonable for lecturers to spend time explaining the technical background. It is indeed unfortunate that this need inevitably tends to discourage extensive discussion of physical issues! Still, if my guess is correct, the problem might be that you are placing unreasonable expectations upon the lecturers--- if you could find a group of physics students who like you already know the math, perhaps you could engage in discussion of the good stuff!
I think someone already mentioned a reading list I put together years ago http://www.math.ucr.edu/home/baez/RelWWW/reading.html#gtr (note that the popular textbook by Hartle came out after the last revision of this list); I'd just add that for someone who wants to see at least as much discussion of physics as of mathematical background, the textbooks by Ohanian and Ruffini, MTW, and Weinberg (at roughly increasing order of mathematical sophistication) would be particularly suitable. I would add that even students who think they know all the math might find the appendices in Carroll or Wald useful. (In another thread, someone recently mentioned some issues with an appendix in the last book, but I don't think we need to worry about that here.)
