How to learn general relativity

[stunner5000pt]

Maybe this isn't the right forum for this ... in which case i am sorry

What kind of background would be necssary to even dream about learning General relativity??

Where would be a good starting point?? Maybe at least learn the mathematics involved such as Tensor Calculus (??)

 

[mathwonk]

i recommend one of einsteins expository essays for the general public.

 

[quasar987]

Judging from your posts in the HW help sections stunner, GR is more accessible to you than you think. From what I've read out of Schutz, his book makes GR accessible to any 2nd or 3rd year physics student.

I've only had a few more classes than you and I'm taking GR right now. With a text ~ of the level of Caroll.

 

[Daverz]

Maybe this isn't the right forum for this ... in which case i am sorry

What kind of background would be necssary to even dream about learning General relativity??

Where would be a good starting point?? Maybe at least learn the mathematics involved such as Tensor Calculus (??)

You might get some introductory GR books from the library and browse through them. Examples in rough order of sophistication are

Hartle, Gravity
Schutz, A First Course in General Relativity
Ohanian, Gravitation and Spacetime
Carroll, Spacetime and Geometry

See also http://math.ucr.edu/home/baez/RelWWW/reading.html#gtr .

We've had long threads on this before that in retrospect got rather sidetracked on all the math that might possibly be relevant.

In my opinion, it's more important to start with an understanding of classical mechanics and electrodynamics (at the level of, for example, The Feynman Lectures vols. 1 & 2), and special relativity (at the level of Taylor & Wheeler). Introductory GR books usually do a good job on tensor calculus.

 

[stunner5000pt]

yea in my search of this forum i found threads wherein it was suggested that one should learn from a large range of mathematical topics like abstract algebra

i know that it is hard to learn GR by yourself that's why we hae school but i would try my best to learn and understand everything that i could along with the generous help this forum provides!

 

[redrzewski]

https://www.amazon.com/dp/020138423X/?tag=pfamazon01-20

This book is a friendly introduction that is easier than Schutz.

 

[mathwonk]

I am a rank beginner, but i like simple explanations. physics is not math. einstein himself made it clear the main point is to try to understand gravitational forces.

His example of a spaceman in an elevator pulled by a cord on top at an increasing rate, being undistinguishable from the force of gravity, from the passengers point of view has stayed with me all my life.

then if you start thinking of "shortest" paths in space, as the paths taken by moving particles, which are attracted by massive bodies, you realize that these paths are curved more near more massive objects.

the mathematical part of general relativity is to me the study of the resulting curvature this imposes on spacetime.

Tensors are just notation for writing down and quantifying curvature. Understanding the physical ideas is something else again. 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.

As a mathematician I have been very disappointed in many lectures supposedly on physics which contained no physics, just math i already knew.

To me the key is to grasp the physical phenomena. The mathematics should be kept in its place as a way to expres that and to calculate with it. Moreover the accuracy and usefulness of the math should always be open to review.

of course if you accept that GR is encoded in a riemannian manifold, then it is obvious you want to learn that geometric theory. But there may well be new phenomena in GR that will require more, and then you should be guided by the physics and not confined by the math.

these are the relatively uninformed opinions of a mathematician, not a physicst.

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.

if you like clear fun explanations with cartoons and intelligent analogies, like the opening "parable of the surveyors", you'll like that.

 

[Tom1992]

of course if you accept that GR is encoded in a riemannian manifold, then it is obvious you want to learn that geometric theory.

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.

 

[mathwonk]

well the lorentz metric is hardly positive definite.

 

[Chris Hillman]

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 $SO(1,n)$.)

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:

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:

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".

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.)

 

[mathwonk]

The lectures I referred to were at the 2001 Park City conference on Quantum Field Theory, super symmetry, and enumerative geometry, sponsored by the Institute for Advanced Studies.

But I recommend you take Chris's advice in preference to mine as he seems to know what he is talking about here, i.e. he seems to believe he understands the topic, while i admit I do not.