Advice needed on a course outline to teach myself General Relativity

[itsthemac]

Advice needed on a "course outline" to teach myself General Relativity

I want to learn it inside and out. I'm tired of settling for dumbed-down versions of GR, and thus dumbed-down versions of how we describe the very universe we live in, just due to my lack of mathematical understanding. But I don't know what the best path to follow would be to overcome this. Specifically, I want to know what math I should master beforehand, and what order I should learn those maths in.

My math background: I have a decent understanding of basic multi-variate/vector calculus. That's really as far as I've gotten. I have no experience with differential equations (and hence differential geometry), linear algebra, tensors, or Reimannian manifolds (I don't even know what most of this stuff is). I only listed these things because I have heard them thrown around when people discuss the math of GR, but I'm sure there are other things that I'm not even aware of that I would need to learn. I want to know what my "course outline" should be. Given my limited background, what specific types of math should I learn? But more importantly, what order should I learn them in (I'm sure I would need to learn some of these before I could move on to more advanced ones)?

Also, if you have any recommendations for any texts on these topics, I would really appreciate that as well. But my main question is just what math I need to learn, and what order I should learn it in. I'm willing to put in the effort and spend the time (even if it takes a few years) as this is has been a goal of mine for a long time. Thanks in advance.

 

[bcrowell]

Make sure you start from a strong foundation in SR. Spacetime Physics by Taylor and Wheeler is good for this.

After that , try Exploring Black Holes by Taylor and Wheeler. It's not a general GR book, but it will teach you a lot at an intellectually rigorous level without any need to learn more math.

There are also various GR books out there intended for upper-division physics majors. I wrote one of this flavor that's free: http://www.lightandmatter.com/genrel/ A well known one is Hartle.

Don't discount what can be learned from a nonmathematical but rigorous book such as Geroch's Relativity from A to B.

 

[atyy]

Learn linear algebra really well. Learn what a vector space is. And learn that a "metric" is not part of the definition of a vector space, but is instead an additional structure.

Then learn some differential equations (actually vector calculus is enough differential equations), paying attention to the idea that these are related to vector fields, and the vector at each point is in a tangent space.

Special relativity is the idea that the metric of spacetime is Minkowskian, and general relativity is the idea that the metric of spacetime is curved and determined by matter. In both special and general relativity, the tangent spaces are vector spaces with metrics.

I liked http://books.google.com/books?id=bT...&resnum=1&ved=0CC4Q6AEwAA#v=onepage&q&f=false

Also http://books.google.com/books?id=iD...&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q&f=false

And http://books.google.com/books?id=Ia...ook_result&ct=result&resnum=1&ved=0CCkQ6AEwAA

And of course use the standard texts by Schutz; Misner, Thorne and Wheeler (a masterpiece, but for reference only); Weinberg; and Wald.

 

[dextercioby]

It's hard to learn a theory by yourself. You need to be taught. Formal education. Some guy there explaining you the physics and the maths at the same time, but at an adequate pace. A university course won't cover a textbook's material for sure, but only part. If the teacher's smart, the course covers the essential.

The book (and the reccomendations of books on GR are never enough, since some guys wrote the same chapter better than the others) should come as a help, not as a basis.

If you can't go to school and get the chance to be taught, then you're only left with the books and the challange you'll have to face to capture all that's really important out of a theory.

Feynman's notes on gravitation should be a must in both circumstances. Easiest math.

 

[PAllen]

It's hard to learn a theory by yourself. You need to be taught. Formal education. Some guy there explaining you the physics and the maths at the same time, but at an adequate pace. A university course won't cover a textbook's material for sure, but only part. If the teacher's smart, the course covers the essential.

The book (and the reccomendations of books on GR are never enough, since some guys wrote the same chapter better than the others) should come as a help, not as a basis.

If you can't go to school and get the chance to be taught, then you're only left with the books and the challange you'll have to face to capture all that's really important out of a theory.

Feynman's notes on gravitation should be a must in both circumstances. Easiest math.

I disagree on the idea that self study is not plausible. It depends on the person, the books, and patience. Note that everyone does self study after some level of mastery - there are no courses. For me, courses didn't work well. The key was finding books with good explanations plus lots of exercises. Personally, I never took a course in calculus but got a perfect score on the harder of the two AP calculus exams via self study.

 

[Newtime]

1. Know linear algebra inside and out.
2. https://www.amazon.com/dp/354052018X/?tag=pfamazon01-20

The above book is great. Although, the section on GR is at the very end, so you'll have to trudge through 11 chapters of intense mathematics before then.

 

[bcrowell]

IMO self-study is a perfectly reasonable way to learn GR.

My favorite linear algebra book is this one: http://joshua.smcvt.edu/linalg.html/ It also happens to be free.

IMO a course in differential equations would be 99% wasted effort as preparation for GR. Courses in differential equations are basically cookbooks, and almost all of the material is useless if you're using it on a problem that none of the recipes applies to. For example, techniques for solving linear differential equations won't help in solving the field equations of GR, which are nonlinear. It's useful to absorb some general ideas about boundary-value problems.

One math book that might be helpful is Coxeter's Introduction to Geometry. (Despite the title, it's not an introductory book for high school kids, it's a book for upper-division math majors.)

GR really doesn't require heavy mathematical knowledge as a prerequisite. Furthermore, many people with strong mathematical backgrounds thoroughly misunderstand the physics of GR. It certainly helps if you have a certain level of mathematical experience and sophistication, but that's orthogonal to an encyclopedic knowledge of specific techniques such as the ones taught in a diffeqs course.

 

[twofish-quant]

It's hard to learn a theory by yourself. You need to be taught. Formal education.

Disagree. I taught myself general relativity. Two of the books which I found useful were Wald's "General Relavity" and Shapiro/Teukolsky's Black Holes, White Dwarfs, and Neutron Stars. I didn't find Misner or Weinberg that useful.

Something that I've found very useful is to read different textbooks, since it helps if people explain it in a different way. Also, I've found that newer textbooks tend to be "better" because people figure out new and better ways of explaining something.

 

[lugita15]

Feynman's notes on gravitation should be a must in both circumstances. Easiest math.

I hope you don't mean the Feynman Lectures on Gravitation, which assumes familiarity with quantum field theory and the associated mathematics.

 

[DiracLandau]

One of the best books for learning GR is through Landau and Lifgarbagez's 2nd volume in course of theoretical physics:The classical theory of fields.First go through the initial chapter on SR and then start with Chapter 10,I suppose ,on GR.There's no need to go through all the chapters in that like those on cosmology etc.But the initial 3 to 4 chapters are the MUST and the best.Then for problems start solving the classic problem book by Lightman,Press,Price and Teukolsky.This will give a strong foundation.People may tell that Landau's book is bit outdated etc as it doesn't use the manifold formalism & all that.But formalism can wait.The physical crux and the beauty of the subject is very well presented in that book.Once this is done you can go for any other book.If you want to learn the formalism etc also sideways Schutz' undergraduate book or Hartle's book will do.But first do the above and the rest is taken care later.

 

[Fredrik]

Some linear algebra is needed. My favorite book is Axler, but most other books will do just fine. I recommend that you choose a book that defines linear operators (also called linear maps, linear transformations or linear functions) early in the book. (I know Anton defines it around page 300, so stay away from that one). You don't need an entire course of linear algebra, but you should at least understand inner product spaces, orthonormal bases, and the relationship between linear operators and matrices.

You didn't say anything about special relativity. It helps to understand SR before GR. You should study a book that emphasizes spacetime diagrams, and doesn't treat the whole thing as an exercise in elementary algebra. Taylor & Wheeler gets the most recommendations around here, so it can't be bad. (I haven't read it). I like the SR section of Schutz's GR book myself. If you buy that one, you can also use it to get started with GR.

For GR, Wald is a good choice for the mathematically mature, but you don't seem to be there yet. Perhaps Nabeshin's post from a few weeks ago will help:

There are really three tiers at which you can really learn anything substantive about general relativity.

Tier I: Primarily algebra based with light calculus. This is suitable for a first or second year university student. Prior knowledge of physics should include the basic rudiments of mechanics from the F=ma point of view. The hallmark textbook is: https://www.amazon.com/dp/020138423X/?tag=pfamazon01-20

Tier II: Calculus based with light differential geometry. This is suitable for a third or fourth year university student. Prior knowledge of physics should include an upper level E&M course and exposure to lagrangians and hamiltonians. The hallmark textbook is: https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20

Tier III: Full general relativity with differential geometry. This is usually relegated to graduate classes, but can be tackled by advanced undergraduates. Prior physics knowledge is basically an entire undergraduate physics/astrophysics education. Some representative textbooks are:
https://www.amazon.com/dp/0226870332/?tag=pfamazon01-20
https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20

For differential geometry, I recommend the books by John M. Lee: "Introduction to smooth manifolds" and "Riemannian manifolds: an introduction to curvature". Read the first one to understand manifolds, tangent spaces and tensor fields. (I think that's less than a third of the book). Read the second to understand connections, parallel transport, covariant derivatives and curvature. (About half of a much shorter book).

The definition of "manifold" includes the words "Hausdorff and 2nd countable topological space". If you can live with not knowing what those things mean (and you really don't need them to understand the physics) you don't have to study topology. If you really want to know, you can probably just look them up at Wikipedia. If that doesn't work for you, there are many topology textbooks you can consult, including Lee's "Topological manifolds", and "Topology" by Munkres.

IMO self-study is a perfectly reasonable way to learn GR.
...
IMO a course in differential equations would be 99% wasted effort as preparation for GR.

I totally agree with these two points.

 

[MathematicalPhysicist]

Well a tip from my advisor who said that while you have the option for a course go to it, because as you get more advanced there won't be anymore courses that you could take, and from there you basically on your own.

Obvisouly you can learn by yourself GR, but I think the best approach is to learn from a textbook and a course. Unless ofcourse the teacher is awful.

I mean how many of you would have opted to learn by yourself GR from a textbook rather than alongside a master of his craft?

 

[Fredrik]

Many of us have never had an opportunity to learn "alongside a master of his craft". The teachers at the university I went to just barely knew enough to be able to hold a presentation on the subjects they were teaching. There was often no point asking them questions.

I agree of course that if you have a chance to take a course held by a real expert, you should definitely do it.

 

[itsthemac]

I just wanted to thank everyone for the awesome replies so far. There's a lot of good stuff in here. So far, it seems that the general consensus is that my first step should be learning linear algebra, so I think that's where I'll start. If anyone has any more advice/suggested texts, please post. Thanks again.

 

[MathematicalPhysicist]

Many of us have never had an opportunity to learn "alongside a master of his craft". The teachers at the university I went to just barely knew enough to be able to hold a presentation on the subjects they were teaching. There was often no point asking them questions.

I agree of course that if you have a chance to take a course held by a real expert, you should definitely do it.

I wonder where did you learn, cause usually they give a course such as on GR to someone who is researching in gravitation theory.

 

[Fredrik]

Stockholm. (The university, not KTH). I wasn't talking specifically about the GR course. I don't know how good or bad the guy who held the GR course was, because I didn't go to any of the lectures. It was mostly because I needed GR for my M.Sc. thesis so I had already begun studying Wald for GR and Spivak for differential geometry. It seemed pointless to go to the first lectures and then I didn't see a reason to start going there later.

They had a really funny way of teaching GR there. Every odd year, a soft course based on Schutz, with a few fairly easy homework problems and an exam at the end. Every even year, those who wanted to study GR studied Wald on their own and had to solve all exercises in the book to pass the course.

 

[bcrowell]

I mean how many of you would have opted to learn by yourself GR from a textbook rather than alongside a master of his craft?

I learned GR the first time in grad school, with a professor who was a well known relativist. I learned very little.

I learned GR for the second time on my own, from books. I learned the subject much better.

 

[atyy]

Rindler, Weinberg, Wald, Misner, Thorne, Wheeler are masters of the craft. So you can self-study and learn from a master of the craft.

But yes, it's probably very hard to reach professional standards (ie. be able to produce a research paper) by self-study (unless you are a grad student, but in which case you are a professional, and learn by self-study and contact with other professionals).

 

[elfboy]

i still think you can duplicate the results of Einstein conceptually.

 

[MathematicalPhysicist]

Rindler, Weinberg, Wald, Misner, Thorne, Wheeler are masters of the craft. So you can self-study and learn from a master of the craft.

But yes, it's probably very hard to reach professional standards (ie. be able to produce a research paper) by self-study (unless you are a grad student, but in which case you are a professional, and learn by self-study and contact with other professionals).

I am not sure how can you learn by yourself from MTW, from Schutz's textbook it seems reasonable.

 

[twofish-quant]

But yes, it's probably very hard to reach professional standards (ie. be able to produce a research paper) by self-study (unless you are a grad student, but in which case you are a professional, and learn by self-study and contact with other professionals).

It really depends on what you mean by "professional standards". The way that GR is used in most supernova studies is to take the full Einstein equations and reduce them to something with many, many fewer variables. Once you've reduced everything to a PDE, then the hard part of screaming at the computer until you have something that you can numerically solve starts.

None of this is out of the reach of a clever physics senior.

I would argue that *no one* fully understands GR, and you can spend your entire career trying to fully understand GR. But if you want to just understand a enough of it so that you can actually start solving problems, then it's not that hard.

 

[twofish-quant]

For GR, Wald is a good choice for the mathematically mature, but you don't seem to be there yet.

Curiously I found Wald a lot easier read because he using somewhat more abstract mathematics than Weinberg or Misner. The fact that Wald uses rather abstract concepts made it easier for me to understand what he was doing.

What I found frustrating about Misner is that it's a very thick book because MTW tries to spend a lot of pages explaining what a "one form" is, and he didn't succeed at least for me. Once I figured out what a differential form was, then a lot of explanations made sense.

 

[redrzewski]

For linear algebra (and algebra in general), I'd go with Artin's Algebra.

The other thing to consider is if you're going to take the math path or the physics path. There is generally a different style and approach depending on if you're working from a math book vs. a physics book.

I started similarly to you many years ago, and I jumped on the physics path, since I wanted to understand GR. I picked up Schutz. I got bogged down in the index gymnastics, and eventually lost interest. Now I'm starting to circle back, but following a primarily math path, via Lee (Smooth Manifolds), Arnold (math methods of classical mechanics), and Spivak (diff geo). I definitely enjoy the mathematical approach more. Next steps after Spivak will be O'Neill's Semi-remannian geometry.

Another more mathematic treatment of GR:
http://www.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf


So spend some time, and see which style appeals to you.

Also consider Bamberg/Sternberg - A course in mathematics for students of physics. This is a good intro to differential forms with many worked out examples.

Also, Spivak's Calculus on Manifolds is another good choice (although I haven't used it personally).