GR: A First Course in General Relativity

[Gza]

I just flipped through a book on GR today called: A First Course in General Relativity by Bernard Schutz, and was instantly fascinated. I was blown away by the physical and mathematical richness of the subject. I always had the impression that learning it would be far down the road in my academic career, and thus never really had interest in the subject, but found I could understand most of the terminology. He even claimed that all one really needed to know was a little special relativity, some electrostatics, and math through multivariable calculus, which are all subjects I've studied pretty thouroughly in the past year. I still haven't started on any of my upper division coursework, so do you think I would be mature enough to actually grasp the material, or would I just be wasting my time? If any of you have any links to some good beginner sites on GR that would be great too. Also a nice primer on Tensor calc would be helpful. Thanks guys.

 

[selfAdjoint]

Yes, with that background you shouldn't have any problem learning basic GR, and then if you were still interested in it you could move into special topics, black holes or gravity waves or numerical calculations of gravity effects or whatever grabs you.

 

[Gza]

The book claimed that it would be suitable for a course of 6 months to a year. And this is supposedly a book for beginners. Would you expect a person of average to above average intelligence to take that long just to learn the fundamentals of the theory?

 

[selfAdjoint]

The book claimed that it would be suitable for a course of 6 months to a year. And this is supposedly a book for beginners. Would you expect a person of average to above average intelligence to take that long just to learn the fundamentals of the theory?

Yes. You don't just read the book like a novel. You allow plenty of time to go over the first definitions and commit them to memory. Even when you are not doing problems, you should be questioning statements in the text in the light of what you have already learned. Don't be shy of asking questions here when you are stumped, either. I would be surprised if anyone not absolutely brilliant or over prepared could do it in under 6 months. Remember you do not have the advantage of a classroom with the instructor's insights and answers to the other student's questions. You have to generate a substitute for all that by your lonesome. You can DO it, but not if you don't take it seriously.

 

[Gza]

I had a question on tensors. I was told in simple terms by someone that you use a tensor when you want to express a linear relationship between two vectors, that may not be necessarily in the same direction. This makes a little bit of sense to me, but when I see how the tensors are computed, simply by multiplying every possible permutation of the vectors' components, and sticking them in a matrix, it all seems kind of contrived to me. What kind of physical basis does that computation have? Can someone give some examples of tensor use in a more easy setting than GR? Thanks for the help by the way Adjoint; I'm motivated now.

 

[selfAdjoint]

This is how it was originally explained to me. Suppose you have a block of Jello(r) put some kind of force, a twist say, onto it (gentle! Don't break it!). Now look at a small region within, you want to describe the effect of the force there. For any orientation there you have a plane, described by a normal vector with its three components. And acting on that plane is the force, described by another vector with three components. Now for every component of the normal, and every component of the force, there's a number you have to keep track of, the stess that that force component generates on that normal component. So nine numbers, write them in a square array with (say) the normal components as the rows and the force components as the columns. Then you can prove that the array obeys a linear algebra and behaves so-and-so when you change coordinates, and it turns out to be a tensor.

 

[homology]

A really basic example of a tensor is a linear transformation on a vector space. It takes vectors and sends them to vectors, i.e. it is a 'mixed' rank tensor. If you work it out in matrices and then work it out with indices you'll see a connection. So in a sense you have been using some tensors for a while.

It can get a little tricky when you move away from rank two mixed tensors to something like a rank 2 covariant tensor. These are still abundant but are hidden early on in physics. An example is the inner product. As you'll see in the beginning of Schutz and as you may already be familiar with, the inner product in special relativity is not the same as the classical dot product. The usual dot product is an inner product induced by the identity so if v and w are vectors then (v,w)=Transpose(v)*I*w where I is the identity matrix, in SR you have another matrix, call it N so that for two 4-vectors v,w you have (v,w)=Tranpose(v)*N*w. But many different sorts of matrices can define innner products and for a matrix A satisfying certain properties the inner product defined by it (at least in the real case) is (v,w)=Transpose(v)*A*w. So I, N and A are all rank two covariant tensors.

For a nice treatment, that is surprisingly deep, on tensors (ranks 0,1,2,3 for both covariant and contravariant) check out Gabriel Weinreich's book Geometrical Vectors.


Cheers,

Kevin

 

[Gza]

For a nice treatment, that is surprisingly deep, on tensors (ranks 0,1,2,3 for both covariant and contravariant) check out Gabriel Weinreich's book Geometrical Vectors.

I got a chance to skim through the book at a bookstore, and didn't really see the word "tensor" mentioned. I bought it anyway because it looked like an enlightening read. What other sources would you recommend for an absolute beginners treatment?

 

[homology]

I got a chance to skim through the book at a bookstore, and didn't really see the word "tensor" mentioned. I bought it anyway because it looked like an enlightening read. What other sources would you recommend for an absolute beginners treatment?

Don't be fooled by its pictures, its not simpled minded. Its very accessible but it's very deep and I have always found that it encourages independent thought. There are tensors in the book, I'll give you a start.

arrows: contravariant vectors
stacks: covariant vectors
thumbtacks: rank 2 contravariant tensors
sheafs: rank 2 covariant tensors

Its a good read. Other books for tensors that are nice are Simmonds
A Brief on Tensor Analysis ()

There's another one I can't quite think of it yet. Oh, well if you like relativity, Schutz's A First Course in General Relativity develops tensors nicely. Be aware that some books will treat covariant and contravariant tensors the same, which they are in an orthonormal basis, but its still useful to use these texts. But Schutz will help you calculate with them and Weinreich will help you see the difference between them in an very general fashion.

Kevin

 

[robphy]

thumbtacks: rank 2 contravariant tensors
sheafs: rank 2 covariant tensors

If I'm not mistaken, those pictures are only for totally-antisymmetric tensors.



A great and inexpensive tensor book (although probably not for beginners) is Schouten's "Tensor Analysis for Physicists"

There are pictures of differential forms like what are described above.

Here's a useful site for a beginner: Peter Dunsby's "Tensors and Relativity" http://vishnu.mth.uct.ac.za/omei/gr/chap3/frame3.html

 

[pmb_phy]

I just flipped through a book on GR today called: A First Course in General Relativity by Bernard Schutz, and was instantly fascinated.

I agree 100%. Schutz's text is highly touted to be one of the best texts to learn GR from. Its the main book I used to learn GR.

I still haven't started on any of my upper division coursework, so do you think I would be mature enough to actually grasp the material, or would I just be wasting my time?

Its definitely worth your time since you have the people in this forum to help you. I can help you as much as you'd like. At the moment I'm disabled and have all the time in the world to discuss GR etc. If you'd like we can even discuss it in e-mail as much as you'd like (I'm not shy about giving my e-mail address out).

You should also check out Schutz's new book Gravity from the ground up. See http://www.gravityfromthegroundup.org/. If your goal is to learn GR then I'd suggest reading this first. I haven't read it yet myself. I just got it and am reading it when I have free time. I looked through it and it seems to be as high quality as his GR text. It'd be nice to have someone to discuss the book with as I read it so if you do pick it up then please let me know.

If any of you have any links to some good beginner sites on GR that would be great too. Also a nice primer on Tensor calc would be helpful. Thanks guys.

I created a website for the sole purpose of facilitating physics discussitions over the interent in e-mail, in physics forums and physics newsgroups. The web page is at

http://www.geocities.com/physics_world/

The intro to tensors is at
http://www.geocities.com/physics_world/ma/intro_tensor.htm

Also see the section Free Online Notes in my web page. There is a lot of good stuff out there. Especially Thorne and Blanchard's new text which is presently online at http://www.pma.caltech.edu/Courses/ph136/yr2002/index.html

Pete

 

[pmb_phy]

Be aware that some books will treat covariant and contravariant tensors the same, which they are in an orthonormal basis, ...

I think I disagree with that statement big time. But first, can you please clarify what you mean by this statement? Thanks.

Pete

 

[sal]

Be aware that some books will treat covariant and contravariant tensors the same, which they are in an orthonormal basis, ...

I think I disagree with that statement big time. But first, can you please clarify what you mean by this statement? Thanks.

Since I'm not a mind reader, I don't know for sure what "homology" meant by that but I think I understand and I think I agree.

First, a qualifier: No general relativity text, nor any tensor calculus text, ever treats them as "the same". But that leaves a lot of other textbooks!

In any basis, the dual vectors (covariant rank 1 tensors) are isomorphic to the vectors (contravariant rank 1 tensors), via the fact that both are vector spaces of the same dimension. The isomorphism is through the correspondence between the basis dual vectors and the basis vectors.

However, there is in general nothing unique or "natural" about this isomorphism, because it depends entirely on the basis chosen.

But -- and here's where the weasel gets in -- if you restrict yourself to orthogonal changes of basis, then ordinary dot product is preserved, and the isomorphism between dual vectors and vectors is fixed and "natural". And in this limited case, the distinction between covariant and contravariant tensors largely melts away.

Note, however, that this means you must use a basis at every point which is orthonormal under ordinary dot-product. But that means that you will often find yourself using a non-coordinate basis! The integrability conditions may not generally be met, in which case you can't extend the basis to a full coordinate system on the manifold.

Now, go pick up just about any intermediate mechanics text, and look at the way they write basis vectors. You'll most likely find they're wearing hats (^). They're using unit basis vectors, even though this means they must give up the convenience of having a coordinate system to back them with. One very large reason for doing this is that it allows them to retain the natural correspondence between vectors and dual vectors, and in consequence, they can totally ignore the distinction between covariant and contravariant tensors.

Symon's "Mechanics" text, for instance, is something of a standard among intermediate-level texts and it does indeed make no distinction between contravariant and covariant tensors.

 

[homology]

Now, go pick up just about any intermediate mechanics text, and look at the way they write basis vectors. You'll most likely find they're wearing hats (^). They're using unit basis vectors, even though this means they must give up the convenience of having a coordinate system to back them with. One very large reason for doing this is that it allows them to retain the natural correspondence between vectors and dual vectors, and in consequence, they can totally ignore the distinction between covariant and contravariant tensors.

This is what I had in mind. Many physics books either use tensors implicitely or explicitely use them without respect for the way they change under coordinate transformations, because many physics books focus on orthonormal basis so the metric tensor is just the identity and you might as well just put all your indices in one place. (This doesn't mean that I advocate such a view, i don't).

Of course, this is not the case in relativity or any situation where the metric is nontrivial. But its worth noting that some tensor books (I'm at a loss for the titles at the moment) do spend a lot of time in the beginning with no distinction between co and contravariant components. They work in orthonomal bases and then, in the later sections generalize to nonorthonormal bases.

Kevin

 

[pmb_phy]

...because many physics books focus on orthonormal basis so the metric tensor is just the identity

The metric tensor, when represented by a matrix, is never the idenintity matrix in relativity.


Pete

 

[HallsofIvy]

The metric tensor, when represented by a matrix, is never the idenintity matrix in relativity.


Pete

That's true. It is always possible to choose a coordinate system in which it is (locally) a diagonal matrix with three negative and one positive (or vice versa) ones on the diagonal.

There are a heck of lot of other things tensors are used for other than relativity. One important application is in non-homogeneous elastic materials. In many applications, one can take orthonormal coordinate systems so that the metric tensor is the identity matrix. As long as one only considers transformations between such orthonormal coordinate systems there is no distinction between "covariant" and "contravariant" tensors. They are referred to as "Euclidean tensors".

 

[homology]

Of course I never said that the metric tensor in relativity was ever the identity. However the metric tensor is a mathematical object without any necessary physical application. The identity is a metric tensor, just not one used in relativity.

Kevin

 

[pmb_phy]

They are referred to as "Euclidean tensors".

They are also known as Cartesian tensors which is an example of something more general called an affine tensor.

Pete

 

[DW]

They are also known as Cartesian tensors which is an example of something more general called an affine tensor.

Pete

They are not tensors in relativity.

 

[Gza]

I agree 100%. Schutz's text is highly touted to be one of the best texts to learn GR from. Its the main book I used to learn GR.

Thanks for the excellent resources pmb. It'd be great to discuss the book with someone who's read it, i'll be sure to email you with any lengthy questions/comments I may have. Thank you for the reference to his new book as well. It would seem to be a good warm-up book before jumping into any serious study. I got a chance to take a look at your site, and was very impressed. I printed out the intro to tensors since I found it to have an excellent explanation of the notation involved (something most books on the subject tend to skip, and assume you know what is happening.)

 

[da_willem]

It's nice to hear other people taking on the subject too. I'm an undergraduate applied physics student and I think I will never be tought the subject at the university. So I as well decided to learn it from Schutz's text.

I am also reading "a short course in general relativity" (Foster and Nightengale) which is (as the title suggests) much more brief than Schutz' text, which makes it ideal to study when you do not wan't to spend too much time on the subject. It does not treat covariant and contravariant tensors the same but uses gradients and normals for it, not mentioning that this is only one example of these tensors. I find the book worth studying but I am glad I combine it with another book.

 

[pmb_phy]

Thanks for the excellent resources pmb. It'd be great to discuss the book with someone who's read it, i'll be sure to email you with any lengthy questions/comments I may have. Thank you for the reference to his new book as well.

You're most welcome. Feel free to e-mail me anytime

I got a chance to take a look at your site, and was very impressed. I printed out the intro to tensors since I found it to have an excellent explanation of the notation involved (something most books on the subject tend to skip, and assume you know what is happening.)

Thanks. Always nice to hear feedback, especially when its good.

Pete

 

[DW]

You're most welcome. Feel free to e-mail me anytime

Thanks. Always nice to hear feedback, especially when its good.

I'm not sure if I mentioned it somewhere a Lorentz tensor is an example of an affine tensor.

Pete

No it isn't.