Getting Into Relativity: Tips for a Physics/Math Major

[SuitCoatBassis]

Hello all.

I'm a third year physics/math major and I'm trying to figure out what area of physics to go into for my graduate studies. I'm interested in something very mathematical, and I would like to learn more about relativity and cosmology in general.

It's all a gigantic wealth of information, and so if anyone has any tips on how to start learning about cosmology and relativity (e.g. particular books, articles), I'd really appreciate it!

Thanks!

 

[yenchin]

Have you taken a course in differential manifold or Riemannian geometry?

 

[SuitCoatBassis]

Not yet, I haven't. I was led to believe that those were generally graduate level courses?

 

[yenchin]

Not yet, I haven't. I was led to believe that those were generally graduate level courses?

I guess it depends on the school. OK. I am going to copy my reply from another thread and modify it

Some people prefer to directly learn abstract manifold theory, but I think it is good to get some geometric intuitions by first learning 3-dimensional differential geometry of surfaces. A standard text of this is Do Carmo's "Differential Geometry of Curves and Surfaces ". Depending on your interest, you don't really have to go over the whole book though. You can then pick up Riemannian geometry. Do Carmo has a book on that too. But my favorite differential geometry text is John M. Lee's "Riemannian Manifolds: An Introduction to Curvature". You probably should start with his other book smooth manifold before you read the one on curvature though. Spivak first two volumes are worth a read as well, though some people don't like his verbose style.

Of course general relativity is really about semi-Riemannian geometry [though physicists like to refer to it as merely 'Riemannian'], so you would want to also refer to texts like O'Neill's Semi-Riemannian Geometry With Applications to Relativity.

For general relativity itself, again, there are many choices. If you prefer to see the "physics first" approach, Hartle is a popular choice. If you are more mathematically inclined, go with Carroll. I personally don't like Hartle's book, and first learned GR via the little book of Foster & Nightingale. General Relativity by Wald is good, but maybe not as your first textbook. You should of course explore many texts and see which one suits you the best. Different book speaks to different soul!

By the way as you are more mathematically-minded, you may enjoy the following book for the structure of special relativity [it won't teach you much physics, but it is a superb math book]: The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity .

 

[George Jones]

For most stuff in general relativity, you don't need a course in differential geometry. Linear algebra, multivariable calculus, and a smattering of differential equations is sufficient mathematical preparation to start learning general relativity, and I recommend that you not wait for a course in differential before starting to dig into general relativity.

If you find that you find the math intrinsically interesting, or if you find it necessary for certain topics like global methods and singularity theorems, then study differential geometry. I am interested these things, and I have studied differential geometry (I own all three of Lee's books).

 

[yenchin]

As for cosmology, there are again many texts [not a surprise I guess]!
Actually for most of the cosmology, you don't really need to master much details from general relativity. I think Liddle's little book An Introduction to Modern Cosmology will be suitable as your introduction into the field. Of course most general relativity textbooks also have a chapter or two on cosmology. But Liddle's book allows you to pick up cosmology with minimal general relativity background, in case you can't wait

You may also enjoy the half-popular science-half-technical book Your Cosmic Context: An Introduction to Modern Cosmology, which really does a good job in explaining many concepts.

 

[WannabeNewton]

If you are interested in the mathematical physics aspects of GR (generating solutions to the EFEs, initial value problem, singularity theorems, topological aspects of causality etc.) then Wald is probably the best introduction for that, followed by the very math heavy Hawking & Ellis. Hopefully, since you're a 3rd year physics and math major, you've already taken a few semesters of analysis and have seen things like analysis on submanifolds in $\mathbb{R}^{n}$ as well as integration theory (measure spaces and Lebesgue integration mostly) because these things will show up in Hawking & Ellis (on the other hand, as far as integration theory goes, you won't need any for Wald as the only integration he ever uses is integration of differential forms using Riemann integrals). If your school is like mine, then you've hopefully also taken a semester of introductory point-set topology because topology is a huge part of the mathematical physics of GR. As an upside, Wald has a ton of in-text calculations, as well as numerous end of chapter problems, that specifically get you well acquainted with tensor algebra / tensor calculus so that is one thing you don't have to have seen at all before with regards to the mathematical physics.

If you are interested in the down to Earth physics of GR, however; then I would say books like Wald and Hawking & Ellis will not be the optimal ones to use by any means. Carroll would be good and there is also the classic MTW, which you've probably heard of. You might also try Schutz "A First Course in General Relativity" which is light on math but very good in terms of the physics. A very recently published book that George Jones recommended to me, and which I recently received in the mail, is Padmanabhan "Gravitation". It is both heavy on math (though not as much as Wald) as well as the actual physics of GR. From what I've seen while skimming, it looks to be quite good (and has tons of exercises).

 

[yenchin]

A very recently published book that George Jones recommended to me, and which I recently received in the mail, is Padmanabhan "Gravitation". It is both heavy on math (though not as much as Wald) as well as the actual physics of GR. From what I've seen while skimming, it looks to be quite good (and has tons of exercises).

Even more recent are
[1] General Relavity by Norbert Straumann,
[2] Einstein Gravity in a Nutshell by Anthony Zee.

But I have yet to take a look at any of them to say anything :-)

 

[Lavabug]

My personal recommendation is the book titled "relativity demystified - a self-teaching guide". Don't be deceived by the cover, it is a serious book that covers a lot of content, pretty much all of entry-level relativity and cosmology and your preparation as a 3rd year student is definitely sufficient (unlike a lot of the big classic GR texts that do assume familiarity with tensor calculus). It has some typos though, but it's dirt cheap (<10$) compared to most of the big titles in the subject and you'll probably get more out of it as a beginner.

IMO it is a better intro than S. Carroll's lecture notes (based on his book, which I don't like either), which get recommended a lot.

Schutz gets recommended a lot, might be a good place to start as well. But from what everyone says about it, it might not give you as much mathematical grounding as my first rec.

As a side note, if you have a more general interest in highly mathematical areas of physics, look into what falls under the generic banner of nonlinear sciences: fluids, atmospheric physics, biophysics, fundamental statistical mechanics to name some. These are all areas of research that are equally if not more mathematical than GR (often intersecting with it, ie: gravitational waves) and are very interdisciplinary, which translates to a large variety of potential summer research positions and grad schools. I'm developing an interest in this myself, I just got me a pair of books on this discipline for self-study (J. Guckenheimer for a more general math approach and O. Regev which is geared towards astrophysical applications).

 

[WannabeNewton]

Even more recent are
[1] General Relavity by Norbert Straumann,
[2] Einstein Gravity in a Nutshell by Anthony Zee.

But I have yet to take a look at any of them to say anything :-)

Omg, Zee wrote a book on GR? If its anything like his QFT book, then I must have this! I love his whimsical writing style.

 

[yenchin]

Omg, Zee wrote a book on GR? If its anything like his QFT book, then I must have this! I love his whimsical writing style.

It is ~300 pages thicker than his QFT book though... that is a coconut shell

 

[jtbell]

How many MTWs is it equivalent to?

 

[WannabeNewton]

How many MTWs is it equivalent to?

Well let's see...a million pages divided by 888 pages equals...O.O

 

[robphy]

This looks like an interesting starting point for undergrads...
A General Relativity Workbook (Thomas A. Moore, 2013)
http://pages.pomona.edu/~tmoore/grw/

(Some of my faculty colleagues (who are non-relativity folks) have been going through some chapters informally... with the goal of possibly teaching it to our students someday.)


In addition, Geroch's 1972 lecture notes on General Relativity have been typeset into LaTeX
https://www.amazon.com/dp/0987987178/?tag=pfamazon01-20
(As Wald says in his text... "(page x) The influence of Robert Geroch should be apparent to readers familiar with his viewpoints on general relativity.")

https://www.amazon.com/dp/0226288641/?tag=pfamazon01-20 General Relativity from A to B is also good... as a supplement.
It was one of my first relativity books... which I did not appreciate until my third re-reading of it.

 

[WannabeNewton]

In addition, Geroch's 1972 lecture notes on General Relativity have been typeset into LaTeX
https://www.amazon.com/dp/0987987178/?tag=pfamazon01-20
(As Wald says in his text... "(page x) The influence of Robert Geroch should be apparent to readers familiar with his viewpoints on general relativity.")

Not to mention Wald cites Geroch's papers in like every other line lol Many of the calculations in Wald's book are pretty much truncations of the ones in Geroch's papers e.g. Geroch's 1971 paper. How useful would you say Geroch's book "Relativity from A to B" actually is with regards to GR? I have never used the book so I don't know if its style is similar to Wald's and I don't know for what purpose it was written. Thanks in advance robphy!

 

[robphy]

"General Relativity from A to B" explained to me the meaning of the radar-coordinates (emphasizing operational measurements) and the meaning of the spacetime interval (emphasizing the causal structure). While I could push symbols around, after re-reading it I understood better what was going on physically and geometrically.

It also addresses some (arguably philosophical [i.e., conceptual foundations of science]) issues in the structure of relativity theory.

It is a remarkably deep book... which might not be evident at a glance... since it uses a lot of words and simple algebra, and draws lots of spacetime diagrams. I would say that GRfromAtoB is more about "spacetime thinking" than it is about "GR" (which in most books is about the Einstein Field Equations).

Note... the book is based on a course for nonscience majors at UChicago.

The lecture notes above, of course, offer more mathematical details in both "spacetime thinking" and "GR"... using coordinate-free methods.

 

[WannabeNewton]

I have his lecture notes but it looks like GR from A to B will definitely be something I have to buy, given your comments. Thankfully it's only 8 bucks! Thanks again robphy. It would be nice to take a break from the sea of tensors and covariant derivatives in Wald's text for once haha.

 

[SuitCoatBassis]

Thanks for all the input! I have a list of "to-read" books ready to go now, and I purchased a copy of the excitingly-cheap "General Relativity from A to B" to read in my spare time.

I'm trying to examine my prospects for the future if I continue on to study relativity/cosmology type physics and from what I hear the job market is quite grim :(

 

[WannabeNewton]

I'm trying to examine my prospects for the future if I continue on to study relativity/cosmology type physics and from what I hear the job market is quite grim :(

Tell me about it. At this point it seems better to just study it for fun and get a job as a stock broker instead :p

 

[vanhees71]

If you are interested in the down to Earth physics of GR, however; then I would say books like Wald and Hawking & Ellis will not be the optimal ones to use by any means. Carroll would be good and there is also the classic MTW, which you've probably heard of. You might also try Schutz "A First Course in General Relativity" which is light on math but very good in terms of the physics. A very recently published book that George Jones recommended to me, and which I recently received in the mail, is Padmanabhan "Gravitation". It is both heavy on math (though not as much as Wald) as well as the actual physics of GR. From what I've seen while skimming, it looks to be quite good (and has tons of exercises).

Perhaps I'm wrong, but from a close look at the chapter on special relativity it seems to be very good conceptional wise but on the other hand quite useless, particularly for the beginner, because it's full of typos. E.g., at one place he writes
\partial_{\mu} J^{\mu} = \partial_t J^0 -\vec{\nabla} \cdot \vec{J} \quad \text{(WRONG!)}
although of course it must read
\partial_{\mu} J^{\mu} = \partial_t J^0 + \vec{\nabla} \cdot \vec{J}.
He seems to have overlooked that
\partial_{\mu} = \frac{\partial}{\partial x^{\mu}},
i.e., there is no sign in the covariant four-divergence. This mistake I've often seen students making when introduced to the four-dimensional vector/tensor calculus. So it would be good if such trivial wouldn't appear in textbooks.

At another place he has a factor of two wrong in the definition of the orbital-angular momentum operator which is \hat{\vec{L}}=-\mathrm{i} \vec{x} \times \vec{\nabla} and not twice this operator.

Another great sin is to write \Lambda^{\mu}_{\nu} instead of {\Lambda^{\mu}}_{\nu} for Lorentz-transformation matrices, which can become awkward, when one raises and lowers indices of such matrices (as usual with the Minkwoski metric \eta_{\mu \nu}).

These are only little things, but can be very confusing, particularly for beginners. It's a pity, because I think it's a pretty good didactical work in principle.

 

[WannabeNewton]

Is this from Padmanabhan's book?

 

[Sigma057]

I'm a 3rd year math\Physics major as well, and this summer I taught myself General Relativity using "Relatavity, Gravitation, and Cosmology" by Robert Lambourne

It isn't the most mathematically rigorous book out there, but it presents the necessary mathematics in a way suitable for undergrads. I find it really bridges the gap between vector and tensor analysis as well.

I highly recommend it as a self study book because it's incredibly well written, concise, and in color! An almost unheard of feature nowadays.