General Relativity by Robert M. Wald

[Greg Bernhardt]

• Author: Robert M. Wald
• Title: General Relativity
• Amazon Link: https://www.amazon.com/dp/0226870332/?tag=pfamazon01-20
• Prerequisities:
• Contents:

 

[bcrowell]

This is a classic text, but "classic" isn't completely a good thing. General relativity is a living field, and 29 years is a long time. The book never had an acceptable amount of contact with observation, and that shortcoming has become even more severe with the passage of time; it predates LIGO, Gravity Probe B, modern studies of CMB anisotropy, and the discoveries of supermassive black holes and the nonzero cosmological constant.

Pedagogically, I would not recommend this book for someone encountering GR for the first time. For a first-time student, a more appropriate text would be Carroll or the also-classic Misner, Thorne, and Wheeler. For someone who is serious about GR, the book is useful because it treats some advanced topics in a more accessible fashion than one can find elsewhere.

 

[VantagePoint72]

For a first-time student, a more appropriate text would be Carroll or the also-classic Misner, Thorne, and Wheeler. For someone who is serious about GR, the book is useful because it treats some advanced topics in a more accessible fashion than one can find elsewhere.

For a very first look at GR without already having a background in differential geometry, I think a quick read the first few chapters of Schutz is a good idea. After that, I second the recommendation for Carroll (and I think it would be wise to read before MTW). Of course, Carroll (and most others) don't assume prior knowledge of differential geometry either, but Schutz takes you through it at such a leisurely pace that it's especially good for first exposure.

I usually suggest the Schutz > Carroll sequence for introductions, followed by either Wald or MTW for further study.

 

[WannabeNewton]

I wouldn't recommend Schutz over Carroll at all. Schutz butchers the beautiful subject of differential geometry and if you've seen in the past on people asking specific questions from the text on this forum, it is usually tied to his horrible exposition of differential geometry. As a golden rule of thumb: never learn a math subject from a physics textbook especially when the subject forms the very core of the underlying physical theory. Wald is top notch - nothing bad to say about it at all and Carroll is also good (less advanced but still covers a lot and his tone is very gentle). Wald is very mathematical which, for me anyways, makes it much more enjoyable. Carroll sometimes handwaves the mathematics but he still explains everything at a nice level of rigor.

 

[Fredrik]

I think Wald is fine as a first book on GR, but I would only recommend it (as a first book) to a very serious student who's studying differential geometry at the same time. I recommend the books by John M. Lee for that, "Introduction to smooth manifolds" and "Riemannian manifolds: an introduction to curvature". These books are excellent. The only problem is that you need both of them.

I also second the recommendation to read the first few chapters of Schutz first. Wald only devotes one page to SR, but Schutz covers it very thoroughly. Schutz also contains a nice introduction to tensors. (Edit: Just to make it perfectly clear, the following two sentences are about Schutz, not Wald). It's a GR book, but what makes it good are the parts about SR and tensors. The part about GR is too thin on differential geometry for my taste.

 

[martinbn]

Just to add to the above that I would also recommend it to a mathematician who wants to study general relativity.

 

[Lavabug]

To those of you who've read/worked with all of the big name GR texts (Wald, MTW, Weinberg, Landau's fields), if you were to buy only one of them which one would you pick? I'm looking for an encyclopedic thing that will keep me coming back but I would like it to have some treatment on gravitational waves.

I've taken a "mathematical methods for GR" course so I wouldn't be spending much time on the first few chapters(but it would be nice if it were self-contained like Landau's classical theory of fields). I've read some of Carroll's but I really didn't like it(too many handwavy explanations, it just doesn't flow well), and a bit of Weinberg's "Gravitation and Cosmology" and I really liked it(notation was identical to my course), but I only read the intro chapters on tensor calculus and not the actual physics.

 

[WannabeNewton]

My personal bias would have me say Wald, without a doubt. Although I do wish his problem sets were less computational and more theoretical, to complement his theoretical discussions. If you ever get your hands on the book, take an immediate look at problem 7.5; it asks you to show, given the twist ##\omega_{a} = \nabla_{a}\omega = \epsilon_{abcd}\xi^{b}\nabla^{c}\xi^{d}## of a time-like killing field ##\xi^{a}##, that ##\nabla_{[e}(2\xi^{c}\xi_{c}\nabla_{a}\xi_{b]} + \omega \epsilon_{ab]cd}\nabla^{c}\xi^{d}) = 0##.

Without a doubt this was the most arduous tensor calculus problem in the entire text. While I find such tensor calculus problems fun, they make up a very large bulk of the problems in the text whereas I would have liked more "physical" problems (the sort you would see in MTW) and more theoretical problems (the kind you will see in chapter 8 of Wald which were basically topology problems haha). In this sense, I would say getting only Wald would be a mistake because you would need to supplement the problem sets in Wald with the problems from other texts e.g. Carroll (which I also like very much) and, as mentioned before, MTW. I wouldn't recommend Landau's classical theory of fields for learning GR at all.

 

[Lavabug]

My personal bias would have me say Wald, without a doubt. Although I do wish his problem sets were less computational and more theoretical, to complement his theoretical discussions. If you ever get your hands on the book, take an immediate look at problem 7.5; it asks you to show, given the twist ##\omega_{a} = \nabla_{a}\omega = \epsilon_{abcd}\xi^{b}\nabla^{c}\xi^{d}## of a time-like killing field ##\xi^{a}##, that ##\nabla_{[e}(2\xi^{c}\xi_{c}\nabla_{a}\xi_{b]} + \omega \epsilon_{ab]cd}\nabla^{c}\xi^{d}) = 0##.

Without a doubt this was the most arduous tensor calculus problem in the entire text. While I find such tensor calculus problems fun, they make up a very large bulk of the problems in the text whereas I would have liked more "physical" problems (the sort you would see in MTW) and more theoretical problems (the kind you will see in chapter 8 of Wald which were basically topology problems haha). In this sense, I would say getting only Wald would be a mistake because you would need to supplement the problem sets in Wald with the problems from other texts e.g. Carroll (which I also like very much) and, as mentioned before, MTW. I wouldn't recommend Landau's classical theory of fields for learning GR at all.

That's kind of what I was trying to avoid, as my course pretty much consisted of exercises like this (mostly simpler things, ie tensor transformation of Christoffel symbols,general tensor identity proofs, grinding out Ricci components and curves from a given metric but no actual derivation of the Schwarzschild or FLRW metrics...)

It appears as if all major GR texts except Carroll are out of date now (and I don't like this one, it's on my shelf gathering dust but I haven't looked at the later chapters on cosmology or GW which might be done better than the basics). Maybe I should go with Weinberg's first?

 

[George Jones]

Lavabug, you might also want to have a look at the more modern "Gravitation: Foundations and Frontiers" by Padmanabhan, which has bothy exercises and more in-depth projects,

https://www.amazon.com/dp/0521882230/?tag=pfamazon01-20

Read the Editorial Reviews, and use the LOOK INSIDE! feature to a careful look at the Table of Contents.

 

[WannabeNewton]

That's kind of what I was trying to avoid, as my course pretty much consisted of exercises like this (mostly simpler things, ie tensor transformation of Christoffel symbols,general tensor identity proofs, grinding out Ricci components and curves from a given metric but no actual derivation of the Schwarzschild or FLRW metrics...)

If you look at chapter 11 of Wald, problem 11.6 is a funny one because the first two parts of the problem are insanely trivial whereas the third part asks you to calculate the Komar integral for the total angular momentum of the charged kerr space-time. It was such a painful calculation that I cried, several times.

EDIT: Just to clarify, Wald does derive the main results (e.g. Schwarzschild metric) and uses very satisfying geometric arguments in doing so. Also, just for fun, here is a worked out example of another typical tensor calculus problem from Wald (problem 7.1, so yeah same chapter as the one I mentioned above lol): https://www.physicsforums.com/showthread.php?t=677782

Lavabug, you might also want to have a look at the more modern "Gravitation: Foundations and Frontiers" by Padmanabhan, which has bothy exercises and more in-depth projects,

I'll have to take a look at this one alongside Lavabug; I remember you mentioning this book in chat once. How did you find it thus far? Thanks George!

 

[Lavabug]

Lavabug, you might also want to have a look at the more modern "Gravitation: Foundations and Frontiers" by Padmanabhan, which has bothy exercises and more in-depth projects,

https://www.amazon.com/dp/0521882230/?tag=pfamazon01-20

Read the Editorial Reviews, and use the LOOK INSIDE! feature to a careful look at the Table of Contents.

That looks fantastic, thanks! Is this book catching on a lot?

 

[WannabeNewton]

That looks fantastic, thanks! Is this book catching on a lot?

Yeah I would like to know too. The exercises look very fun, they have like an MTW feel to them (haven't seen any of the projects though); unfortunately they don't have any solutions but neither does Wald so no complaint there . It is quite steep in price so it would be nice to check it out somewhat more carefully before buying. We should work through it together haha

 

[WannabeNewton]

So I ordered the Padmanabhan text. I can't wait to dig into it during the summer; hopefully it is as good as it seems from the Amazon preview. Thanks George!

 

[1ndranil]

Robert M. Wald (general relativity) prerequirments

I wish to read ch 1 to ch 6 from the book "General Relativity" by Robert M. Wald in full mathematical generality. [I do not have any considerable background in differential geometry. I'm a physics graduate student with some background in topology, advanced calculus, linear algebra]. Suggest me a book for mathematical parts.

 

[Jorriss]

I would recommend David Malament's text.

http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf

He fills in the gaps left by Wald's treatment of differential geometry.

 

[WannabeNewton]

I have personally found the notes linked by jorriss to be absolutely amazing for GR; in addition to jorriss's recommendation, check out this thread:

https://www.physicsforums.com/showthread.php?t=722285

 

[pervect]

If you have the time, I'd suggest reading an undergraduate GR text first (I'm fond of Taylor, so I'd suggest Exploring Black Holes - there may be better choices out there). And I would read a tensor treatment of electromagnetism (Jackson's is traditional, but you might do just as well or better with the appropriate sections of Griffiths). I assume you've already had something like Goldstein's "Classical Mechanics".

I haven't read Malement, alas.

 

[bcrowell]

Wald has almost no applications and makes almost no connection with experiment, which makes it a poor choice for a student's first introduction to GR. I would start with Taylor and Wheeler's Spacetime Physics, which, although it's an SR text, actually has quite a bit of material in it that explicitly prepares you for GR. Pervect's suggestion of reading an undergrad GR text is also excellent. Hartle is good. Exploring Black Holes is a fine book as far as it goes, but its focus is very narrow, and it won't prepare you with any of the mathematical techniques. Rather than Wald, which is extremely out of date, I would suggest Carroll. There is even a free preliminary version of Carroll online.

 

[Jorriss]

Rather than Wald, which is extremely out of date, I would suggest Carroll. There is even a free preliminary version of Carroll online.

Wald is extremely out of date?

 

[jtbell]

It was published in 1984.

 

[WannabeNewton]

Carroll is a carbon copy of Wald with less sophisticated mathematics. I don't see how it is in any way more modern than Wald. There's little to nothing presented in Wald that is out of date because there are, as already noted, little to no applications of GR in the book; as an aside, it's not like Carroll has any applications either.

Anyways, if one is seeking a book that is both modern in its use of differential geometry as well as its presentation of applications (and existence of applications in the first place) then check out Straumann: https://www.amazon.com/dp/9400754094/?tag=pfamazon01-20 (if your university has Springer access then you can download the e-book version for free).

 

[robphy]

Wald's text is certainly not meant to be a first introduction to GR (for undergraduates)...
but it could be regarded [at the time] as a first course to a modern treatment, emphasizing geometrical methods.In response to 1ndranil's recent post...
for more materials to prepare for Wald, I would suggest:
Geroch's recently published
lectures notes on GR
https://www.amazon.com/dp/0987987178/?tag=pfamazon01-20
and
lecture notes on Differential Geometry
https://www.amazon.com/dp/1927763061/?tag=pfamazon01-20
which likely influenced Malament (when he was at Chicago) and Wald.For a first introduction to GR, here is a new text by Tom Moore:
https://www.amazon.com/dp/1891389823/?tag=pfamazon01-20
which is more application-driven, but less geometrical.

 

[K^2]

This is not a good introduction to General Relativity. If you don't know anything about GR, this should not be the first book you read. But it probably should be the second one. It takes a very rigorous differential topology approach to General Relativity, and it does well in filling in gaps that somebody not from pure mathematics background might have on subjects of topology and linear algebra. Unlike many texts, it abandons embedded manifolds from the getgo, which makes it that much harder to follow if you are new to the subject, but allows for treatment of more general problems.

 

[WannabeNewton]

I will say though that Wald has very few instructive problems; honestly off of the top of my head the only instructive problems in the text that I can think of are the ones in chapter 4 on the frame dragging effect in the weak field regime and gravitational radiation as well as the one in chapter 6 on the redshift factor for mechanical forces in stationary asymptotically flat space-times.

A lot of the problems are heavy index manipulation ("index gymnastics") calculations; the goal of a lot of the problems in Wald seems to be focused on getting you to use coordinate-free abstract index calculations whenever possible (and it definitely does that well) but there isn't much in the way of actual physics. To be honest, this seems to be characteristic of a good number of advanced GR texts (e.g. Straumann, Carroll, d'Inverno off the top of my head) and in my opinion the books that do have very instructive problems on the actual physics of GR are:

https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20 (goes without saying)

https://www.amazon.com/dp/0521882230/?tag=pfamazon01-20

https://www.amazon.com/dp/069108162X/?tag=pfamazon01-20 (uses very old fashioned notation just like that in MTW but awesome problem sets nonetheless)

https://www.amazon.com/dp/0750627689/?tag=pfamazon01-20

 

[Lavabug]

A few months later after my post in this thread, I have to say I am in full agreement with the above. Wald's problems are not very instructive IMO, there is much more practical GR to be learned from those 4 texts.

For a first encounter though, the "Relativity Demystified" -despite the silly cover and being a Schaum's edition- is a very excellent serious intro to the subject. Better than Carroll's book, notes, and the myriad of lecture notes I've found on introductory courses.

 

[WannabeNewton]

Hey Lavabug if you're interested, I used the following book scrupulously to practice for my GR exam(s) because it has a wealth of end of chapter problems that are in my opinion very instructive in terms of the actual physics of GR and figured it would be of great use to you as well: https://www.amazon.com/dp/0521829518/?tag=pfamazon01-20

You'll notice as you go through the end of chapter problems that a good number of them also appear in Padmanabhan's book.

 

[Lavabug]

Thanks, I'll keep it in my rapidly unmanageable "to check out" list...

I am in the midst of reading/working through Dirac's GR "pamphlet" just for revision before cracking open MTW over the holidays (might finally get around to Padmanabhan maybe...). Largely annoying and not too much fun, but some of the sentences he drops here and there feel like a sledgehammer of insight to the face.

On a somewhat related note, I'm thinking of getting this book:
https://www.amazon.com/dp/0201416255/?tag=pfamazon01-20