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Textbook for General Relativity

  1. Jun 19, 2006 #1
    Hi. I was wondering if you guys can refer me to the best possible text that teaches the computations in tensor analysis. I hate pure math. I just want to get a feel for the mathematics before engaging the physics.

    As far as I know this is a widely referred text, however, it doesn't come with a solution manual.

    http://www.amazon.com/gp/product/00...002-3399608-9341634?s=books&v=glance&n=283155

    Thanks
     
  2. jcsd
  3. Jun 23, 2006 #2
    In those Schaum books, the solutions come with the problems. But I don't learn very well by that.

    I started by reading Einstein's original GR paper, just to see how much tensor stuff he actually used. Then I jumped to the last chapter of Goldstein. Field Theory by landau and lif****z came next.

    I'm sure in retrospect someone else can recommend a better path!
     
  4. Jun 24, 2006 #3
    can einstein's original GR paper be viewed via the internet?
     
  5. Jul 1, 2006 #4
    Can someone give me a comparison between the two texts?

    Tensor Analysis on Manifolds by Bishop and Goldberg, Dover Pub

    Tensors and Manifolds by Wasserman.
     
  6. Jul 3, 2006 #5

    gvk

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    If you have a 'physical' mind, I would recommend the book Dubrovin, Novikov, Fomenko,
    Modern Geometry-Methods and Applications, Part I,II,III (Universitext)
    Springer-Verlag (1990).
    This is very good text with plenty of solved examples.
     
  7. Jul 10, 2006 #6

    mathwonk

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    as i have recommended elsewhere, ideally one should learn coordinate free mathematics to discuss relativity. einstein did not have this available so his papers are not written in that style.

    the best books in my view are those with authors who are themselves experts, which, except for the old style math, of course starts with einstein. (but actually einstein was borrowing from riemann, if you really want to start at the beginning.)

    i would suggest therefore the book by thorne misner? and wheeler.

    but to understand that you need to understand coordinate free calculus on m,anifolds, and befoire that coordinate free lienar algebra.

    so i recommend reading an abstract linear algebra book, like maybe halmos, then an abstract calculus on manifolds book, like spivaks little one, or guillemin and pollack, which borrows from spivak, and finally the book by wheeler et al.


    but this is a suggestion to a bright, young, idealistic person, from an old guy who has never actually doine this himself.

    better is to get a start with some lectures from a knowledgable person. thats why school is useful. and will never be replaced by the internet.
     
  8. Dec 10, 2006 #7
    I have both books, and Wasserman's book is much more thorough. Here are my solutions to Chapters 1 and 2:
     
    Last edited: Jan 22, 2008
  9. Dec 10, 2006 #8
    Wasserman does require more abstract algebra (I have the 1st edition). And your average physics student would probably have to crack open a linear algebra text as well (a "real" math text, not one aimed at engineers.)

    Given that he hates pure math, I suspect Nusc would not like Wasserman, Bishop & Goldberg, or Spivak's dense little book, but the Schaum's may be just right (well, by now he probably can tell us). Or just get a good GR text like Carroll, who is very practical about the math, from the library.
     
    Last edited: Dec 10, 2006
  10. Dec 10, 2006 #9
    I am taking a class in general relativity now and we are using weinbergs gravitation and cosmology.
    I have no other general relativity book to compare it to, but so far its one of the best textbooks I have ever had.
     
  11. Dec 10, 2006 #10

    chroot

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    MTW's Gravitation is a fantastic book, and you probably won't finish it for some 18 years. Wald's General Relativity is a much more modern, but also much more demanding text.

    If you need help with the mechanics of tensors, Schaum's Outline of Tensor Calculus is quite good.

    - Warren
     
  12. Dec 30, 2006 #11
  13. Dec 31, 2006 #12

    Chris Hillman

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    Shouldn't this thread be in the gtr board?

    Hi, Nusc,

    You hate pure math!? How on Earth can you hate pure math but love (I guess) theoretical physics?

    About your question, I am fairly confident that most textbook authors would tell you not to worry about tensor calculus. You can pick up what you need as you go along.

    Since you are apparently worried about understanding the mathematical background more than many students, though, I would probably recommend MTW. Some of the other books would also be good choices for you, I think: Weinberg is certainly a gold mine of information, as is MTW, but the easiest book for you might be Ohanian and Ruffini. Schutz makes a particular effort to introduce the idea of tensors gently, BTW, so you might also want to look at that. Carroll has some excellent material on mathematical background in some appendices which can be read semi-inpendently. All of these are excellent books, although some claim to find MTW or Weinberg daunting (I feel such judgements are mostly wrong). Most people seem to find O&R, Schutz, and Carroll to be very readable. Schutz does have partial solutions to the (excellent) problems, and you can always consider the classic problem book coauthored by Lightman, Press, Price, and Teukolsky.

    There are several other excellent textbooks http://www.math.ucr.edu/home/baez/RelWWW/reading.html#gtr (to that list, add Hartle, which appeared after I compiled it.)

    In general, there is no reason to avoid the most widely used textbooks, but many good reasons to use one of them as your first book. Many of these should be available in inexpensive paperback editions via on-line booksellers.

    Schaum's Outlines books are often quite good, but you really should not need to study such a book before plunging into gtr. Far more essential is that you have a solid grasp of the spacetime interpretation of str (see Spacetime Physics, by Taylor and Wheeler, FIRST edition). It seems especially unwise to subject yourself to studying a book on tensor calculus for its own sake if you truly "hate pure math".
     
    Last edited: Dec 31, 2006
  14. Dec 31, 2006 #13

    mathwonk

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    i wouod recommend petes website on tensors (sear ch for threds on tensors), or ruslan sharipov's book . it seems to be pleasing to people who have tried it for computational facility with tensors. if you then also want conceptual treatments there are many possible sources in pure math books and general relativity books like wheeler, misner, thorne?
     
  15. Jan 1, 2007 #14
    The answer to your question surely depends upon how seriously you want to study GR. If, as you say, you hate pure maths, you're probably going to be better served by reading a book which will give you a good qualitative introduction to the theory. At the most elementary level, Bob Geroch's General Relativity from A to B provides a nice non-mathematical account of the theory. At the next level I would probably choose to use Sean Carroll's book - it doesn't use a great deal of "difficult" mathematics and is certainly a far from rigorous treatment of the subject, but it does discuss a great deal of interesting material. It's certainly what I would choose to start with were I in your position.

    As these threads seem to pop up so often, it might be worth pointing out a couple of books which would suit the other camp, i.e., those who aren't afraid of getting their hands dirty by considering mathematical rigour. In my opinion, MTW, while in many respects an interesting book, is an absolutely horrible text to learn GR from. It has a sort of chatty, informal style which I feel contributes little and often gets in the way of a clear discussion of the material. It is, however, a great book to dip into as bedtime reading once you're already competent in the subject (if your idea of fun bedtime reading involves manipulating a couple of kilograms of paper, that is). If you're interested in learning the mathematical aspects of GR, there are probably no books out there at the moment which deal with the mathematics in an acceptable way alongside the physics. My own advice would be to read, then re-read, then read again, the relevant chapters in Choquet-Bruhat/Dewitt-Morette/Dillard-Bleick's Analysis, Manifolds, and Physics, vol. I (the first author may or may not have been credited as Fouret-Bruhat, depending upon which edition you can lay your hands on). As a companion to this it's probably a good idea to have a more heuristic understanding of topology and Riemannian geometry to the level of, say, Nakahara's book. Only then would you be well-served by moving to Wald (Wald's book, by the way, contains a small but significant number of very serious mistakes, both conceptual and formal, which one needs to watch out for carefully. His appendix on Lagrangian and Hamiltonian field theory, in particular, contains serious errors and, if I recall correctly, his chapter on spinors needs a serious rewrite. Unfortunately none of these mistakes seem to be being corrected through reprints.) Apart from Hawking & Ellis, Wald should provide those of a mathematical bent with all they need to know about GR to start reading the literature and, as far as I'm concerned, there isn't a single other book out there which comes close to these two.
     
    Last edited: Jan 1, 2007
  16. Jan 2, 2007 #15
    I imagine it would be pretty easy for many to come to such a conclusion.

    Pure Mathematics:
    A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.

    Not So Pure Mathematics:
    A subset of a space is compact if it is closed.

    You'd be surprised how many people are put off by the first explanation, even if it is more correct. The key point is that to understand the first explanation, you would have to have first grasped the second completely, along with its drawbacks.
     
  17. Jan 2, 2007 #16

    George Jones

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    On page 350, Wald states

    In other words, Wald says that the universal covering group of the proper Poincare group is (isomorphic to) the semidirect product of SL(2,C) with C^2, when, actually, the universal covering group of the proper Poincare group is (isomorphic to) the semidirect product of SL(2,C) with R^4.

    Greg Egan noticed this a few years ago.

    You might, however, have other, more egregious, examples in mind.
     
  18. Jan 2, 2007 #17

    mathwonk

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    but the second statement in post 15 is false. is that why some people like it better, or is that allowed as a drawback???
     
  19. Jan 2, 2007 #18

    George Jones

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    A bit of elaboration on what mathwonk means.

    I suspect that ObsessiveMathsFreak meant to state what is sometimes (e.g., blue Rudin) called the Heine-Borel theorem: a subset of R^n is compact iff it is closed and bounded.

    As it stands, ObsessiveMathsFreak's second statement doesn't even work for R: [0, infinity) is closed but not compact.
     
  20. Jan 2, 2007 #19
    Yes, that's what I was getting at. For finite domains in R^n, compact means closed. I left out the bounded part.

    Basically, you have to first know that compactness is an extension of closed boundedness. But without knowing about closed boundedness, I can't imagine that many people would appreciate compactness.

    I imagine most people don't like pure mathematics much because it is a subject forever running before it walks. Introducing compactness without ever discussiing closed boundedness. Without referring to the latter, the former is relatively meaningless upon presentation. I myself encountered the pure definition of compactness and was mislead by it for a fairly substantial amount of time. I proved theorems with it without ever knowing what compactness actually was. The proofs were perfectly valid, but devoid of semantics for me.

    Other less "pure" forms of mathematics would start at the begining and might never get to compactness. But at least they would get somewhere someone can appreciate. Mathematics is about concepts as well as definitions. Definitions are just the way we present our concepts, not the concepts themselves. Pure mathematics often forgets that.
     
  21. Jan 2, 2007 #20

    Chris Hillman

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    Huh?

    We are wandering farther and farther off topic, I think, but I don't like to let that pass. Hmm... where to begin...

    First, those aren't "explanations" but, in essence, definitions. As you know, mathematicians call an explanation of why some statement holds true a "proof"! And I guess we might call other types of explanations "context" or "discussion" or even "exposition".

    Why do I say the statements quoted above are in essence definitions?
    Well, the first definitions of "topology" and "compact" were not logically equivalent to the definitions which later became standard, and which really are preferable for many reasons (some general topology textbooks take the time to discuss why this is the case!). There are in fact many competing ways in which one could try to capture intuitive ideas of "nearby point", "continuous function", and so on, and the fathers of topology proved theorems relating some of these to each other. To some extent what concept you choose to call "topology on a set" or "continuous function" or "compact set" without qualification is a matter of convention, but as I say, the standard choices are well motivated. At this point, let me say that the best short introduction to general topology I know is Chapter 4 in the superb textbook by Gerald Folland, Real Analysis, Wiley, 1984.

    Second, as you probably know, the Heine-Borel theorem states that in [itex]\bold{R}^p[/itex] with the standard metric topology, a modification of your second proposed definition ("closed and bounded") is equivalent to the first. This is an example of what I said above, "the fathers of topology proved theorems relating some of these to each other". The added condition "bounded" is crucial since for example [itex]\bold{R}^p[/itex] itself is closed, but not bounded, and it is not compact according any reasonable definition.

    Third, there exist important topologies on important sets which do not behave anything like metric topologies.
     
    Last edited: Jan 2, 2007
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