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A good book in Differential Geometry FOR GR?

  1. May 2, 2007 #1
    My understanding of GR is very coordinate oriented which kind of drags me down when I try to answer more general questions.

    Can somebody recommend a book in Differential Geometry with application to GR?

    Here are my preferences. I don't like hand waving typical for some books 'written for physicists' - I need to see clear logical connection between the concepts defined not 'plausability arguments'. On the other hand, I don't like the hidden logic (the DaVinchy code lol) in many math books that just give you definitions and theorems without explaining the intuitive logic behind the scene. Such books, frankly speaking, do not anticipate the logical process of the reader and apparently do not care. I don't need monographies, 'phone books', 'bibles' or summaries of current research that leave me 'very informed' about stuff I don't actually understand.

    The book I am looking for must start with the most basic conceptual layer of Diff Geometry. It should anticipate that I am a beginner and not assume that I already know what it is supposed to teach me. It should contain lots of diagrams (cause I tend to think in terms of pictures) and lots of examples of application of Diff Geom to real life GR problems. I tend to learn most from examples of actual calculations. Also, the book needs to have exercises with answers or at least hints. The book has to be explicitly oriented towards applications to GR not a general monography in Diff Geom.

    The final goal is to intuitively and rigorously understand and work in practise with stuff like Lie derivatives, one forms, Killing vectors, foliation of spacetime into space and time etc.

    Does that perfect book exist?
  2. jcsd
  3. May 2, 2007 #2


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    Maybe this will appeal to you:
    http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/tc.html [Broken]
    Introduction to Differential Geometry and General Relativity
    Lecture Notes by Stefan Waner,
    Department of Mathematics, Hofstra University

    In addition, try
    A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry
    Peter Szekeres, University of Adelaide

    Other suggestions:
    http://www.math.ucla.edu/~bon/ [Broken] Barrett O'Neill's texts
    http://www.math.harvard.edu/~shlomo/index.html Shlomo Sternberg's texts

    If you do find one, please let the rest of us know.
    Last edited by a moderator: May 2, 2017
  4. May 2, 2007 #3

    Chris Hillman

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    Good books, plural

    Hi, smallphi,

    There are many wonderful theories in the world (especially if you include pure math, not just physics); gtr is just one which is well worth studying simply for the sheer joy of learning something beautiful. Fortunately for those who want to learn this particular theory, students of gtr are almost uniquely blessed with a plethora of superb textbooks; I can think of few other subjects upon which so many experts have bestowed such labors of love.

    Actually, the gtr textbook by Carroll should help most students with similar complaints, but who wish to minimize the effort spent preparing the ground before assaulting the mighty fortress of gtr itself :-/ since it has some mathematical appendices which stand alone as a minitext of the type you seek.

    But that said, for students who prefer a systematic approach to plunging right in, I certainly recommend that before studying gtr one should have exposure to manifold theory, and before studying manifolds one should know some curve theory and surface theory. So my reccommendation for you is Millman & Parker for curve theory and surface theory, then O'Neill for semi-Riemannian geometry with applications to gtr, then one of the dozen or so superb gtr textbooks for a detailed study of gtr. See http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html [Broken] for full bibliographic citations.

    My gosh, smallphi, with desiderata like this, you are setting yourself up for eternal disappointment. I'll assume you are just pouring out some temporary frustration and will be willing to be sensible once you calm down.

    One suggestion is that you shouldn't seek one book. I have studied dozens of gtr and geometry textbooks, and all of them offer uniquely valuable insights. The reading list at "Relativity on the Web" (cited above) is intended to help students choose books most likely to meet their needs and to satisfy their interests.

    Well, I think Carroll's textbook actually does a very good job of this.

    Which is why good students "read actively", i.e. think up problems and work them as they read. If by "real life GR problems" you simply mean "good problems which make you think or teach valuable computational skills", then you might like the book by Poisson, and of course you can try the problem book by Lightman et al. If by "real life GR problems" you mean research problems, then you'll have to start reading classical papers in back issues of Gen. Rel. Grav., Phys. Rev. D, and Class. Quant. Grav.. To find out what the classical papers are you either need a professor guiding you, or you need to proceed iteratively by starting in reading the literature and looking up citations which sound interesting--- after a few years you'll have a sense for who writes good papers.

    The reading list cited above does say whether I think a given textbook has good exercises. (Most of them--- of course, newbies might not fully appreciate that an exercise is valuable until they've mastered enough material.)

    Well, Carroll's gtr textbook addresses all of these issues.

    Ultimately the responsibility for learning is yours. While manifold theory and gtr are certainly a more subtle subjects than might first appear, they are blessed with many superb textbooks. Any good student should be able to master these subjects by studying these books. Any good student will, I think, instinctively appreciate that encountering new perspectives from an author new to you infuses additional joy into the process of learning, so I hope and trust that once you get over your current frustration you will be eager to consult, not just one book, but many.
    Last edited by a moderator: May 2, 2017
  5. May 3, 2007 #4
    How do you think the textbook and the notes by Carroll compare? Is the book a huge improvement?
  6. May 3, 2007 #5

    George Jones

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    One major difference is that the book includes a chapter, Quantum Field Theory in Curved Spacetime (on the Unruh and Hawking effects), that is not included in the notes. In my opinion, this difference alone is worth the price of admission.
  7. May 3, 2007 #6
    So far I'm liking these:

    the lecture notes by Stefan Waner
    http://people.hofstra.edu/faculty/St...f_geom/tc.html [Broken]
    very GR oriented

    "Elements of Diff. Geometry" by Millman and Parker
    written for undergrads (love such books), with lots of diagrams examples and exercises, mostly about curves and surfaces in 3D Euclidean space but I recognize elements I know from GR courses like Christophels, curvature etc., doesn't get into one-forms though

    "A relativistic toolkit" by Eric Poisson
    this is a real workout of diff geometry in GR, amazingly clear style every sentence hits right at target

    I have the Carroll book, usually use it as reference and usually have a lot of unanswered questions after I read a topic from it. At first it 'makes sense' then when your mind starts rolling 'but what if ...' it turns out the sense is not that solid. The O'Neill semi-Riemannian geometry doesn't excite me either - looks like a dry math book with too little diagrams and weak contact with GR.

    I will need introductory Diff. Geom. book to take me higher than Millman and Parker's. I love books for undergrads so let me know if any exists that discusses one forms etc... Is Chris Isham's "Modern Differential Geometry for Physicists" good enough?

    Thanks everybody for the suggestions by the way!
    Last edited by a moderator: May 2, 2017
  8. May 4, 2007 #7

    George Jones

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    This a great book, and soon (maybe Monday), I plan to start going through this book line by line, but it doesn't really seem to fit in with the theme outlined in your original post.

    This is a book that I would like to have on my shelf.

    Another book that I like.

    This is an abstract book that has no exercises, and few examples of real physics. It purpose is to explain the mathematics (i.e., Lie groups and fibre bundles) behind gauge field theory. It doe not talk about GR directly.
  9. May 11, 2007 #8
    try this set of lecture notes (I already posted a link in another thread), maybe you will like it better.
    http://www.theorie.physik.uni-muenchen.de/~serge/T7/ [Broken]

    I would also agree with Chris Hillman's comment about your expectations being too high. The kind of book you are looking for does not exist, and this is to be expected, since every individual person would ideally need an individual book written just for them at the right level. Moreover, when I imagine a book with examples, exercises, but also some calculations shown in full, and also good motivation, intuition, and also rigorous approach and everything - too much work and time would be needed to write such a book at an advanced level, and the result would be 2000 pages long. Misner-Thorne-Wheeler tried to do this, but they had their idiosyncratic style and the result is not quite what you want. But you might want to try MTW as well.

    In any case, I wouldn't expect a single book to do everything for me. A single-book-that-has-everything is possible for beginner's courses in analysis or mechanics, and you have seen them - huge talmuds with lots of pictures and so on. But this is not really possible for more advanced stuff.

    As for figures and diagrams - have you tried to make one? It takes an hour just to produce a single nice-looking diagram with a few vectors and arrows, a sphere, and some lines and shades. No wonder books like O'Neill's Semi-Riemannian geometry have no figures.
    Last edited by a moderator: May 2, 2017
  10. May 11, 2007 #9
    Those lecture notes you suggested look excellent. Exactly what I was looking for. Thanks !
  11. May 11, 2007 #10


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    i enjoyed appendix B, "how not to learn tensor calculus", complete with the traditional coordinate dominated approach.
  12. May 11, 2007 #11
    I felt that this was a bit offensive, - don't you think? Especially considering that most GR books do things this way. For example, Landau&Lifschitz Classical theory of fields.
  13. May 11, 2007 #12
    I posted I didn't like the Barrett O'Neill's Semi-Riemannian geometry. The same author has another book "Elementary Differential Geometry" that deals with one forms, has pictures examples and exercises and seems to fit my bill to take me higher than Millman and Parker.

    My reading list so far goes like this:

    0. "Introduction to Differential Geometry and General Relativity" by Stephan Waner (online)
    1. "Elements of Diff. Geometry" by Millman and Parker
    2. "Elementary Diff. Geometry" by Barrett O'Neill
    3. "Topics in Advanced General Relativity" by Sergei Winitzki (online)
    4. "A Relativist's Toolkit" by Eric Poisson

    The "Modern mathematical physics" by Peter Szekeres is good probably as a summary after one learns the stuff.
    Last edited: May 11, 2007
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