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A good book in Differential Geometry FOR GR? 
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#1
May207, 09:48 AM

P: 443

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
May207, 10:13 AM

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PF Gold
P: 4,120

Maybe this will appeal to you:
http://people.hofstra.edu/faculty/St...f_geom/tc.html Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, Department of Mathematics, Hofstra University In addition, try http://www.cambridge.org/uk/catalogu...=9780521829601 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/ 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. 


#3
May207, 10:33 AM

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P: 2,340

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. 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 semiRiemannian 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 for full bibliographic citations. 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. 


#4
May307, 07:38 AM

P: 191

A good book in Differential Geometry FOR GR?
How do you think the textbook and the notes by Carroll compare? Is the book a huge improvement?



#5
May307, 08:18 AM

Mentor
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#6
May307, 09:48 PM

P: 443

So far I'm liking these:
the lecture notes by Stefan Waner http://people.hofstra.edu/faculty/St...f_geom/tc.html 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 oneforms 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 semiRiemannian 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! 


#7
May407, 08:04 AM

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P: 6,232

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. 


#8
May1107, 06:54 AM

P: 28

smallphi
try this set of lecture notes (I already posted a link in another thread), maybe you will like it better. http://www.theorie.physik.unimuenchen.de/~serge/T7/ 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. MisnerThorneWheeler 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 singlebookthathaseverything 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 nicelooking diagram with a few vectors and arrows, a sphere, and some lines and shades. No wonder books like O'Neill's SemiRiemannian geometry have no figures. 


#9
May1107, 09:01 AM

P: 443

Those lecture notes you suggested look excellent. Exactly what I was looking for. Thanks !



#10
May1107, 11:17 AM

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P: 9,453

i enjoyed appendix B, "how not to learn tensor calculus", complete with the traditional coordinate dominated approach.



#11
May1107, 12:16 PM

P: 28




#12
May1107, 02:43 PM

P: 443

I posted I didn't like the Barrett O'Neill's SemiRiemannian 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. 


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