I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free. The idea is to use only a clean abstract purely geometrical language without encoding operations with indices or matrices of coordinates.
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4$\begingroup$ For a physicist's perspective on why coordinate-free language is inherently not practical, see here. $\endgroup$– knzhouCommented May 20, 2018 at 12:39
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6$\begingroup$ What do you have against abstract index notation, which is a coordinate-free notation and superior to "mathematician" notation in conciseness and expressiveness? $\endgroup$– user21349Commented May 21, 2018 at 2:35
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17$\begingroup$ It seems worth emphasizing that the abstract index notation is coordinate free. The notation is well explained in the book by Penrose and Rindler, but the basic idea is that indices are labels that serve to indicate valency, covariance/contravariance, ordering, and symmetries. No choice of coordinates or local frame is necessary. So $X^{i}$ indicates a vector field, while $\alpha_{i}$ indicates a one-form. This is no different than writing simply $X$ or $\alpha$; the indices are simply decorations that indicate the nature of the tensorial object. $\endgroup$– Dan FoxCommented May 21, 2018 at 10:53
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4$\begingroup$ @AlexM.: for the first problem usually one introduces additional notation. Common variants include the use of latin indices $a, b, c,\ldots$ for abstract ones and greek indices $\mu,\nu, \ldots$ for concrete indices, or the use of $a, b, c, \ldots$ for abstract indices and adorned versions (like $(a), (b), (c), \ldots$ for concrete ones). // For the second problem:I agree the notation may become quite cluttered for tensors of high order, but is there an alternative that is not cluttered for high order tensors? $\endgroup$– Willie WongCommented May 21, 2018 at 16:04
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4$\begingroup$ @knzhou The article you linked here physics.stackexchange.com/questions/15002/… is extremely interesting. Now I understand why I never liked the parenthesis and the notation in the coordinate-free language. The abstract index notation is more suitable to represent tensor contraction, that is intrinsically a graph and not a sentence with arbitrary annoying parenthesis. At least that's what I understand. $\endgroup$– Bruno MartelliCommented May 30, 2018 at 12:10
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7 Answers
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Try "General Relativity for mathematicians" by R. Sachs and H. Wu. Also, "Gravitation" by C.W. Misner, K.S. Thorne, J.A. Wheeler - it's so famous that it's got its own Wikipedia page. Finally, "The large scale structure of space-time" by S.W. Hawking and S.F.R. Ellis - another "star" with a Wikipedia page. All of them were published in the '70s, so they might not be up to date with the experimental part (that you are probably not interested in, given your question) or with black hole cosmology. But once you're proficient in the subject, you'll be able to find your way further by yourself.
Also, notice that a coordinate-free approach is an extremist dream, that I warmly invite you to get rid of as soon as possible (I've been there too, but now I'm cured). If you're unhappy with the above books, I'm afraid that they are as coordinate-free as it gets. Even Riemannian geometry is often done with a mature mix of coordinate and invariant methods. Good luck defining volume forms without coordinates!
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2$\begingroup$ My personal experience with Hawking--Ellis was in the opposite direction of what the OP asks for. Sachs-Wu a bit better. $\endgroup$ Commented May 20, 2018 at 6:12
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1$\begingroup$ The extremist dream part is a bit funny , but I don't quite consider it to be a fools errand. I'm not just asking this question for an exercise of math overformality, I see many practical benefits of such an approach too. Check my answer below , also I found a very interesting article by David Hestenes on mathematical viruses("coordinitis", he also has a lot of articles on algrebraic coordinate free approaches) but you do make a good point also pointed out in Misner's book that a mature blend of coordinate and intrinsic abstract methods might be optimal. $\endgroup$– LeoCommented May 21, 2018 at 12:02
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3$\begingroup$ don't see a problem defining volume forms thought $\endgroup$– LeoCommented May 21, 2018 at 12:03
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4$\begingroup$ I disagree with the "extremist dream" reproach. I am not against using the coordinates but, I think, it is always possible and always helpful to translate a coordinate-bound definition into coordinate-free language. The $n$-th exterior power of the metric tensor is the square of an $n$-form, well-defined up to the sign. This form is the volume form, the choice of the sign is the orientation. $\endgroup$ Commented May 22, 2018 at 6:52
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$\begingroup$ That's what I meant , it's no problem to define a volume form naturally without coordinates, you can use coordinates if you like, when its appropriate to do so. $\endgroup$– LeoCommented May 22, 2018 at 11:31
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You might want to check out the classic paper by Tullio Regge: General Relativity without Coordinates (it is discussed in the Misner/Thorpe/Wheeler phonebook, but it is usually better to go to the source).
answered May 19, 2018 at 21:12
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2$\begingroup$ Maybe not exactly what the OP wants: Regge introduces some sort of "triangulation" of the space-time, which is now used Regger calculus and which some doing quantum gravity find useful. Definitely not an expository text about general relativity. $\endgroup$– Alex M.Commented May 19, 2018 at 21:46
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$\begingroup$ @AlexM. I disagree. The article is very accessible. $\endgroup$ Commented May 20, 2018 at 1:06
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R. Penrose, Structure of space-time (Benjamin, NY, 1968).
answered May 19, 2018 at 21:30
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3$\begingroup$ A Google search does not return anything about this title, are you sure it exists? The closest thing is en.wikipedia.org/wiki/The_Nature_of_Space_and_Time. Anyway, If I remember correctly, Penrose is a great proponent of the abstract index notation (Weinberg's "Gravitation and Cosmology" uses it too), and the OP clearly wants to avoid it. $\endgroup$– Alex M.Commented May 19, 2018 at 21:43
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$\begingroup$ @Alex M. I corrected the title. $\endgroup$ Commented May 20, 2018 at 5:28
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$\begingroup$ @AlexM. Abstract index notation is not the same as working in local coordinates, even though it might look like it. It's coordinate independent, and nicer than any other notation I have seen for working with tensors. $\endgroup$ Commented May 20, 2018 at 20:14
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2$\begingroup$ @MichaelBächtold: I know, but the OP said "no abstract indices". $\endgroup$– Alex M.Commented May 20, 2018 at 22:21
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$\begingroup$ @AlexM. oh, I didn’t see that. $\endgroup$ Commented May 21, 2018 at 7:58
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I am wondering why nobody mentions the book of Barrett O'neill "Semi-Riemannian Geometry With Applications to Relativity". I think this is the closest you can get into a coordinate free introduction to general relativity.
answered Jun 26, 2021 at 9:19
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Although it does not focuses too much on special/general relativity, and is sometimes sloppy, after years of looking for something I like Tensor Geometry: The Geometric Viewpoint and its Uses by Dodson, Christopher T. J., Poston, Timothy
answered May 20, 2018 at 6:17
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After a lot of searching I came across Advanced general relativity lecture notes by Sergei Winitzki and an intro to differential geometry and curvature by Hestenes ,link below,. Misner's Gravitation is likely the best relativity book but it's only partly coordinate and index free.
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$\begingroup$ That link should be geocalc.clas.asu.edu/pdf/Shape%2520in%2520GC-2012.pdf, but it's broken $\endgroup$– David Roberts ♦Commented Jan 20, 2022 at 11:28
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I have written a book called A Mathematical Introduction to General Relativity,
which is meant for advanced undergraduate/graduate level mathematics students. The only physics background needed is familiarity with Newton's laws of motion and the Newtonian gravitational law. PDF files of the preface and the first chapter can be found on the publisher's website above, and a Google preview can be found here. It introduces (starting from scratch) and uses the coordinate-free language of differential geometry.
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$\begingroup$ I was looking at this thread and noticed you here; I bought your book a few months ago. I haven't started it yet but it looks VERY nice. Thank you! $\endgroup$ Commented Dec 10, 2022 at 3:59
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1$\begingroup$ Thank you. It is the book I wanted to read while I was learning the subject. $\endgroup$ Commented Dec 11, 2022 at 10:40