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.

2$\begingroup$ @SergeiAkbarov: Despite their name, Einstein manifolds are Riemannian manifolds, therefore not what the OP is after. $\endgroup$ – Alex M. May 19 '18 at 19:46

2$\begingroup$ For a physicist's perspective on why coordinatefree language is inherently not practical, see here. $\endgroup$ – knzhou May 20 '18 at 12:39

3$\begingroup$ What do you have against abstract index notation, which is a coordinatefree notation and superior to "mathematician" notation in conciseness and expressiveness? $\endgroup$ – Ben Crowell May 21 '18 at 2:35

9$\begingroup$ 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 oneform. 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 Fox May 21 '18 at 10:53

2$\begingroup$ @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 Wong May 21 '18 at 16:04
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 spacetime" 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 (thet you are probably not interested in, given your question) or with the backhole cosmology. But once you're proficient in the subject, you'll be able to find your way further by yourself.
Also, notice that a coordinatefree 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 coordinatefree 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!

2$\begingroup$ My personal experience with HawkingEllis was in the opposite direction of what the OP asks for. SachsWu a bit better. $\endgroup$ – Filippo Alberto Edoardo May 20 '18 at 6:12

$\begingroup$ 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. $\endgroup$ – Leo May 21 '18 at 12:02

1$\begingroup$ don't see a problem defining volume forms thought $\endgroup$ – Leo May 21 '18 at 12:03

1$\begingroup$ 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 coordinatebound definition into coordinatefree language. The $n$th exterior power of the metric tensor is the square of an $n$form, welldefined up to the sign. This form is the volume form, the choice of the sign is the orientation. $\endgroup$ – Ivan Izmestiev May 22 '18 at 6:52

$\begingroup$ 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. $\endgroup$ – Leo May 22 '18 at 11:31
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).

1$\begingroup$ Maybe not exactly what the OP wants: Regge introduces some sort of "triangulation" of the spacetime, 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. May 19 '18 at 21:46

$\begingroup$ @AlexM. I disagree. The article is very accessible. $\endgroup$ – Igor Rivin May 20 '18 at 1:06

R. Penrose, Structure of spacetime (Benjamin, NY, 1968).

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. May 19 '18 at 21:43


$\begingroup$ @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. $\endgroup$ – Michael Bächtold May 20 '18 at 20:14

1$\begingroup$ @MichaelBächtold: I know, but the OP said "no abstract indices". $\endgroup$ – Alex M. May 20 '18 at 22:21

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
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.