Is there a mathematical book on general relativity that uses exclusively a coordinate free language even in practical computations? 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.
 A: 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!
A: 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).
A: R. Penrose, Structure of space-time (Benjamin, NY, 1968).
A: 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.
A: 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 
A: 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.
https://www.google.com/url?sa=t&source=web&rct=j&url=http://geocalc.clas.asu.edu/pdf/Shape%2520in%2520GC-2012.pdf&ved=2ahUKEwivp570x5bbAhXiMewKHerqB2YQFjAIegQIABAB&usg=AOvVaw2S2zym_MEHOhv7OpFYJOwt
A: 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.
