I am looking for a mathematical precise introductory book on general relativity. Such a reference request has already been posted in the physics stackexchange [here](https://physics.stackexchange.com/questions/363/getting-started-self-studying-general-relativity). However, I'm not sure whether some physicists know what "mathematical precise" really means, that's why im posting it here. Anyway, Wald's book [General Relativity](https://books.google.de/books/about/General_Relativity.html?id=9S-hzg6-moYC&redir_esc=y) seems to have that mathematical rigorosity (I have seen in a preview that he introduces manifolds in a mathematical way), and also O'Neill's [Semi-Riemannian Geometry](https://books.google.de/books/about/Semi_Riemannian_Geometry_With_Applicatio.html?id=CGk1eRSjFIIC&redir_esc=y) seems to be mathematically flavoured as far as I have seen from the contents. However, both are more than 30 years old.


So are there any other more recent books out there. As I said, its language should be mathematically rigorous and modern, it should contain physics (not only a text on the math behind general relativity), and an introduction to semi-Riemannian geometry would not be bad (since it is not as common as Riemannian geometry).


**Edit:** I guess I have found the perfect fit to my question: [An Introduction to Riemannian Geometry (With Applications to Mechanics and Relativity)](https://books.google.de/books?id=oV4qBAAAQBAJ&hl=de&source=gbs_navlinks_s) by Godinho and Natario. However, I did not read it yet.



Modern mathematical books on general relativity
[Curvature in Mathematics and Physics](http://store.doverpublications.com/0486478556.html) (2012), by Shlomo Sternberg, based on an earlier book
*Semi-Riemann Geometry and General Relativity* [free [download](http://www.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf) from the author's website] covers much of the same material as O'Neill but is much more recent.



> 
> This original text for courses in differential geometry is geared
>  toward advanced undergraduate and graduate majors in math and physics.
>  Based on an advanced class taught by a world-renowned mathematician
>  for more than fifty years, the treatment introduces semi-Riemannian
>  geometry and its principal physical application, Einstein's theory of
>  general relativity, using the Cartan exterior calculus as a principal
>  tool. Starting with an introduction to the various curvatures
>  associated to a hypersurface embedded in Euclidean space, the text
>  advances to a brief review of the differential and integral calculus
>  on manifolds. A discussion of the fundamental notions of linear
>  connections and their curvatures follows, along with considerations of
>  Levi-Civita's theorem, bi-invariant metrics on a Lie group, Cartan
>  calculations, Gauss's lemma, and variational formulas. Additional
>  topics include the Hopf-Rinow, Myer's, and Frobenius theorems; special
>  and general relativity; connections on principal and associated
>  bundles; the star operator; superconnections; semi-Riemannian
>  submersions; and Petrov types. Prerequisites include linear algebra
>  and advanced calculus, preferably in the language of differential
>  forms.
> 
> 
> 


