Modern mathematical books on general relativity 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. 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 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 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) by Godinho and Natario. However, I did not read it yet.
 A: Here is a selection of some other sources which seem not have been mentioned yet. I will include some lecture notes and review papers which seem to me to be either comparable in breadth and precision to a textbook, or worth knowing about due to the inclusion of very recent results.
Textbooks


*

*Any textbook on the topic written by Yvonne Choquet-Bruhat, the "founding mother" of the rigorous Cauchy problem in General Relativity: particularly (and in increasing order of "difficulty") her "Introduction to General Relativity, Black Holes, and Cosmology" (OUP, 2015) and "General Relativity and the Einstein Equations" (OUP, 2008)

*John Stewart's "Advanced General Relativity" (CUP, 1991)


Lecture notes


*

*"The Geometry of Black Holes" (2015) by Piotr T. Chruściel -- very up-to-date and quite rigorous, with lots of introductory material in Part II

*"Lectures on Mathematical Relativity" (2008) by by Piotr T. Chruściel
Review papers


*

*"Mathematical General Relativity: A Sampler" (2010) by Chruściel, Galloway and Pollack

A: One reference which is fairly recent is

Hans Ringström, The Cauchy Problem in General Relativity (2009).

True, not really a physics reference, but aimed at both physicists and mathematicians. It focusses on the formulation of the Einstein equations as initial value problem and includes introductions to PDE and Lorentzian geometry as well as a chapter on (some) spatially homogeneous models. Check out the errata on the author's web page for the corrected proof of existence of a maximal globally hyperbolic development. 
I have heard from several mathematicians in GR now that they use this as introductory book for there PhD students.
A: It's a while ago but I used to study the books:

An Introduction to General Relativity, Hughston and Tod (1990)

and

General Relativity With Applications to Astrophysics , Straumann (2004)

I remember both books to be  "mathematical precise" and contain enough physics to connect it with our physicist general relativity lecture, at least from my point of view. However the first book is more written like a math book than the second one.
Both books do not treat semi Riemannian Geometry, however.
A: You may be interested in Winitzki, Topics in Advanced General Relativity, which is free online. It's recent and mathematically rigorous. It uses index-free notation. I think you would need some preparation before tackling it.
As you noted in the question, Wald is extremely out of date. But what has changed a lot in GR since 1984 is not the mathematical foundations. What's changed is (1) observational data, and (2) theoretical developments on topics that are at a much higher level than an introductory book. What I've been recommending to people who want a more recent alternative to Wald is Carroll, Spacetime and Geometry: An Introduction to General Relativity. There is a free online version. However, I haven't looked carefully to see how Carroll compares with Wald in level of mathematical precision.
Wald has a basic introduction to global methods, and it would prepare you well to move on to Hawking and Ellis, The Large Scale Structure of Space-Time, which is the standard book on that topic.
A: Maybe this one: A Mathematical Introduction to General Relativity
A: I would recommend An Introduction to Mathematical Relativity (2021) by Natário. It also contains many suggestions for further reading.
A free version of it is available on arXiv.
A: Curvature in Mathematics and Physics (2012), by Shlomo Sternberg, based on an earlier book
Semi-Riemann Geometry and General Relativity [free download 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.

