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.