Since I want to provide a few comments to my suggested references, I'll answer separately. One is free to add the references below to the CW answer above, of course.
A general modern reference which is is complementary to O'Neill's book cited in the CW answer is
- J. K. Beem, P. E. Ehrlich, K. L. Easley, "Global Lorentzian Geometry" (2nd. ed., CRC Press, 1996);
Among other things, it intends to be a Lorentzian counterpart of the landmark book by J. Cheeger and D. G. Ebin, "Comparison Theorems in Riemannian Geometry", which was the first book on modern global methods in Riemannian geometry. Unfortunately, it lacks important developments in Lorentzian geometry which came after 1996, such as the smooth version of Geroch's splitting theorem, settled by Bernal and Sánchez, and also later by Chrusciel, Grant and Minguzzi, the clean-up of concepts and results such as the causal ladder and the causal boundary of space-times, and the area theorem for black holes by several authors, just to quote a few. For those, a newer book on global Lorentzian geometry is long overdue.
Texts on more restricted examples of Lorentzian geometries are
- S. Chandrasekhar, "The Mathematical Theory of Black Holes" (Oxford, 1983);
- G. L. Naber, "The Geometry of Minkowski Space-Time" (2nd. ed., Springer, 2012);
- B. O'Neill, "The Geometry of Kerr Black Holes" (Dover, 2014);
For the Minkowski space-time geometry, the book of Naber is quite comprehensive, including a complete proof of the fundamental Zeeman theorem which characterizes causality-preserving transformations in this space-time.
Mathematical texts on general relativity (which can be safely read by mathematicians while learning some of the involved physics along the way):
- Y. Choquet-Bruhat, "General Relativity and the Einstein Equations" (Oxford, 2009);
- S. W. Hawking, G. F. R. Ellis, "The Large Scale Structure of Space-Time" (Cambridge, 1973);
- M. Kriele, "Spacetime - Foundations of General Relativity and Differential Geometry" (Springer, 2001);
- R. M. Wald, "General Relativity" (Chicago University Press, 1984);
Texts focusing on the theory of the Einstein equations (a quite complicated quasi-linear hyperbolic system with constraints) are actually a mixture of Lorentzian geometry and PDE theory, but since this is quite a central topic and linked to important open problems in Lorentzian geometry, here they go:
- The book of Choquet-Bruhat cited above;
- D. Christodoulou, S. Klainerman, "The Global Nonlinear Stability of Minkowski Space" (Princeton, 1993);
- S. Klainerman, F. Nicolò, "The Evolution Problem in General Relativity" (Birkhäuser, 2003);
- D. Christodoulou, "The Formation of Black Holes in General Relativity" (European Mathematical Society, 2008);
- A. D. Rendall, "Partial Differential Equations in General Relativity" (Oxford, 2008).