Gathering books on Lorentzian Geometry

Question

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm talking about the Lorentzian manifolds and Lorentz-Minkowski spaces (some notations of it are $\Bbb L^n$, $\Bbb E^n_1$, etc). I know that the subject is recent (about $15$ years or so?), so we might not have a lot of texts about it anyway, but it costs nothing to try.

I am not talking about isolated papers and articles, but of texts which make a systematic approach of the subject.

I thought of making a list here, so we can gather some material, the most we can.

I'll make a CW answer, and I invite everyone who knows something about it to give their two cents.

Thanks.

Answer 1

List of books and texts on Lorentzian Geometry:

• Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space (also 1st arXiv version) - Rafael López;

• Semi-Riemannian Geometry With Applications to Relativity - Barret O'Neill; (does a bit about the specific subject asked)

• An Introduction to Lorentz Surfaces - Tilla Weinstein;

• Curves and Surfaces in Minkowski Space - Johan Walrave; (PhD thesis, hard to find)

• Linear Algebra and Geometry - Igor R. Shafarevich, Alexei O. Remizov; (from pages $265$ to $288$)

• An Introduction to Lorentzian Geometry and its Applications - Miguel Javaloyes, Miguel Sánches.

Answer 2

Also consider books on general relativity since the theory models spacetime as a 4-dimensional Lorentz manifold. Hence, these books often develop the theory of Lorentz manifolds systematically.

Answer 3

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).

Answer 4

Roger Penrose has a short book: Techniques of differential topology in relativity (1972)