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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.

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    $\begingroup$ The correct answer to this would depend a great deal on what type of physics you are interested in. If you really just want the math, then read the old books (the ones you've mentioned plus Beem, Erlich and Easley, the Penrose monograph and, Hawking and Ellis). The mathematics hasn't changed that much and what is missing is all on the arxiv. The modern books tend to come in specific "flavours", e.g. string theory, cosmology, black hole thermodynamics, numerical relativity. Are you looking for a specific "Flavour"? $\endgroup$
    – Ben Whale
    Commented Sep 7, 2015 at 7:41
  • $\begingroup$ Just an introduction/the basics on the physics side, but as modern as possible on the math side. $\endgroup$
    – Werner Thumann
    Commented Sep 7, 2015 at 8:21
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    $\begingroup$ Why not just stick with Wald? The basic ideas have not changed very much since its publication, and his presentation is very clean. $\endgroup$
    – user1504
    Commented Sep 18, 2015 at 19:51
  • $\begingroup$ @Werner, a comment on your latest edit: by all means, chapter 6 in the textbook you indicate might serve you well as a first brief introduction to some important concepts, and it does look mathematically rigorous. I would simply warn you against regarding the book as a "modern mathematical book on general relativity", as it seems very limited in scope. $\endgroup$
    – Umberto Lupo
    Commented Sep 20, 2015 at 17:29

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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.

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    $\begingroup$ +1 for a link to Shlomo Sternberg web page. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 6, 2015 at 14:30
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    $\begingroup$ Sternberg's book (earlier version) is interesting, but it's not an intro to GR as requested by the OP. It contains a lot of material on differential geometry which, while interesting, is often only tangentially related to GR. For example, the first chapter is a cute treatment of intrinsic and extrinsic curvature using thickening of surfaces, but it's limited to the Euclidean case (e.g., he assumes that unit normals exist). There is no discussion of important topics in GR such as the equivalence principle or gravitational waves. Nothing wrong with the book, but it's not an intro to GR. $\endgroup$
    – user21349
    Commented Sep 7, 2015 at 18:40
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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

Lecture notes

Review papers

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    $\begingroup$ Piotr Chruściel recently published two books, Elements of General Relativity (2020) and Geometry of Black Holes (2020), which AFAIK built upon and extend his lecture notes and might also serve as an introduction. $\endgroup$
    – balu
    Commented Mar 2, 2021 at 23:07
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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.

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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.

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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.

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  • $\begingroup$ Straumann's book looks interesting. I will have a closer look at it. I accepted the other answer because it is also legit and has more up-votes. $\endgroup$
    – Werner Thumann
    Commented Sep 11, 2015 at 7:11
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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.

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Maybe this one: A Mathematical Introduction to General Relativity

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R.K. Sachs and H. Wu, General Relativity for Mathematicians is a valuable source.

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