Advanced General Relativity Pdf

[Eva Dunckel]

Iam 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.

Curvature in Mathematics and Physics (2012), by Shlomo Sternberg, based on an earlier bookSemi-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.

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.

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.

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.

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.

General relativity is the theory of spacetime and gravity developed by Einstein. This module aims to develop a geometric understanding of general relativity and explore advanced applications of GR. In particular, we will study the physics of black holes (regions of spacetime which are causally disconnected from the rest of the Universe), uncovering their surprising properties.

General relativity is a theory of gravity that explains how massive objects interact with each other in the universe. It is important for physicists to study because it has been proven to accurately describe the behavior of large objects and has led to important discoveries such as the bending of light around massive objects and the existence of black holes.

While it is not necessary for all physicists to take a course in general relativity, it is highly recommended for those who are interested in studying the behavior of massive objects in the universe. It is also important for those who want to work in fields such as astrophysics, cosmology, and theoretical physics.

Yes, physicists can specialize in other areas without studying general relativity. There are many other subfields of physics, such as particle physics, condensed matter physics, and biophysics, that do not require an in-depth knowledge of general relativity.

General relativity is a complex theory and can be challenging to understand, but with proper instruction and dedication, it can be learned by physicists at any level. It requires a strong foundation in mathematics and physics, so it may be more difficult for those without a background in these subjects.

Yes, there are several benefits to taking a course in general relativity, even if it is not directly related to a physicist's field of study. Studying general relativity can improve problem-solving skills, critical thinking abilities, and mathematical proficiency. It can also provide a deeper understanding of the fundamental principles of physics and how they apply to the universe as a whole.

Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

This is a rigorous and encyclopedic treatment of special relativity. It contains pretty much everything you will ever need in special relativity, like the Lorentz factor for a rotating, accelerating observer. It is not an introduction, the author does not bother to motivate the Minkowski metric structure at all.

This is one of the best physics books ever written. This can be comfortably read by anyone who knows $F=ma$, vector calculus and some linear algebra. Zee even completely develops the Lagrangian formalism from scratch. The math is not rigorous, Zee focuses on intuition. If you can't handle a book talking about Riemannian geometry without the tangent bundle, or even charts, this isn't for you. It's rather large, but manages to go from $F=ma$ to Kaluza-Klein and Randall-Sundrum by the end. Zee frequently comments on the history or philosophy of physics, and his comments are always welcome. The only weakness is that the coverage of gravitational waves is simply bad. Other than that, simply fantastic. (Less advanced than Carroll.)

The classic book on spacetime topology and structure. The chapter on geometry is really meant as a reference, not everything is given a proper proof. They present GR axiomatically, this is not the place to learn the basics of the theory. This text greatly expands upon chapters 8 through 12 in Wald, and Wald constantly references this in those chapters. Hence, read after Wald. For mathematicians interested in general relativity, this is a major resource.

This is a proof graveyard. Some of the proofs here are not found anywhere else. If you're willing to skip 70 pages of pure math and take the results on faith, skip this. It overlaps with Hawking & Ellis a lot.

This is really a toolkit, you're assumed to know basic GR coming in, but will leave with an idea of how to do some of the more complicated computations in GR. Includes a very good introduction to the Hamiltonian formalism in GR (ADM).

This is an extremely rigorous text on GR for mathematicians. If you don't know what "let $M$ be a paracompact Hausdorff manifold" means, this isn't for you. They do not explain geometry (Riemannian or otherwise) or topology for you. Put aside the strange notation and (sometimes stupid) comments on physics vs. mathematics and you have a solid text on the mathematical foundations of GR. It would be most helpful to learn GR from a physicist before reading this.

A mathematically sophisticated text, thought not as much as Sachs & Wu. The coverage of differential geometry is rather encyclopedic, it's hard to learn it for the first time from here. If you're a mathematician looking for a first GR book, this could be it. Besides the overall "mathematical" presentation, notable features are a discussion of the Lovelock theorem, gravitational lensing, compact objects, post-Newtonian methods, Israel's theorem, derivation of the Kerr metric, black hole thermodynamics and a proof of the positive mass theorem.

The standard graduate level introduction to general relativity. Personally, I'm not a fan of the first four chapters, the reader is much better off reading Wald with a basic understanding of GR and geometry. However, the rest of the text is excellent. If you can only read one text in the "advanced" list, it should be Wald. Some topology would be good, the appendix on it is not very extensive.

3a8082e126