Carroll Lecture Notes On General Relativity

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Casio Bauman

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Aug 3, 2024, 5:59:33 PM8/3/24

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General relativity is one of the towering achievements of modernphysics, the best answer we currently have to the question 'what isgravity?'. Here is where you get to grips with it - the maths can be abit gory, but by the end you should understand the Einstein equations!These enable us to describe how mass/energy curves spacetime, whichgives rise to the effects we used to call 'gravity'! After eachlecture I will link material to this page - so keep checking to seewhat is here (and hit reload to make sure you are getting the most upto date version!). These lecture notes are NOT a substitute forattending the lectures. But do look at them because I sometimes edit themAFTER the lecture, so I emphasise and try to find other ways ofexplaining any points which were obviously an issue in the lecture.

There is a highly recommended web sit of Sean Carroll's lecture notes on general relativity.I especially like hisNo-Nonsense Introduction to General Relativity. Only thing towatch is that he uses the opposite sign convention on his metric!His links are worth checking out as well.A very different approach (much more along the pure mathematics,differential geometry line) is anIntroduction to Differential Geometry and General Relativity.But its got some good pictures in it. And an excellent essay onfundamental meaning of GR (and quantum mechanics) I once did an experimental DU astrosoc talk onBlack holes - this was midway between a lecture and my usual'edutainment' approach to public talks.

There are also some fun relativity pages on the web
Popular science (non technical sites) include spacetime wrinkles.There are also some good visualisation sites like falling into ablack hole and a make your own orbits around a black hole (java appletsite).

These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology.

This is a course on general relativity, given to Part III (i.e. masters level) students. It covers advanced material, but is designed to be understandable for students who haven't had a first course in the subject. Please do email me if you find any typos or mistakes.

2. Introducing Differential Geometry: PDF
Manifolds: Topological spaces, differentiable manifolds and maps between manifolds. Tangent Spaces: tangent vectors, vector fields, integral curves and the Lie derivative. Tensors, covectors and one-forms. Differential Forms: the exterior derivative, de Rahm cohomology, integration and Stokes' theorem.
3. Introducing Riemannian Geometry: PDF
The metric; Riemannian and Lorentzian manifolds, the volume form and the Hodge dual. The Maxwell action. Hodge theory. Connections and the covariant derivative, curvature and torsion, the Levi-Civita connection. The divergence theorem. Parallel transport, normal coordinates and the exponential map, holonomy, geodesic deviation. The Ricci tensor and Einstein tensor. Connection 1-forms and curvature 2-forms.
4. The Einstein Equations: PDF
The Einstein-Hilbert action, the cosmological constant; diffeomorphisms and the Bianchi identity; Minkowski, de Sitter and anti-de Sitter spacetimes; Symmetries and isometries, Killing vectors, conserved quantities; Asymptotics of spacetime, conformal transformations and Penrose diagrams; Coupling matter, the energy-momentum tensor, perfect fluids, spinors, energy conditions; Cosmology.
5. When Gravity is Weak: PDF
The Linearised theory, gauge symmetry, the Newtonian limit; Gravitational waves, de Donder gauge, transverse traceless gauge, LIGO; Gravitational wave production, binary systems, the quadrupole formula, gravitational wave sources.
6. Black Holes: PDF
The Schwarzschild solution, Birkhoff's theorem, Eddington-Finkelstein Coordinates, Kruskal diagrams and Penrose diagrams, weak cosmic censorship; The Reissner-Nordstrom solution, Cauchy horizons and strong cosmic censorship, Extremal black holes; The Kerr solution, global structure, the ergoregion, the Penrose process and superradiance, no hair theorems.

Problem SheetsJoo Melo has put together a preparatory worksheet, based on Chapter 1 of the lectures notes, to help refresh your understanding of geodesics before the course begins. It can be downloaded here.

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.

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.

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.)