Spacetime : foundations of general relativity and differential geometry

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One of the most concise introductions available. Covers much less than Stephani or d'Inverno , but clear and well written.

Advanced undergraduate to beginning gradate level. The textbook of choice for the discerning graduate student. Well written, with a good selection of topics, including careful discussions of tensor formalism, the basic singularity, stability, and uniqueness theorems, as well as black hole thermodynamics. But I think every serious student must own this at least as a supplementary text and dip into it on a regular basis. MTW was the first "modern" GR textbook, and has inspired two generations of students. While in many respects it is now rather out of date, and in a few places is pretty darn confusing, this beautifully illustrated book features fascinating insights found nowhere else on almost every one of its odd pages.

All of these books have exercises; DINV is particular well suited for self study since it also has solutions in the back. And I'd recommend MTW to anyone, anywhere, any time. For the convenience of the rank beginner who wants to purchase one or more of these textbooks, here is a very rough guide to the coverage: all of these books introduce tensors, including the matter and Riemann and Ricci tensors. All discuss geodesics, connections and covariant derivatives.

All discuss the Equivalence Principle, weak field theory, and at least one interpretation of the field equations. All discuss the classic predictions such as light bending, perihelion advance, gravitational redshift. Among the exact solutions, all discuss in some detail the "usual suspects" Schwarzschild vacuum and Friedmann dust. All discuss the linearized theory of gravitational waves and Cartan's method of curvature forms. Five of the six textbooks also discuss at length various of the following important topics: spinors, algebraic symmetries of tensors, the variational principle formulation of GR, the initial value formulation of GR, the Petrov classification of curvature types, EXACT gravitational wave solutions, the singularity theorems, Penrose diagrams conformal compactification , Hawking radiation, and thermodynamics of black holes.

Among exact solutions beyond "the usual suspects", DINV features detailed discussions of the Kerr-Newman vacuum, Reissner-Nordstrom electrovac, Tolman fluid, de Sitter and anti-de Sitter cosmological solutions. HT also features a particularly clear and concise treatment of the Bianchi classification of homogeneous spacelike hyperslices. While I think the six books listed above are among the best currently available textbooks, there are several others worthy of special mention. Since the only way to learn a mathematical theory is by doing problems, the more the merrier, this book is an invaluable resource for serious students.

Presents the bare essentials geodesics, curvature, the field equation, "the usual suspects" in a concise and accessible manner. But it uses coordinate notation exclusively, and thus cannot be considered a "modern" introduction despite the date of publication , but it can be good place to learn the essential! Yes, that Dirac.

Spacetime : Foundations of General Relativity and Differential Geometry -

In his inimitable, incredibly concise style, Dirac offers a sixty page sketch of GR, with all the math but not a single picture. First published in , this book doesn't cover any of the modern developments in the subject. If you are very impatient and have a very strong background in advanced calculus and some differential geometry, this just might be the right book for you.

Otherwise it will sail right over your head. No exercises. Most GR books follow more or less in Einstein's footsteps in motivating the field equation. These authors take a different approach which has become increasingly important in recent years; they motivate the linearized field equation by a careful formal analogy with Maxwell's theory of electromagnetism, and then argue their way to the full field equation. Strong on the important formal analogies with EM, but weak on geometry. It also has one of the best treatments to be found among introductory GR texts of the experimental and observational consequences of the theory, along with a nice discussion of newtonian gravity.

I don't know this book but I've seen it somewhere; as I recall it looked somewhat forbidding. A concise and readable introduction, emphasizing modern coordinate free notation. Has some good exercises. Features a particularly comprehensive introduction to the geometric meaning of the field equation, and a detailed introduction to relativistic optics. Features a discussion of LeMaitre coordinates for the Schwarzschild solution and some other things not found in many other books. Some good exercises. Dover has also reprinted books on relativity by the youthful Wolfgang Pauli, the mature Max Born, Peter Bergman, and Richard Tolman, which I feel are of marginal utility today, since they are very out of date and the topics they do discuss are IMO better explained in more modern language elsewhere.

I would strongly recommend that students spend their money on more expensive but up-to-date textbooks. I'll begin with several books in the Schaum's outline series, which, if read with discipline, can actually be a very effective way, I think, to learn some problem-solving skills. If you really are starting with linear algebra, however, you should expect to spend many months in hard labor working through these books before you are ready to being your study of GR. I am not familiar with all of the following books, but consider the one I own the last to be a good book.

A good introduction to classical differential geometry. Note well; for GR you need more advanced notions, including modern notions of manifolds, covariant, Lie, and exterior derivatives, connections, and curvature tensors. An introduction to coordinate basis tensor computations, including the metric tensor, geodesics, the Riemann tensor, with applications to classical mechanics and SR but not GR.

The Geometry of Causality - Space Time

This won't entirely get you up to speed for GR, but like the previous book it may be useful as a supplementary text. This book can probably serve as a substitute for all of the Schaum's books mentioned above save the last two , with the additional bonus of introducing exterior forms early on and properly emphasizing the fact that these objects are natural, easy to understand, and easy to compute with.

General Relativity’s Influence and Mysteries

This book is simply gorgeous. It offers a thorough and beautifully illustrated introduction to everything from riemannian geometry, Cartan geometry, symplectic geometry, differential topology and Morse Theory to vector bundles and Pontryagin and Chern classes. Speaking of manifolds and differential geometry, I think that one of the best all around introductions is:. One book which is particularly well suited for background reading in GR is the outrageously expensive.

This book covers not only manifolds, tensors, metrics, connections, curvature, calculus of variations, homogeneous spaces, and covering spaces, but also Minkowski spacetime, the Friedmann and Schwarzschild solutions, and the singularity theorems. This book has a somewhat fussy notation, and tends toward the verbose, but it is engaging and full of insight. Boothby is shorter but covers more, although the last volume of Spivak is a gentle introduction to Chern classes. I am not familiar with the following book, but I like an elementary GR text by the second author :.

Here are two pricey and extremely concise outlines of the basics of differential geometry and topology as they are used in modern physics:. These are so dense I wouldn't recommend them for anyone without a strong background in modern physics. Dover has reprinted books by Levi-Civita, Schouten, and Synge on tensor calculus.

Spacetime: Foundations of General Relativity and Differential Geometry

These were all essential references in their day but they are now hopelessly out of date and I recommend that students spend their money on more expensive but more modern texts. Here are some books that may help the student place relativity theory into the grand scheme of things, physically speaking:. I like this book very much. Lawrie quite properly emphasizes the formal analogies between hamiltonian mechanics and quantum theory; the variational principle formulations of GR ties this relativity theory to both these subjects. Lawrie also emphasizes the fact that newtonian theory is not simply "wrong"; by a mere change of interpretation and a factor of i here and a factor of h bar there the equations of newtonian theory as rewritten by Hamilton go over to their quantum analogs.

Needless to say these formal analogies are a great help to the working physicist. One of the great scientific expositions of all time. Full of enthusiasm and overflowing with fabulous ideas. Feynman's geometric explanation of the physical meaning of Maxwell's equation is a joy; so is his discussion of action at a distance his revolutionary work with Wheeler. The first two volumes are particularly recommended. Note well: in volume 2, the section on SR is one of the few weak points in the book; I advise that you skip it altogether. If you must read it, not, RPF is not saying that spacetime has a Euclidean metric!

Well translated and quite readable for the most part. Initiated by the great Russian physicist Lev Landau and continued after his untimely death by his disciple Lifshitz. The Russian approach to physics and math is significantly different from American ideas in many respects and it is worthwhile gaining some familiarity with Landau's vision.

Unlike say Feynman's great books , this series offers many excellent exercises. Another heroic attempt to survey all of modern theoretical physics at the advanced undergraduate to second year graduate level, this time with a European perspective. Space-time symmetry groups and partially background-independent space-times.

Supplementary Information

General relativity as a gauge theory. The hole argument for elementary particles. The problem of quantum gravity. Conclusion: The Hole Argument Redivivus. Nijenhuis on the Formal Definition of Natural Bundles.