Semi-Riemannian Geometry With Applications to Relativity (Volume 103) (Pure and Applied Mathematics, Volume 103)

Professor O'Neill was my teacher and PhD supervisor. He was the best teacher I or anyone else ever had. First of all he knew the material as it should be known, namely through clean coordinate-free geometric understanding; afterwards the coordinates can be chosen when specific local calculations are needed. Secondly, he was able to assemble the material and express it cleanly and clearly.Everybody should know the basics of this material, so that the geometric content of the Calculus will finally become common knowledge. This knowledge is beautiful, powerful, clean and simple. (I mean you professional mathematicians and pedagogues across the world, as well as students and all people who need to use mathematics to solve problems, should know these basics. They were known to the classical mathematicians, even if they could not make it explicit or communicate it directly.)Modern-day pedagogues are still writing and teaching all about coordinates even when they have nothing to do with the discussion at hand. Mathematics needs to catch up to where it was in the past.

This is a classic book on what the title is about. Others have dealt adequately with the content, so I will focus instead on the production value. In short, the book looks and feels like a set of photocopies glued together by a little kid - the cover is just like it was written up in a page of word, printed out and glued together with the rest. For a book that costs 125 dollars, that's absolutely unacceptable. Unfortunately, customers have no way of knowing these details, and greedy, unscrupulous publishers (Academic Press, An Imprint of Elsevier) get away with what, in my mind, amounts to theft. Amazon should hire a professional book binder to comment on the quality of the books (at least of expensive ones) - such lack of professionalism should be punished.

Was hoping for some material about holonomy groups, development of curves, Minkowski structures etc. Also, should take about Lorentz coordinates more in the chapter on special relativity.The book is however well written, well planned and informative otherwise.

Yep, it’s the book he wanted.

This book is the real deal when it comes to semi-Riemannian geometry and in particular the Lorentz manifolds for general relativity. The only book I have ever seen which gives the topic a complete and rigorous treatment.

If you want to engage in a serious study of general relativity, then you must master the mathematical language of semi-Riemannian manifolds in which it is cast. Sadly, the development of classical Riemannian geometry as studied by pure mathematicians only parallels the development of semi-Riemannian geometry in the early stages; eventually, the two subjects diverge rather drastically. For example, the famous Hopf-Rinow Theorem, one of the cornerstones of modern Riemannian geometry, simply has no Lorentzian analogue at all; every single equivalence in the theorem fails in Lorentzian geometry. Thus, one could master all five volumes of Spivak's definitive treatment of Riemannian geometry and still be unprepared to deal with light cones, timelike, null and spacelike geodesics, and the multitude of other uniquely semi-Riemannian constructs that appear in general relativity.O'Neill's wonderful book, which first appeared in 1983, provides the well-prepared reader with a mathematically rigorous, thorough introduction to both Riemannian and semi-Riemannian geometry, showing how they are similar and pointing out clearly where they differ. After developing the mathematical machinery in the early chapters, the last part of the book turns to general relativity by offering lucid introductions to the Robertson-Walker cosmological models (Big Bang singularities), the Schwarzschild model for a single non-rotating star (including black holes), and a brief introduction to Penrose-Hawking causality theory.If you would like to study a pure Riemannian text in parallel with this one, I would recommend the text by Boothby, written at a comparable level of difficulty, which remains one of the clearest and most accessible Riemannian geometry texts on the market. For the serious reader who wishes to continue on with a study of the Kerr solution to Einstein's equations, modeling the exterior spacetime of a rotating star, O'Neill wrote an entire book on the subject in 1995, now difficult to find but well worth tracking down. This 1995 text contains one of the clearest, most accessible introductions available to the difficult subjects of the algebraic classification of the Weyl curvature tensor and the corresponding Petrov classification of spacetimes.I studied from O'Neill's 1983 text when it first came out and I have continued to use it as the primary text for an advanced undergraduate course I have taught over the past 20 years. It is not an "easy" text to read, but then, I have never found the "easy" introduction to differential geometry and general relativity. The reviewer who says this is not a suitable first text is simply in error; there is no better first text on the subject. If you have studied linear algebra, advanced calculus, and a little topology, then with dedication and hard work, you can learn more from O'Neill's text than from many of the far more popular recent texts, written by physicists, which attempt to circumvent the mathematics insofar as is possible while introducing general relativity. This is a perilous course for which the serious student will pay dearly later on, when she/he wants to study any of the many areas of modern physics in which differential geometry (differential forms, bundle theory, connections on a principle fiber bundle, gauge theory, etc.) plays an essential role.

There are far too many fundamental problems with this book.The first chapter is mostly definitions upon definitions with hardly any motivation. Similar books by MTW, Schutz, Carroll, and Kreyszig do just fine without defining things such as atlases or immersions or diffeomorphisms. If this book wants to use these pedantic ideas, they need to be motivated.. It's far too difficult to decide what proofs and definitions are important. A one-form is fundamental to understand this subject, and a submersion is not; however, both concepts are presented in a similar fashion. I spent about 25 hours memorizing the vocabulary of Chapter 1 and I feel like that time has been wasted.Here are a few examples of more objective issues:Page 9: A mysterious unbolded "R" is used that was never referred to before and is never referred to after.Page 67: The geodesic equation is not derived.Page 147. "The lemma" is referred to a few times. 33 lemmas and definitions were previously used in this chapter.Page 148: Early into Lorentz geometry, the author gives an example with 2 time dimensions and 1 spacial dimension: bold R_2^3. Not only has 2 time dimensions never been referred to before and the obvious complications of such are not explained, but Lorentz geometry is defined to have 1 time dimension.I don't know what type of person can benefit from this book.

The binding is horrible with very poor materials. The content is horrible too. There are a lot of theorems and propositions without proofs and explanations; in them the author only puts something like "using the proposition ... the result is immediate", but when you try to do it by youself, it is impossible to obtain the dessired the result. I wanted to use this book to learn about singularity theorems, but I cannot advance enough fast because the lack of reasonable explanations.

Was den Inhalt des Buchs angeht ist klar und wurde bereits in anderen Rezensionen diskutiert. Das ist ohne Zweifel ein hervorragendes Buch. Bei meiner Rezension geht es darum zu betonen, dass das gebundene Ausgabe vom Academic Press-Verlag einen hochwertigen Eindruck macht. Manche schlechte Bewertungen bemängelten die Qualität der broschierte Ausgabe von Elsevier. Man macht nichts falsches bei dem Einkauf der gebundenen AP-Ausgabe.

This book is very well-written and exceptionally clear. Moreover, it deals with a large variety of topics, including many applications to the special and general theory of relativity. It employs both the coordinate and the intrinsic notation for tensors. It includes a concise but precise and very clear introduction to manifolds and tensors, before facing geometry.

LO QUE BUSCABA

The best book on Differential Geometry. Ever.