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Spacetime and Geometry: Pearson New International Edition: An Introduction to General Relativity Paperback – 12 Aug 2013
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Yes. This book is really a very clear and thorough introduction to modern General Relativity thanks to the very up to date chapters, although the author does not deepen enough in most of the long steps and computations or examples as a real beginner would like but anyway not many books do so. Special Relativity is reviewed but obviously assumed and differential geometry is introduced and developed quite enough but again some familiarity manipulating tensors is assumed (for this the best is to try Schutz's "Geometrical methods"). The layout is really nice and the book is more or less self-contained in every aspect with a "introduction-to-less-than-advanced-level" approach in most aspects.
There are no solutions for any exercise (a thumb up for Schutz's book on the same subject) but the worst flaw could be the number of typos for being the first edition, not too many or too important though (and reported in his web).
Overall you should buy this book if you know special relativity, undergraduate electrodynamics and notions of tensors because reading and understanding it through is more than enough for having a semi-advanced level in general relativity allowing you to tackle more tough stuff like Wald or Hawking's books. It's one of the latest best books on the subject.
If you want a good story, however, and if you would like to see the connection with mechanics, I would recommend Einstein Gravity in a Nutshell by A. Zee. For me this book appeals more to physical intuition and is less mathematical.
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I purchased this book after taking a class with Cheng's book and find myself quickly going through it. The book is really making a lot of concepts more rigorous and thorough, but as a second read it is a bit too easy.
That being said, I really love the book and find myself learning tons and making a bunch of connections that slipped past me my first time learning GR.
Unless you start with a strong math background to the level of tensors, you should expect to spend a lot of time in Wikipedia and Wolfram learning the math. But if you do, it is an incredibly rewarding book that leaves you comfortable reading the current literature and thinking on quantum gravity. No small feat!
In particular, the first chapter is a review of special relativity: a brief but clear summary, useful to become familiar with the use of the 4-vector notation, too. The second and third chapters are committed to manifolds and curvature, and you have to learn the fundaments of differential geometry. The chapters from fourth to seventh are focused on the "real" general relativity, from Einstein's equation to gravitational waves: this is a quite advanced dissertation, and I think it is necessary to have a basic background from an introductory book. The last two chapters are an introduction to cosmology (brief, but pretty good) and an introduction to quantum field theory in curved spacetime (but I never read this chapter, sorry!).
Remark that the book contains ten (10!) very useful appendixes on additional topics that are not debated in the ordinary chapters: they are a good extension to examine in depth some themes (in particular on a second reading).
Very good binding and hardcover: it's durable and solid, with a good value for money.
The math chapters 2 and 3 will teach you tensor analysis on manifolds in much clearer way than other books. The book makes a clear distinction between assumptions, choices (like working with a metric compatible connection), or derived facts. It also makes a difference between a Christoffel connection and a generic connection. The appendices will give you a feeling for some new to you math on manifolds like pullbacks, Lie Derivatives, hypersurfaces etc.
Chapter 4 is worth reading too cause it makes clear that Einstein's equations are just the simplest guess out of many other possibilities. It shows how we generalize physical laws from special relativity to GR making it clear our choices are the simplest ones but not the only ones possible.
The chapters after that discuss applications of GR like black holes, gravitational radiation, cosmology etc. Of these, I've read only the black holes chapters 5 and 6 and I wasn't able to understand 100% what was goin on. The problem was that the book uses concepts that you still don't quite understand if you are a beginner like 'spacelike singularity' or 'conformal diagrams'. That is informative but the book doesn't provide the necessary level of detail and examples for beginners so you could really master such concepts and use them in your practice.
There are problems after each chapter but not the necessary beginners problems that increase your conceptual understanding of the theory. Instead, some are just tedious algebra of type 'find the curvature for some general form of the metric' for which specialists in the field use symbolic programs like Mathematica. Solving these by hand proves that you can take derivatives and you are a mazochist but not that you understand GR. Other problems are really relevant to your education but are not dirrectly connected to the discussion in the text. Because of that you have to solve them from scratch and it will take you ages ...
In retrospective, Carroll's book is a middle level GR, I sometimes use it as a starting reference for my research (GR applications to Cosmology). It is a book written to inform you and give you the general logical outline of GR together with the differential geometry. It is not constructed to train you to actually apply this stuff in practice - you end up "understanding" indices and geometrical constructs but when the time comes to apply them, you can't solve a simple physical problem. Being informed well does not equal understanding does not equal mastery.
The books that covers the conceptual beginner level and will actually teach you how to apply GR in simple physical situations are James Hartle's "Gravity: An Introduction to Einstein's General Relativity" and Bernard Schutz's "A first course in General Relativity". The Inverno text is with more diff. geometry like Carroll. Is is not as diverse in topics but is more focused and will teach you applications instead of just informing you.