Spacetime and Geometry: An Introduction to General Relativity

My comments come with a few caveats.

1. This is my fourth GR book.

2. I'm not hardcore into physics. I'm not a physic grad and I'm reading GR for fun. I have a decent graduate math background but I've been corrupted with 10+ years in working in various roles software engineering, electronics engineering and marketing.

3. I assume that since you're considering buying this book, you're goal is to get at the "real" GR, not the watered down discover channel version.

With these caveats in mind, here are my comments.

First, on a scale of 1-5, I rank Carroll at level 3 in terms of math/physics maturity and thoroughness. Here is my full ranking of authors from my limited reading: 1. schutz 2. hartle 3. penrose 3. carroll 4. wald 5. physics journal articles

Second, using the rankings above, I recommend Carroll as the second port of entry. If you're comfortable with multivariable calculus, start with schutz (#1). You'll get warm fuzzies doing the toy exercises. But Schutz is tensor/math-lite. If you've had advanced calculus and geometry already, jump in with carroll (#3). But you'll be hard-pressed to find anyone else as polite to the reader. He won't prepare you for 80 percent of what's published. If you're ready to throw off the training wheels and jump dive into mainstream GR go with Wald (#4).

Note that Hartle (#2) is a good "tweener" book with feel-good exercises and some of the full-on GR equations at the end. I bet most instructors teaching a first year grad course would go with Hartle along with a dose of supplementary material.

Third, don't expect Carroll to be your last GR book purchase if you want to reach the promised land (see caveat #4). Living and breathing GR is found in physics journals and for that you'll need Wald or another advanced GR book.

Last year, when I was at my Dad’s and stepmom’s house, I found the book “An Introduction to General Relativity, Spacetime and Geometry”, by Sean M Carroll. I was struck by the promise at the beginning of the preface: “.. it is an unalloyed joy to finally reach the point in one’s studies where these phenomena may be understood in a rigorous quantitative level. If you are contemplating reading this book, that point is here.” For me the book has lived up to this promise. I have been reading the book off and on since then and working some of the problems. I studied general relativity a little in college several decades ago, but there were many vague ideas floating in my head. This book has sharpened up the concepts, explained new ones, and connected them in a good way.

This book has helped me long before it was ever published! It is based off of lecture notes that Carroll gave for a graduate level General Relativity course. These notes are still freely available at:
[...]
But you miss out on extras like better diagrams, more examples and exercises, so this is still a great buy!

This is an outstanding textbook on general relativity. It's very detailed, well written and the order of the topics is very well chosen, covering a wide range of themes. The level is appropriate to graduate student, with a quite decent mathematical background.
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.

I am a physics graduate student without pre-knowledge about GR, and I must say:

The Ta-Pei's book called is MUCH better than Sean Carroll's textbook , simply because the Ta-Pei's book provides much more detailed and rigorous explanation with more illustrative diagrams than what the Sean's book does.

For example, Ta-Pei book provides two different ways to derive the Geodesic equations on the page 88, 106 and 320, whereas Sean is only able to provide one way to derive it on the page 105. I always want to know how the Geodesic equation is related to the Lagrange. and Ta-Pei's book explains this very well, whereas Sean does not explain anything about it.

Many description on Sean is unclear. He tends to omit many intermediate steps, for example, on the page 137, he omits a lot steps to derive the equation (3.178). Another example, on the page 161, he says that it is straightforward to have equation (4.56). At least not straightforward to me. He assumes that every readers are geniuses or have plenty of time to figure out what he says. Clearly, he excludes those non-geniuses, like me, from reading his book.

He tries to teach me how to do GR by using Differential Geometry without systemically teaching me how to do Differential Geometry. This causes me to waste lots of time to figure out what Sean's book actually means. For example, Sean is awful to explain what "One-form" means. Rather than introducing such complicated concept from Differential Geometry, Ta-Pei teaches me how to do GR by introducing highly self-contained mathematical concept. In Ta-Pei's way, I do not have to look for more advanced math textbooks in order to understand what he really means. This saves me enormous amount of time. Some description on Ta-Pei is marvellous. For example, on the top of the page 147, "the roles of time and space are interchanged when crossing over the r=r*." This concise description captures the whole feature of the event horizon! That is awesome!

In contrast to Sean's book, I also like the diagrams on Ta-Pei's book, for example on the page 106 the fig 6.3, which is simple but means a lot to me. Another example, fig 7.1 on the page 123, which helps me to understand worm hole.

Ta-Pei's book provides detailed solutions to almost all the exercise questions (Including the Review questions in the end of each chapters). This is so great. I learnt a lot how to solve GR problems from him. In contrast, Sean's textbook does not provide any solution - clearly, Sean does not care about teaching, and is a lazy or busy teacher.

Ta-Pei's book is an intermediate level on GR. Sean's book is a little more advanced than Ta-Pei's book. However, Readers who finished Ta-Pei's book could continue to read more advanced GR book without touching Sean's book. So please do not waste time on Sean's book IF you had no pre-knowledge about GR before! My honest suggest! If you had some experience or exposure to GR or Differential Geometry, then Sean's book may be for you to read.

In my graduate studies in physics, I had never taken a course in general relativity or differential geometry. Carroll's book is the right place to start. It is very clearly written and it has a wealth of diagrams to help when the discussion tends to get somewhat abstract. I found it enlightening, entertaining, at times deep and always worth the effort. The material on differential geometry and the appendices are examples of textbook writing at its best. If you have the proper background, go here before attempting Wald's General Relativity or any other more advanced treatise. Joseph R. Dell'Aquila, PhD

This is a serious Graduate level text on GR; well above the level of his many YouTube appearance. Suggest that you also check-out Mike Guidry's 2019 book Modern GR. It focuses on all the many applications of GR. Buy both books!

This is an advanced text, but all the same it is not particularly rigorous or dense, so it is in principle accessible to the beginner. With an easy authority, Carroll leads us on a wandering journey through the mystical lands of general relativity. This is very different from, and compliments nicely, the clarity and directness of Wald. As a student of GR, I use Wald for the bottom line on any subject, and Carroll for the random physical or computational insights that I invariably find in any section of the book. Carroll's prose is like music to the ear and I always enjoy myself when I decide to open up this book.

Be warned that there are lots of mistakes in this first edition--you might want to wait for the second one.

Also, his chapter on cosmology is better than any I've seen.

I am currently on the 4th chapter of Carroll's "Spacetime and Geometry" and thus far I am amazed at how clear it is. Sure there is a lot of math in it however that also is very clearly explained. In fact, I think that Carroll explains the differential geometry material better than any mathematician has in any book on the subject. If you want to learn general relativity, there is no getting around the math; sooner or later you'll have to learn it. I'd suggest, especially if you are self-studying the subject, to rather pick up this book and go through it than pick up a more "elementary" text and a book on Riemannian geometry to look at later.

(Although I do also highly recommend Kay's (Schaum outline) "Tensor Calculus" for self study. The prima donnas don't like Kay's book because it "doesn't have enough theory." I suppose if a freshman calculus book does not have the Lebesgue integral defined in ti they'll complain about that too.)

Because, you can always skip through certain sections if the math is too heavy and go back through it later. And like I wrote earlier, you won't find a better introduction to the mathematical material than here.

Carroll should be given the Nobel prize for this book. If not in Physics, then in literature. I'd give this textbook 10 stars if I could.

Kudos to Carroll.

This book is an excellent INTRODUCTION to SR and GR for the graduate physics student as well as the graduate mathematics students.

Pure mathematics often loses sight of the ideas which motivated it and physics often loses the mathematical foundations from which it is built.

This book offers some level of mathematical formalism to the physics student while exposing the ideas motivating the mathematical concepts.

I particularly like how he builds up the mathematical machinery of GR by introducing sets then topology on this set giving a topological space. Now he adds in the ideas of a manifold which make this topological space look like Rn locally with the patches sewn together smoothly. The manifold comes equipped with tangent space, cotangent spaces and their product spaces giving tensor spaces. These are defined nicely with reference to component formalism as well as the multilinear algebra approach as maps from products spaces to the reals, etc. He delves into forms and tantalized the reader with deRham cohomology although doesnt go into it. He shows how these can be differentiated ( exterior derivative ) and integrated.

Now the metric is introduced giving a geometry. To this is added a connection which is independent of the metric and leads to notions of parallel transport and differentiation of tensors ( covariant derivative ). One sees that in a special case one can derive a unique connection from the metric ( Levi-Cevita ) which is used in GR.

Fibre bundles, Lie derivatives, pullbacks etc are introduced as needed.

He then presents some introductory GR material by applying the mathematics.