This book is an extensive revision and expansion of Dr. Straumann's earlier book, General Relativity with Applications to Astrophysics (2004). The new version contains an additional 61 pages which are accounted for by revisions and additions throughout the text and the inclusion of an entirely new Chapter 10 on Friedmann-Lemaitre cosmological models. As Straumann remarks in the Preface, the new text represents such a thorough revision of the 2004 work that it cannot be considered merely a new edition.
Straumann's book joins the roster of a number of currently available introductions to general relativity written for the beginning graduate student: a list of authors for some of the more popular books includes Wald, Carroll, Ohanian & Ruffini, Stephani, Ryder, Plebanski & Krasinski, Choquet-Bruhat, Rindler, and Gron & Hervik, among others. And of course, many students are still drawn to the venerable book Gravitation by Misner, Thorne & Wheeler.
How is one to choose intelligently from among these references? Is there anything that recommends one book over all the others? Those of us who study mathematics and physics independently are drawn to books that appeal to us on an individual level--books that fit our background preparation, our goals for independent study, and our unique learning style. There is seldom one single text for an advanced topic such as GR that is "the best" resource for every potential reader. With that in mind, I would suggest that Straumann's book will appeal very strongly to any reader who fits the following profile. He/she is mathematically sophisticated and appreciates seeing the mathematics of general relativity done with the same kind of rigor that mathematicians routinely demand in their advanced courses. He does not feel enlightened by attempts to reduce tensor analysis to unjustified sequences of index manipulations. He understands the need to master both index-based and index-free notation and computations within differential geometry and GR. And most importantly, he appreciates that Einstein's GR is a GEOMETRIC theory and wishes to fully understand the beautiful interplay between gravitational physics and Lorentzian geometry on the spacetime manifold. In my opinion, Straumann's book stands out as a superior choice for most students who fit this profile; mathematics and physics are developed together at an extremely advanced level, and neither gets slighted in the presentation.
Straumann's emphasis on mathematical rigor is reflected through the inclusion of Part III: Differential Geometry at the end of the book. Including the appendices that continue the development, this mathematical material occupies pages 579--706; it is a substantial portion of the book, representing an extremely dense distillation of all the necessary background in differential geometry and tensor analysis. Straumann remarks that he encourages his students at Zurich to consult Part III on an as-needed basis each time an advanced mathematical topic is first encountered in the text. I would respectfully suggest that Straumann's book will be more effective for a student who has already had an introductory course in differential geometry on manifolds, preferably one that studied semi-Riemannian (specifically Lorentzian) geometry as well as Riemannian. There are very few books that serve as introductions to both Riemannian and semi-Riemannian geometry; the classic text by Bishop & Goldberg is still available and serves well, but a superior choice is Barrett O'Neill's Semi-Riemannian Geometry with Applications to Relativity. O'Neill's book has become such an acknowledged classic that many instructors use it for purely Riemannian courses. A reader who has mastered the first four chapters of O'Neill will find Straumann far more accessible and will have a far better chance of understanding the more advanced mathematical material in Part III.
Amazon's "Look Inside" feature permits prospective buyers to browse through the chapters, so I need not discuss the contents of the book. The reader should notice, however, that in addition to the standard topics to be found in nearly all introductions to GR, Straumann's book also contains thorough discussions of a number of more specialized topics. I am especially impressed with Chapter 9, which contains an accessible discussion of the work of Schoen, Yau and Witten on the Positive Mass Theorem. I would also like to point out that it has been well known since the early 1980s that many exact solutions to Einstein's equations are most naturally expressed in the form of warped product manifolds [the Schwarzschild model and the Robertson-Walker-Friedmann-Lemaitre cosmological models are notable examples]. Despite the substantive simplifications that result from the use of the warped product formalism, warped products have been slow to appear in the physics literature. Straumann devotes Appendix B to computations of Ricci curvature on warped products, and he refers to warped products in the text when appropriate. Just another sign of the progressive, modern nature of the text.
Straumann's book offers a thorough, modern introduction to GR with unrivaled mathematical integrity. It is written so clearly that it can be used quite profitably for independent study. I teach a one-year undergraduate sequence in differential geometry and general relativity in which we use O'Neill's book as our primary reference. If I am ever given the chance to offer a sequel at the graduate level, Straumann's book will certainly be my choice for primary course text.