In 1985, Bernard F. Schutz, Professor of Applied Mathematics and Astronomy at University College, Cardiff, Wales (now Cardiff University) published the first edition of his textbook *A First Course in General Relativity*. The book was very well received, undergoing numerous reprints and a second edition, published in 2009. In a general relativity-textbook world dominated by the insightful, albeit voluminous, *Gravitation* by Charles Misner, Kip Thorne, and John Wheeler, Bernard's book represented a lighter, more tractable introduction to Einstein's realm.

In the spirit of full disclosure relevant to this review, I was Charlie Misner's teaching assistant the first year he taught from the bound edition of *Gravitation,* and I wrote solutions to numerous homework problems in the book for his students; they were handed out after the problems were graded. Indeed, I still have the mimeographs (you are showing your age if you remember that technology) of many of them. I was also Bernard Schutz' first graduate student and some of our work in theoretical general relativity was cited by Chandrasekhar in his 1983 Nobel Prize lecture. I have taught general relativity here at the University of Maine using *A First Course in General Relativity* several times to good effect. Furthermore, I am also a textbook author (introductory astronomy texts for non-majors).

Robert B. Scott's *A Student's Manual for A First Course in General Relativity* is written to provide solutions to nearly half of the homework problems in the second edition of Bernard's textbook, as well as to provide further exercises and their solutions. According to Scott, *A Student's Manual* is written with several audiences in mind, including “self-learners,” and both undergraduate and graduate students. In other words, *A Student's Manual* was written for use by essentially anyone with the background calculus and the motivation and aptitude for understanding the more advanced mathematics presented in Schutz' book. The solutions are stated clearly and, for the most part, each step is presented and explained. When there are many components to calculate, several are done in detail and the rest are covered with, “after a lot of simple calculus and algebra….” For further clarity, the solutions are all presented in grey backgrounds, while the questions are presented in white. Useful “hints” are provided for several problems. There are also appendices of acronyms, symbols, and “useful results.” So far, so good.

It is only when one steps back and looks at the bigger picture of textbooks and their contents that the clouds begin to thicken in regard to this book. As a textbook writer, I am leery of books of solutions to the questions in my texts. My reasoning is that many students are taking the introductory courses in astronomy because they need the credit to graduate, rather than because of a deep-seated interest in the course content. These students are more than happy to have access to solutions that will help them get better grades on their homework, even if those solutions will not help them understand the material as fully as they should. This often shows up when such students do not perform as well on exams as they would if they had made the effort to read the book and use it to answer the questions.

For more advanced courses, such as introductory general relativity, one could reasonably make the case that the students are highly motivated to understand the material. Therefore, they will make a “best effort” to read the book, parse the content, and work the exercises on their own or with their peers, until they solve all the problems they can. For problems they cannot solve, reference to a book or online file with answers to the exercises, used judiciously and with the instructor's permission, would indeed be a good thing. Bernard feels the same way, as he states in a preface to the solutions he released online in 2009 to half the exercises in his text. He says, “Solutions are provided to those problems which might present particular conceptual challenges….”

Robert Scott's book also provides general relativity-related questions and answers that are not in Schutz' book. Such additional exercises, along with their solutions, can be instructive for people who are interested in gaining a deeper understanding of a topic. Indeed, Scott's book is not the first providing new exercises and solutions in general relativity. Schutz cites another book filled with exercises for students to solve: *Problem Book in Relativity and Gravitation* by Alan Lightman, William Press, Richard Price, and Saul Teukolsky (Princeton University Press, 1975). I notice that this book is not cited in the references provided by Scott.

So, who benefits intellectually from a book that provides detailed solutions to exercises set in someone else's advanced-level textbook? Plausibly, students who cannot solve problems and whose other resources have not helped them, as noted above, and who could use published answers as a last resort. Also, non-students interested in a field, who have tried the problems on their own, and if necessary gone to an online chat room to work with others doing the same problems, but who still cannot solve them could find detailed published answers insightful. However, in the case of *A First Course in General Relativity*, both students and non-students can look up Bernard's solutions online and work backward from them until they understand the problems. A related issue with having access to step-by-step solutions as in Scott's book is if students submit homework based on those answers, their solutions will not necessarily reflect how well they really understand the material.

I could envision one type of ancillary book that could help in this process, for introductory general relativity and many other advanced topics; namely, books that give *effective hints* for many of the textbook exercises, without giving the answers. Such a book could help guide students in ways that their instructors would find helpful without disrupting the best practices for students working to learn advanced materials.

*Neil F. Comins teaches astronomy, astrophysics, and physics at the University of Maine. He has had 19 trade and textbooks published. His research focuses on general relativity, galactic dynamics, and correcting misconceptions about astronomy.*