Learning theoretical general relativity requires students to first become immersed in the diverse mathematics that is essential for understanding that field. In my experience, students who have had at least two, preferably three, years of undergraduate math courses relevant to science in general, and physics in particular, face two semesters of introductory general relativity courses. The first semester is dedicated to expanding that mathematics into the tools of the trade, for example, introducing the concepts of differential forms and covariant differentiation. The second semester provides students with the opportunity to apply that mathematics to the real world. That second semester can be filled with satisfying “ah ha!” moments when the physics of black holes, gravitational radiation, curved spacetime, and cosmology comes into focus.
As with many things in life, students would benefit from going into that first course in general relativity having already seen the “big picture” of what mathematics they will need to know and how the various topics, such as vectors, one-forms, tensors, manifolds, and covariant differentiation, among other things, fit together in the context of general relativity. Even more appealing: what if they could get that big picture in a couple of weeks? Given that the students have already learned the essential mathematical concepts, that appears to be what Norman Gray had in mind in writing A Student's Guide to General Relativity.
The book is short, with four chapters totaling 109 pages, plus three appendices totaling 37 pages. The first chapter briefly introduces the history and principles of gravitational physics. The second chapter explores the relevant properties of vectors, tensors, and functions. The third chapter takes that material to mathematics of curved spacetime. The final chapter gets to the science in the form of energy, momentum, and Einstein's equations. The first appendix reviews special relativity; the second appendix briefly (15 pages) provides solutions of Einstein's equations; while the final appendix summarizes the notation used throughout the book.
One of the particular strengths of this book is his effective use of words to describe the mathematics. The presentation of both the mathematical and physical concepts is concise and focused, accompanied by figures that help students visualize some of the mathematical properties.
Can such a short introduction to general relativity provide students with a meaningful understanding of the essence of that field? As a stand-alone source of the mathematics and underlying physical concepts, I would say no. However, that clearly was not what Norman Gray had in mind. Rather, I see this book as providing students planning to take general relativity courses with a primer of the material so that they will be in a much better position to understand and process the underlying concepts they will then be learning in their formal courses. Put another way, A Student's Guide to General Relativity will help students jump-start their understanding of the field.
Gray acknowledges the need for further information on most of the concepts that he presents. He does this by frequently (quite frequently) referring readers to specific pages of standard textbooks for more information. Not surprisingly, he most frequently cites Bernard Schutz's A First Course in General Relativity 2nd edition (Cambridge University Press, 2009). Reference is also made to Sean Carroll's Spacetime and Geometry (Pearson Education,2004) and Misner, Thorne, and Wheeler's Gravitation (W.H. Freeman, 1973), among others. (Full disclosure: Bernard Schutz was my thesis advisor, and I was the T.A. for Charlie Misner the first year he used the bound version of Gravitation.)
I read through Gray's book over the course of three days, and I expect that students who want to get oriented to their upcoming general relativity courses could get through it in a week or two, especially if they also had a copy of the Schutz text for reference. I have one minor bone to pick concerning A Student's Guide to General Relativity. I find that using abbreviations, such as LIF for Local Inertial Frame or EP for equivalence principle can slow down the reading. Using abbreviations fluently requires more repetition than occurs in the book. Several times I had to go back in the text to find out what the abbreviations mean. The full words used throughout would have made the reading flow a little better but that should not be a deterrent to considering the book for those of your students who want to get started early and quickly.
Neil F. Comins is an astrophysicist on the Faculty of the University of Maine. He has published works in theoretical general relativity, computational galactic dynamics, radio and optical observational astronomy, and science education. He is the author of 21 published books and since 2005 has been a cartoon character in Japan.
