A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics

This new textbook is intended for students familiar with general relativity at the introductory level of Bernard Schutz's book A First Course in General Relativity (1985 Cambridge: Cambridge University Press) and not yet accomplished at the advanced level of Robert Wald's book General Relativity (1984 Chicago, IL: University of Chicago Press), upon which it nevertheless draws rather heavily. What is distinctively new in this book is that it is a real toolkit, and yet it is not short of detailed applications. As such, it is a helpful book to recommend to students making the transition for which it is intended. The idea of a new textbook on general relativity usually delights me, as the field is still changing rapidly. New perspectives find new ways to present old things to new students. They also have totally new things to present to us all, based on the interests of the current research from which they have grown. This new book presents a wealth of useful tools to students in just five, well integrated chapters, starting with a quick review of the fundamentals and ending with an extensive application of general relativity to black hole spacetimes. In his own words, Eric Poisson has striven to present interesting topics and common techniques not adequately covered in readily available existing texts. This has certainly been accomplished, in a synthesis extracted from many sources. Congruences of geodesics, a staple analytical tool, occupy a whole chapter, and in greater depth and clarity than can be found elsewhere. A thorough, and lengthy, presentation on hypersurfaces, including a careful treatment of the null case, carries the author's unique perspective. This treatment of hypersurfaces is put to practical use in the chapter on Lagrangian and Hamiltonian formulations, which also leans on recent quasilocal energy discussions and includes an elegant treatment of the Bondi-Sachs mass in a unified context. Many of us have become familiar with the careful, well thought out, pedagogical style of the author, and this book certainly lives up to that reputation. It has developed from a course originating from Poisson, but now already given a number of times by several different instructors, so it is well battle-tested. Since Poisson has worked extensively in many of the areas he covers, the book also carries a personal touch, with an emphasis on clarity. As intended, the influence of Werner Israel, to whom the book is dedicated, shows through, implicit in many places, and at times explicit as well. Probably my strongest quibble with the content is the absence of a comprehensive discussion of isolated and dynamical horizons, which Ashtekar and coworkers have done so much to develop recently. Students equipped with the skills Poisson intends to impart would do well to be prepared in this one particular complementary area too. The potential reader should also be cautioned that there is no treatment of black hole perturbations. Though their role in gravitational wave discussions is becoming increasingly significant, their absence from this book is justified on the grounds of space and compatibility with the techniques presented. Typographically, the book uses a clear, adequately sized font, an essentially uniform notation, and includes 40 line drawings which helpfully illustrate the text. It is exceptionally well proofread, as one might expect from the publisher concerned. I believe it will give a thorough, advanced preparation to any suitably prepared student using it, and I will definitely recommend it for students matching its intended audience.