Introduction to 3+1 Numerical Relativity

This is the first major textbook on the methods of numerical relativity. The selection of material is based on what is known to work reliably in astrophysical applications and would therefore be considered by many as the 'mainstream' of the field. This means spacelike slices, the BSSNOK or harmonic formulation of the Einstein equations, finite differencing for the spacetime variables, and high-resolution shock capturing methods for perfect fluid matter. (Arguably, pseudo-spectral methods also belong in this category, at least for elliptic equations, but are not covered in this book.) The account is self-contained, and comprehensive within its chosen scope. It could serve as a primer for the growing number of review papers on aspects of numerical relativity published in Living Reviews in Relativity (LRR). I will now discuss the contents by chapter. Chapter 1, an introduction to general relativity, is clearly written, but may be a little too concise to be used as a first text on this subject at postgraduate level, compared to the textbook by Schutz or the first half of Wald's book. Chapter 2 contains a good introduction to the 3+1 split of the field equations in the form mainly given by York. York's pedagogical presentation (in a 1979 conference volume) is still up to date, but Alcubierre makes a clearer distinction between the geometric split and its form in adapted coordinates, as well as filling in some derivations. Chapter 3 on initial data is close to Cook's 2001 LRR, but is beautifully unified by an emphasis on how different choices of conformal weights suit different purposes. Chapter 4 on gauge conditions covers a topic on which no review paper exists, and which is spread thinly over many papers. The presentation is both detailed and unified, making this an excellent resource also for experts. The chapter reflects the author's research interests while remaining canonical. Chapter 5 covers hyperbolic reductions of the field equations. Alcubierre's excellent presentation is less technical than Reula's 1998 LRR or the 1995 book by Gustafsson, Kreiss and Oliger, but covers the key ideas in application to the Einstein equations. The reviewer (admittedly riding a hobby-horse) would argue that the hyperbolicity of the ADM and BSSNOK equations should have been investigated without introducing a specific first-order reduction. Chapter 6 covers gauge problems in numerical black hole spacetimes, black hole excision, and apparent horizons. Like chapter 4 it is both exhaustive and pedagogical. Perhaps more space than necessary is given here to work the author was involved in, while the section on slice stretching could have been more detailed, given that there is no good overview in the literature. Chapter 7 on relativistic hydrodynamics is, quite simply, excellent. Among many other useful things it contains some elementary material on equations of state that is not written up at this level elsewhere, a good mini-introduction to weak solutions of conservation laws, and a brief review of imperfect fluids in GR (Israel--Stewart theory). This chapter complements Font's 2008 LRR. Chapter 8 on gravitational wave extraction provides a welcome pedagogical introduction to a topic in which the original research papers are less than inviting and where notation is not uniform. The mathematical techniques described here are in constant use in numerical relativity codes, but are never fully described in research papers. Chapter 9 on numerical methods covers finite difference and high-resolution shock capturing methods. It is similar in presentation to Leveque's 1992 book and Kreiss and Busenhart's 2001 book, but gives a good selection of that material, concisely presented. It certainly impresses the importance of convergence testing on the reader. Chapter 10 covers methods for spherically symmetric and axisymmetric spacetimes. The former is excellent, reflecting the author's recent research work. The axisymmetry section would have been better if it had been based on a formal Geroch reduction, the method that has been the key to recent progress. This book is bound to become a standard text for beginning graduate students. In an overview for this audience, I would have liked to see a little more detail on null slicings and on the conformal field equations, and brief introductions to the theory of elliptic equations and to pseudo-spectral and finite element methods. One may also regret the many typographical errors. Nevertheless, this excellent book fills a real gap, and will be hard to follow.