Numerical Relativity: Solving Einstein's Equations on the Computer

Numerical relativity is one of the major fields of contemporary general relativity and is developing continually. Yet three years ago, no textbook was available on this subject. The first textbook devoted to numerical relativity, by Alcubierre, appeared in 2008 [1] (cf the CQG review [2]). Now comes the second book, by Baumgarte and Shapiro, two well known players in the field. Inevitably, the two books have some common aspects (otherwise they would not deal with the same topic! ). For instance the titles of the first four chapters of Baumgarte and Shapiro are very similar to those of Alcubierre. This arises from some logic inherent to the subject: chapter 1 recaps basic GR, chapter 2 introduces the 3+1 formalism, chapter 3 focuses on the initial data and chapter 4 on the choice of coordinates for the evolution. But there are also many differences between the two books, which actually make them complementary. At first glance the differences are the size (720 pages for Baumgarte and Shapiro vs 464 pages for Alcubierre) and the colour figures in Baumgarte and Shapiro. Regarding the content, Baumgarte and Shapiro address many topics which are not present in Alcubierre's book, such as magnetohydrodynamics, radiative transfer, collisionless matter, spectral methods, rotating stars and post-Newtonian approximation. The main difference regards binary systems: virtually absent from Alcubierre's book (except for binary black hole initial data), they occupy not less than five chapters in Baumgarte and Shapiro's book. In contrast, gravitational wave extraction, various hyperbolic formulations of Einstein's equations and the high-resolution shock-capturing schemes are treated in more depth by Alcubierre. In the first four chapters mentioned above, some distinctive features of Baumgarte and Shapiro's book are the beautiful treatment of Oppenheimer–Snyder collapse in chapter 1, the analogy with Maxwell's equations when discussing the constraints and the evolution equations in chapter 2 and the nice illustration of the 3+1 formalism by different slicings of Schwarzschild spacetime. Chapter 3, devoted to initial data, presents the York–Lichnerowicz conformal method with many details and examples, along with its descendants (extended conformal thin-sandwich). A very instructive illustration is provided by a boosted black hole. This chapter also introduces the recent waveless approximation and presents a rather detailed discussion of mass, momentum and angular momentum in the initial data. Chapter 4 contains a very pedagogical discussion of the choice of coordinates, via the lapse and shift functions, again with many examples. In particular, it provides the derivation of all maximal slicings of Schwarzschild spacetime, which is hardly found in any textbook. Chapter 5, devoted to matter sources, goes well beyond the ideal fluid generally discussed in the context of 3+1 numerical relativity: it also covers dissipative fluids, radiation hydrodynamics, collisionless matter and scalar fields. Chapter 6 provides a self-consistent introduction to the two main numerical methods used in numerical relativity: finite differences and spectral methods. It is followed by a very nice chapter about the various horizons involved in black hole spacetimes: event and apparent horizons, as well as dynamical and isolated horizons. One may, however, regret that there is no mention of Hayward's trapping horizons, which embody both dynamical and isolated horizons. Chapter 8 discusses in depth spherical spacetimes, including dynamical slicings of Schwarzschild, gravitational collapse of collisionless matter (26 pages! ), collapse of fluid stars and scalar fields and critical phenomena. The main outcome of numerical relativity, gravitational waves, are introduced in a very pedagogical way in chapter 9, with the basic theory and a review of the astrophysical sources and detectors. Chapter 10, entirely devoted to the axisymmetric collapse of collisionless clusters, reflects clearly the research work of one of the authors, but it is also an opportunity to discuss the Cosmic Censorship conjecture and the Hoop conjecture. Chapter 11 presents the basics of hyperbolic systems and focuses on the famous BSSN formalism employed in most numerical codes. The electromagnetism analogy introduced in chapter 2 is developed, providing some very useful insight. The remainder of the book is devoted to the collapse of rotating stars (chapter 14) and to the coalescence of binary systems of compact objects, either neutron stars or black holes (chapters 12, 13, 15, 16 and 17). This is a unique introduction and review of results about the expected main sources of gravitational radiation. It includes a detailed presentation of the major triumph of numerical relativity: the successful computation of binary black hole merger. I think that Baumgarte and Shapiro have accomplished a genuine tour de force by writing such a comprehensive and self-contained textbook on a highly evolving subject. The primary value of the book is to be extremely pedagogical. The style is definitively at the textbook level and not that of a review article. One may point out the use of boxes to recap important results and the very instructive aspect of many figures, some of them in colour. There are also numerous exercises in the main text, to encourage the reader to find some useful results by himself. The pedagogical trend is manifest up to the book cover, with the subtitle explaining what the title means! Another great value of the book is indisputably its encyclopedic aspect, making it a very good starting point for research on many topics of modern relativity. I have no doubt that Baumgarte and Shapiro's monograph will quicken considerably the learning phase of any master or PhD student beginning numerical relativity. It will also prove to be very valuable for all researchers of the field and should become a major reference. Beyond numerical relativity, the richness and variety of examples are such that the reading of the book will be highly profitable to any person interested in black hole physics or relativistic astrophysics. This is not the least among all the merits of this superb book. References [1] Alcubierre M 2008 Introduction to 3+1 Numerical Relativity (Oxford: Oxford University Press) [2] Gundlach C 2008 Review of Introduction to 3+1 Numerical Relativity Class. Quantum Grav. 278 1270