Last year (2015) was a very good year for physics; it saw both the 100th anniversary of the discovery of general relativity (GR) by Albert Einstein and the first detection on Earth of the gravitational waves (GW) that were one of the key predictions of this theory. Members of the Laser Interferometer Gravitational-Wave Observatory (LIGO) collaboration indeed announced that, after decades of exquisite experimental work, they had measured the minute (up to 10^-29 m) space distortions generated by the merger of two massive black holes that took place 1.3 billion years ago. But this major scientific achievement could not have happened without the advances that took place in another field: computer science. In particular, researchers in the scientific and high-performance computing (HPC) communities joined their physics colleagues to come up with ways of simulating the nonlinear Einstein equations that model such large cosmic events; these simulations were used to generate the signal templates that were searched for in the terabytes of data LIGO generates every observation day.

Numerical relativity, written by Masaru Shibata, who is one of the key actors in this long-running endeavor, provides all the elements needed to grasp this new and exciting research field. This massive book (more than 800 pages) is structured into three main parts; eight appendices, which present rather advanced mathematical results usually not discussed in traditional GR textbooks; a bibliography of about 800 entries; and a detailed index. The first part, “Preliminaries for Numerical Relativity,” provides, in 100 pages, a precise and formal review of all the mathematical developments behind the physics of GR, neutron stars, and black holes, with a special emphasis on their influence on the generation of GW.

The second part, “Methodology,” with six chapters and more than 400 pages, bridges the gap between the continuous world of physics and the discrete framework that computer simulations live in. The first issue addressed is how GR equations can be adapted, using the ADM or N+1 formalisms, to foliated space-time, in which time evolution is distinguished from the space-related one, a first step toward time-dependent computer simulation. The various ways of handling both the discretization of the resulting space-time, with possibly adaptive meshes and GR-specific integration approaches (finite differences or spectral techniques), are then discussed; the importance of numerical error handling is stressed. In addition to space-time itself, the celestial bodies also have to be simulated, which requires a good understanding of their physical properties: hydrodynamical, electromagnetohydrodynamical, and radiation formulations are thus introduced, together with techniques to properly handle them numerically. A key issue is testing: to ensure that such simulations closely approximate actual phenomena, simulations of problems for which analytical solutions exist have been performed, and the discrepancies analyzed, to provide stronger correctness claims when simulating real-world problems lacking analytical solutions. This preliminary work ends with the formulation of initial data for GR simulations, together with equilibrium and quasi-equilibrium solutions, from which simulations will start, yielding, in particular, GW characteristics.

The last, four-chapter part of the book (about 250 pages) is an equation-free analysis of various astronomical evolution simulations described in the literature (mostly since the beginning of the 2000s) and based on numerical relativity methods. A significant section is dedicated to the coalescence of binary compact objects such as neutron stars and black holes; the challenges raised by the simulation of the various phases, including inspiraling, matter shedding, torus formation, and effective merger, are thoroughly addressed. The signatures of the resulting GW are discussed at length, and fitting functions provided for them. Other processes, for example, the gravitational collapse of black holes or the deformations of the tori surrounding them, can also be the source of powerful GW, as discussed in the two following chapters. The final chapter discusses an even more exotic situation, namely the simulation of GR in higher-dimensional space-times, something that could have nonetheless practical applications, for instance regarding the possible creation of mini black holes in large particle accelerators such as the CERN Large Hadron Collider.

Even though Numerical relativity was written just prior the actual discovery of GW, it provides an amazing wealth of information that should already ensure its success in both the physics and physics-oriented scientific computing communities. It explains how GW-based astronomy will pave the way to a better understanding of the physical properties of cosmic entities such as neutron stars (for example, their currently poorly understood equations of state, linking pressure, and matter density). Practically minded HPC researchers may regret the lack of information regarding the actual implementations and performance of the numerous simulations, but the references given in the book (which can, in fact, be seen as a gigantic survey paper) will certainly discuss this point. I definitely recommend this book to physics scientists and also to scientific computing researchers, at least those who have a good understanding of GR formalisms.