General Relativity is perhaps one of the most beautiful and elegant of physical theories. It combines deep insights into the nature of space, time and matter into one single theoretical and mathematical framework. Unfortunately, that framework turned out to be quite formidable to work with, and except a few very special cases there have not been many solutions to the equations of General Relativity – Einstein’s equations. That has, fortunately, started to change. Thanks to the increasingly more sophisticated modern computers and the ever-refined numerical algorithms General Relativity has become tractable in recent years. The advances in numerical Relativity have been quite impressive, and “Elements of Numerical Relativity and Relativistic Hydrodynamics” is an excellent overview of this exciting research field.

The book is configured as a set of lecture notes – an accessible and pedagogical intro to the subject. It can serve as a starting point for advanced graduate seminars. The second edition is greatly revised and enhanced, due to the fact that since 2005 there have been some major new technical numerical relativity breakthrough, the most significant of which is the binary black hole simulations.

The book provides insight into numerical techniques, but also into evolution systems, gauge, initial and boundary conditions, even for the numerical algorithms. The book is intended for beginners who want to get into the field of Numerical Relativity. It is based on 3+1 formalism, although this second edition also covers some Z4 formalism. It tries to convey the message that Numerical Relativity is based on insight, and not a matter of brute force.

The topics covered in the book include the overview of standard General Relativity, exact solutions to Einstein’s equations, Harmonic formalism, the Evolution formalism, foliation of spacetime, gravitational waves, the free evolution framework, the Z4 evolution system, symmetry breaking, first order hyperbolic systems, numerical methods, black hole simulations, dynamical time slicing, matter spacetimes, magnetohydrodynamics, and many others. Each chapter of the book, and oftentimes each topic, could be a topic of separate treatise. All the topics are covered in a very accessible and legible manner, but due to the constraints of length most of them are not covered at much depth. To fully do the justice to this extensive field would probably require a textbook that is two to three times the length of this one.

This book is a very accessible and a good first exposure to the “big picture” of Numerical Relativity. However, it is not a very good teaching resource for learning numerical relativity techniques. It hardly has any examples, and no exercises or other standard fare of Physics textbooks. It is best used as a reference guide to other resources in the field and a handy overview of the major techniques and research topics.