Numerical Relativity: Solving Einstein’s Equations on the Computer

Numerical Relativity: Solving Einstein’s Equations on the Computer, Thomas W. Baumgarte and , Cambridge U. Press, New York, 2010. $90.00 (698 pp.). ISBN 978-0-521-51407-1

The application of numerical methods to relativity and gravitation has grown in intensity and scope in the past 20 years, thanks to a corresponding explosion in the power of computers and computational techniques. Those methods have perhaps had their greatest impact in simulations of binary systems of such compact objects as neutron stars and black holes and in models of the gravitational waves such systems produce as they spiral toward a collision.

Colliding binary systems are expected to be major sources of gravitational waves that could be detected and analyzed with ground-based interferometric detectors including LIGO in the US and VIRGO in Europe. Gravitational waves that result from the collision of supermassive black holes might be detected by the space-based LISA observatory, a joint proposal of NASA and the European Space Agency. Detailed models of the gravitational waveforms not only will aid in their initial detection but, perhaps more importantly, will be vital to establishing the field of gravitational-wave astronomy as wave detection becomes routine and wave detectors are used alongside ordinary telescopes to survey the sky.

A large community of researchers worldwide has been working in numerical relativity since the late 1980s. For a long time, it faced serious challenges in simulating stable evolutions of binary systems. That task is subtle because the binaries lose a very small fraction of energy in the emission of gravitational waves; as a result, their separation decreases on a time scale much greater than one orbital period. In 2005, however, a series of breakthroughs led to computer programs that have successfully evolved binary systems. And now that the field has matured, books on the subject are starting to appear.

Thomas Baumgarte and Stuart Shapiro have been collaborating on several frontline topics since 1996 and are major players in the field. Their book, Numerical Relativity: Solving Einstein’s Equations on the Computer, is a well-written overview that includes a brief introduction to general relativity, a primer on setting up initial data from the theory, and tips on dealing with matter sources of a gravitational field. The authors also introduce the most commonly used numerical-relativity tools, including spectral methods and the mesh refinement techniques used to normalize the many scales involved in relativistic simulations. They particularly address the issue of how to cast the Einstein equations in manifestly hyperbolic form, which would make them treatable by well-established mathematical and numerical techniques.

Numerical Relativity starts with a review of general relativity and Einstein’s field equations, including an introduction to gravitational waves and black holes, which both receive their own chapters later in the book. After a brief detour to discuss rotating stars and spherically and cylindrically symmetric collisionless clusters, the book moves on to discuss the central topics of binary black hole and neutron-star evolution. By including exercises, the authors aim to make Numerical Relativity useful as a graduate-level textbook and not just a reference. That feature, and the text’s coverage of neutron stars, distinguishes it from the other comprehensive treatments of the subject, including Introduction to 3 + 1 Numerical Relativity (Oxford University Press, 2008) by Miguel Alcubierre and Elements of Numerical Relativity and Relativistic Hydrodynamics: From Einstein’s Equations to Astrophysical Simulations (Springer, 2009) by Carles Bona, Carlos Palenzuela-Luque, and Carles Bona-Casas.

The text assumes that the reader has a solid understanding of general relativity, but in their preface, the authors wonder whether their book would also be useful to readers—mathematicians or computer scientists, for example—who might be interested in numerical relativity but who don’t have a solid grounding in Einstein’s theory. It is difficult to imagine that a book covering a field at the intersection of multiple disciplines could please all possible audiences. Nonetheless, Numerical Relativity hits the mark in its quite comprehensive coverage; it will be useful to practitioners in the field and especially to graduate students wishing to join them in this active and exciting area of research.