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"The book starts by establishing the mathematical background (differential geometry, hypersurfaces embedded in space-time, foliation of space-time by a family of space-like hypersurfaces), and then turns to the 3+1 decomposition of the Einstein equations, giving rise to the Cauchy problem with constraints, which constitutes the core of 3+1 formalism. The ADM Hamiltonian formulation of general relativity is also introduced at this stage. Finally, the decomposition of the matter and electromagnetic field equations is presented, focusing on the astrophysically relevant cases of a perfect fluid and a perfect conductor (ideal magnetohydrodynamics). The second part of the book introduces more advanced topics: the conformal transformation of the 3-metric on each hypersurface and the corresponding rewriting of the 3+1 Einstein equations, the Isenberg-Wilson-Mathews approximation to general relativity, global quantities associated with asymptotic flatness (ADM mass, linear and angular momentum) and with symmetries (Komar mass and angular momentum). In the last part, the initial data problem is studied, the choice of spacetime coordinates within the 3+1 framework is discussed and various schemes for the time integration of the 3+1 Einstein equations are reviewed. "

"Summary:"First published in 1916; English edition first published in the United Kingdom 1920 by Methuen & Co. Ltd; First published by Routledge 1993.""

"Aimed at students and researchers entering the field, this pedagogical introduction to numerical relativity will also interest scientists seeking a broad survey of its challenges and achievements. --Assuming only a basic knowledge of classical general relativity, this textbook develops the mathematical formalism from first principles, then highlights some of the pioneering simulations involving black holes and neutron stars, gravitational collapse and gravitational waves. --Applications include calculations of coalescing binary black holes and binary neutron stars, rotating stars, colliding star clusters, gravitational and magnetorotational collapse, critical phenomena, the generation of gravitational waves, and many more. --"...an excellent introduction...covering both mathematical aspects and numerical techniques... The authors are world leaders in numerical relativity...This book will soon become the standard advanced text for younger researchers entering the field and will also serve as the authoritative reference for senior researchers..." Abhay Ashtekhar, Director, Institute for Gravitation and the Cosmos, The Pennsylvania State University --"Numerical relativity has come of age...Two leading experts give a lucid as well as richly detailed account building a bridge from the basics to current research-highly recommended." Bernd Brugmann, Friedrich-Schiller-Universitot --"...what a book this is!...a genuine learning manual...exceedingly well written...covers virtually all aspects of numerical relativity...replete with beautiful and helpful diagrams...a useful reference to the researcher and a source of enlightenment to many a student...." Eric Poisson, University of Guelph --"Baumgarte and Shapiro are established leaders in this subject. Their book is a timely contribution to the literature, and the ideal primer for researchers newly attracted to the burgeoning field of computational relativity." Martin Rees, Astronomer Royal and Master of Trinity College, Cambridge --Book Jacket."

"The general theory of relativity : a mathematical exposition will serve readers as a modern mathematical introduction to the general theory of relativity. Throughout the book, examples, worked-out problems, and exercises (with hints and solutions) are furnished. Topics in this book include, but are not limited to tensor analysis, the special theory of relativity, the general theory of relativity and Einstein’s field equations, spherically symmetric solutions and experimental confirmations, static and stationary space-time domains, black holes, cosmological models, algebraic classifications and the Newman Penrose equations, and the coupled Einstein-Maxwell-Klein-Gordon equations appendices covering mathematical supplements and special topics."