This graduate-level, course-based text is devoted to the 3+1 formalism of general relativity, which also constitutes the theoretical foundations of numerical relativity. 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. The prerequisites are those of a basic general relativity course with calculations and derivations presented in detail, making this text complete and self-contained. Numerical techniques are not covered in this book.

This book introduces the modern field of 3+1 numerical relativity. The book has been written in a way as to be as self-contained as possible, and only assumes a basic knowledge of special relativity. Starting from a brief introduction to general relativity, it discusses the different concepts and tools necessary for the fully consistent numerical simulation of relativistic astrophysical systems, with strong and dynamical gravitational fields. Among the topics discussed in detail are the following: the initial data problem, hyperbolic reductions of the field equations, gauge conditions, the evolution of black hole space-times, relativistic hydrodynamics, gravitational wave extraction and numerical methods. There is also a final chapter with examples of some simple numerical space-times. The book is aimed at both graduate students and researchers in physics and astrophysics, and at those interested in relativistic astrophysics.

An accessible introductory textbook on general relativity, covering the theory's foundations, mathematical formalism and major applications.

The Tenth International Conference on General Relativity and Gravitation (GR10) was held from July 3 to July 8, 1983, in Padova, Italy. These Conferences take place every three years, under the auspices of the International Society on General Relativity and Gravitation, with the purpose of assessing the current research in the field, critically discussing the prog ress made and disclosing the points of paramount im portance which deserve further investigations. The Conference was attended by about 750 scientists active in the various subfields in which the current research on gravitation and general relativity is ar ticulated, and more than 450 communications were sub mitted. In order to fully exploit this great occur rence of experience and creative capacity, and to pro mote individual contributions to the collective know ledge, the Conference was given a structure of work shops on the most active topics and of general sessions in which the Conference was addressed by invited speakers on general reviews or recent major advance ments of the field. The individual communications were collected in a two-volume publication made available to the participants upon their arrival and widely distributed to Scientific Institutions and Research Centres.

This book is about the general theory of relativity which is concisely labeled as general relativity. The book is the result of a rather extensive view to the literature of this theory over most of its lifetime reflecting various stages of its development. The book contains 129 solved problems as well as 606 exercises whose detailed solutions are published in another book that accompanies the present book. The book also includes a detailed index and many cross references. The book can be used as an introduction to general relativity at undergraduate and graduate levels. Unlike most other books on general relativity which are mostly dedicated to the presentation, justification, application and validation of the formalism of the theory (and hence rather minor attention is usually paid to the interpretation and epistemology of the theory), this book is primarily interested in the interpretative and epistemological aspects of the theory.

This 2004 textbook fills a gap in the literature on general relativity by providing the advanced student with practical tools for the computation of many physically interesting quantities. The context is provided by the mathematical theory of black holes, one of the most elegant, successful, and relevant applications of general relativity. Among the topics discussed are congruencies of timelike and null geodesics, the embedding of spacelike, timelike and null hypersurfaces in spacetime, and the Lagrangian and Hamiltonian formulations of general relativity. Although the book is self-contained, it is not meant to serve as an introduction to general relativity. Instead, it is meant to help the reader acquire advanced skills and become a competent researcher in relativity and gravitational physics. The primary readership consists of graduate students in gravitational physics. It will also be a useful reference for more seasoned researchers working in this field.

Early Universe cosmology is an active area of research and cosmic inflation is a pillar of modern cosmology. Among predictions of inflation, observationally the most important one is the generation of cosmological perturbations from quantum vacuum fluctuations that source all inhomogeneous structures in the Universe, not to mention the large-scale structures such as clusters of galaxies.Cosmological perturbation theory is the basic tool to study the perturbations generated from inflation. There are a few different approaches to primordial cosmological perturbations. In the conventional approach one perturbs the field equations and after quantizing the perturbations by the use of the corresponding action, one calculates the power spectrum of cosmological observables. This approach extends to higher order perturbations such as bispectrum etc., but the analysis becomes increasingly difficult.The delta N formalism, the topic of this book, is an alternative approach. The novelty of this approach is that, under the condition that the scale of interest is very large so that the spatial derivatives may be ignored in the dynamics, it can be applied to all orders in perturbation theory and has a rigorous foundation in general relativity. Thanks to the fact that one can evaluate perturbations with only the knowledge of background solutions, it is proved to be much easier than the conventional approach in evaluating higher order effects in many cases.