3+1 Formalism in General Relativity

3+1 Formalism in General Relativity : Bases of Numerical Relativity

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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.
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Product details

  • Paperback | 294 pages
  • 155 x 235 x 22.86mm | 480g
  • Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Berlin, Germany
  • English
  • 2012
  • 29 Illustrations, black and white; XVII, 294 p. 29 illus.
  • 3642245242
  • 9783642245244
  • 1,205,133

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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.
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Table of contents

Basic Differential Geometry.- Geometry of Hypersurfaces.- Geometry of Foliations.- 3+1 decomposition of Einstein Equation.- 3+1 Equations for Matter and Electromagnetic Field.- Conformal Decompositon.- Asymptotic Flatness and Global Quantities.- The Initial Data Problem.- Choice of Foliation and Spatial Coordiinates.- Evolution Schemes.- Conformal Killing Operator and Conformal Vector Laplacian.- Sage Codes.
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Review Text

From the reviews:

"The monograph originating from lectures is devoted to the 3+1 formalism in general relativity. It starts with three chapters on basic differential geometry, the geometry of single hypersurfaces embedded in space-time, and the foliation of space-time by a family of spacelike hypersurfaces. ... With the attempt to make the text self-consistent and complete, the calculations are ... detailed such that the book is well suitable for undergraduate and graduate students." (Horst-Heino von Borzeszkowski, Zentralblatt MATH, Vol. 1254, 2013)

"This book is written for advanced students and researchers who wish to learn the mathematical foundations of various approaches that have been proposed to solve initial value problems (with constraints) for the Einstein equations numerically. ... Even for experts it may be useful, as it includes an extensive bibliography up to 2011." (Hans-Peter Künzle, Mathematical Reviews, January, 2013)
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Review quote

From the reviews:

"The monograph originating from lectures is devoted to the 3+1 formalism in general relativity. It starts with three chapters on basic differential geometry, the geometry of single hypersurfaces embedded in space-time, and the foliation of space-time by a family of spacelike hypersurfaces. ... With the attempt to make the text self-consistent and complete, the calculations are ... detailed such that the book is well suitable for undergraduate and graduate students." (Horst-Heino von Borzeszkowski, Zentralblatt MATH, Vol. 1254, 2013)

"This book is written for advanced students and researchers who wish to learn the mathematical foundations of various approaches that have been proposed to solve initial value problems (with constraints) for the Einstein equations numerically. ... Even for experts it may be useful, as it includes an extensive bibliography up to 2011." (Hans-Peter Kunzle, Mathematical Reviews, January, 2013)
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About Eric Gourgoulhon

Eric Gourgoulhon is Researcher at CNRS (directeur de recherche). His research focuses on black holes, neutron stars, gravitational waves, Einstein's equations and numerical relativity. He is an experienced lecturer in relativity and relativistic astrophysics.
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