The Evolution Problem in General Relativity: 25 (Progress in Mathematical Physics, 25) - Hardcover

Klainerman, Sergiu; Nicolo, Francesco

Hardcover

ISBN 10: 0817642544 ISBN 13: 9780817642549

Publisher: Birkhäuser, 2002

Synopsis

The main goal of this work is to revisit the proof of the global stability of Minkowski space by D. Christodoulou and S. Klainerman, [Ch-KI]. We provide a new self-contained proof of the main part of that result, which concerns the full solution of the radiation problem in vacuum, for arbitrary asymptotically flat initial data sets. This can also be interpreted as a proof of the global stability of the external region of Schwarzschild spacetime. The proof, which is a significant modification of the arguments in [Ch-Kl], is based on a double null foliation of spacetime instead of the mixed null-maximal foliation used in [Ch-Kl]. This approach is more naturally adapted to the radiation features of the Einstein equations and leads to important technical simplifications. In the first chapter we review some basic notions of differential geometry that are sys­ tematically used in all the remaining chapters. We then introduce the Einstein equations and the initial data sets and discuss some of the basic features of the initial value problem in general relativity. We shall review, without proofs, well-established results concerning local and global existence and uniqueness and formulate our main result. The second chapter provides the technical motivation for the proof of our main theorem.

About this title

Review

"The book . . . gives a new proof of the central part of the theorem of Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space . . . The authors prove, working in terms of double null foliations, a nonlinear stability, or global existence for small data, result for exterior domains."

―Mathematical Reviews

"...Important results in this book are presented in a more ‘digestible’ form [than] in the preceding book [‘The global nonlinear stability of the Minkowski space’] and thus scientists and graduate students working in relativity are recommended to read at least the introduction and the conclusions."

―Applications Of Mathematics

"...This important monograph, presenting the detailed proof of an important result in general relativity, is of great interest to researchers and graduate students in mathematics, mathematical physics, and physics in the area of general relativity."

―Studia Universitatis Babes-Bolyai, Series Mathematica

"The main purpose of this book is to revisit the global stability of Minkowski space as set out by D. Chrostodoulou and S. Klainerman (1993). Here the authors provide a new self-contained proof of the main part of that result, which concerns the full solution of the radiation problem in vacuum, for arbitrary asymptotically flat initial data sets."

―BookNews 

Synopsis

This monograph examines the global aspects of the problem of evolution equations in general relativity. Central to the text is a new self-contained proof of an extremely important concept: the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). The text focuses on a new self-contained proof of the main part of that result which concerns the full solution of the radiation problem in vacuum for arbitrary asymptotic flat initial data sets. While technical motivation is clearly and systematically provided for this proof, many related concepts and results, some well-established, others new, unfold along the way. Features of the work include: a presentation in chapter one of the basic notions of the differential geometry used throughout the text; methods introduced for proving global existence results, stressing the role of symmetries; the concept of double null foliation of spacetime; full decay estimates for the electromagnetic and Weyl fields; and two chapters on the Cauchy problem in general relativity.

Principal topics include: introduction of the Einstein vacuum equations and initial data sets; basic features of the initial value problem in general relativity; and review of local and global existence results and uniqueness.

Edition details

• Publisher: Birkhäuser
• Publication date: 2002
• Language: English
• ISBN 10: 0817642544
• ISBN 13: 9780817642549
• Binding: Hardcover
• Number of pages: 400