Ebook: The Evolution Problem in General Relativity

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.

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This monograph examines the global aspects of the problem of evolution equations in general relativity. Central to the work is a revisit of the proof of the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). This book 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 important related concepts and results, some well established, others new, unfold along the way. A comprehensive bibliography and index complete this important monograph, aimed at researchers and graduate students in mathematics, mathematical physics, and physics working in the area of general relativity.

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This monograph examines the global aspects of the problem of evolution equations in general relativity. Central to the work is a revisit of the proof of the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). This book 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 important related concepts and results, some well established, others new, unfold along the way. A comprehensive bibliography and index complete this important monograph, aimed at researchers and graduate students in mathematics, mathematical physics, and physics working in the area of general relativity.
Content:
Front Matter....Pages i-xii
Introdution....Pages 1-29
Analytic Methods in the Study of the Initial Value Problem....Pages 31-54
Definitions and Results....Pages 55-114
Estimates for the Connection Coefficients....Pages 115-202
Estimates for the Riemann Curvature Tensor....Pages 203-240
The Error Estimates....Pages 241-294
The Initial Hypersurface and the Last Slice....Pages 295-346
Conclusions....Pages 347-373
Back Matter....Pages 375-385

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This monograph examines the global aspects of the problem of evolution equations in general relativity. Central to the work is a revisit of the proof of the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). This book 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 important related concepts and results, some well established, others new, unfold along the way. A comprehensive bibliography and index complete this important monograph, aimed at researchers and graduate students in mathematics, mathematical physics, and physics working in the area of general relativity.
Content:
Front Matter....Pages i-xii
Introdution....Pages 1-29
Analytic Methods in the Study of the Initial Value Problem....Pages 31-54
Definitions and Results....Pages 55-114
Estimates for the Connection Coefficients....Pages 115-202
Estimates for the Riemann Curvature Tensor....Pages 203-240
The Error Estimates....Pages 241-294
The Initial Hypersurface and the Last Slice....Pages 295-346
Conclusions....Pages 347-373
Back Matter....Pages 375-385
....