Synopses & Reviews
The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to the Einstein Vacuum Equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. The work contains a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions. It thus provides the mathematical framework in which we can give a rigorous derivation of the laws of gravitation proposed by Bondi. Moreover, it establishes other important conclusions concerning the nonlinear character of gravitational radiation. The authors obtain their solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the flat initial data set corresponding to the Minkowski space-time. They thus establish the global dynamic stability of the latter.
Originally published in 1994.
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Review
Winner of the 1999 Bocher Memorial Prize, American Mathematical Association
Review
"This book presents the authors' theorem on the stability of Minkowski space, a landmark in the development of mathematical relativity. The book is quite self-contained.... The book is not easy to read, due to the very technical nature of its contents, but under the circumstances the quality of the exposition is excellent."--Mathematical Reviews
Review
Winner of the 1999 Bocher Memorial Prize, American Mathematical Association
Table of Contents
| Acknowledgments | |
1 | Introduction | 1 |
I | Preliminary Results in 2-and 3-Dimensional Riemannian Geometry | |
2 | Generalized Hodge Systems in 2-D | 31 |
3 | General Results in 3-D Geometry | 53 |
4 | The Poisson Equation in 3-D | 78 |
5 | Curvature of an Initial Data Set | 110 |
6 | Deformation of 2-Surfaces in 3-D | 121 |
II | Bianchi Equations in Space-Time | |
7 | The Comparison Theorem | 135 |
8 | The Error Estimates | 205 |
III | Construction of Global Space-Times. Proof of the Main Theorem | |
9 | Construction of the Optical Function | 261 |
10 | Third Version of the Main Theorem | 284 |
11 | Second Fundamental Form | 311 |
12 | The Lapse Function | 341 |
13 | Derivatives of the Optical Function | 351 |
14 | The Last Slice | 411 |
15 | The Matching | 443 |
16 | The Rotation Vectorfields | 466 |
17 | Conclusions | 491 |
| Bibliography | 513 |