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 spacetime, 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.
Author(s): Christodoulou, D. and Klainerman, S.
Series: Martin Classical Lectures. New Series
Publisher: Princeton University Press
Year: 1993
Language: English
Pages: 514
Acknowledgments ......Page 7
1. Introduction ......Page 9
I. Preliminary Results in 2-and 3-Dimensional Riemannian Geometry ......Page 37
2 Generalized Hodge Systems in 2-D ......Page 39
2.1 Isoperimetric Inequality and Radius of Injectivity ......Page 40
2.2 The L^2 Theory for Hodge Systems ......Page 44
2.3 The L^p Theory ......Page 51
2.4 Proof of the Uniformization Theorem ......Page 55
3.1 Preliminaries ......Page 61
3.2 Sobolev and Poincaré Inequalities ......Page 64
3.3 The Action of SO(3) on (E, g) ......Page 74
4.1 Preliminaries ......Page 86
4.2 Degenerate and Nondegenerate L^2 Estimates ......Page 89
4.3 L^2 Estimates for the Principal Error Term ......Page 104
4.4 Symmetric Hodge Systems in 3-D ......Page 110
5 Curvature of an Initial Data Set ......Page 118
6 Deformation of 2-Surfaces in 3-D ......Page 129
II. Bianchi Equations in Space-Time ......Page 141
7.1 Preliminary Results ......Page 143
7.2 The Electric-Magnetic Decomposition ......Page 151
7.3 Null Decomposition of a Weyl Field ......Page 154
7.4 The Null-Structure Equations of a Space-Time ......Page 173
7.5 Ricci Coefficients and the Vectorfields K, S, T, and O ......Page 178
7.6 The Statement of the Comparison Theorem ......Page 188
7.7 Proof of the Comparison Theorem ......Page 190
8.1 Preliminaries ......Page 213
8.2 Statement of the Boundedness Theorem ......Page 230
8.3 Proof of the Theorem ......Page 237
III. Construction of Global Space-Times. Proof of the Main Theorem ......Page 267
9.1 Construction of the Exterior Optical Function ......Page 269
9.2 Interior Construction of the Optical Function ......Page 283
9.3 The Initial Cone Co ......Page 290
10.1 Basic Notations, Norms ......Page 292
10.2 Statement and Proof of the Main Theorem ......Page 306
11.1 Preliminaries ......Page 319
11.2 The Exterior Estimates ......Page 328
11.3 The Interior Estimates ......Page 340
11.4 Estimates for the Time Derivatives ......Page 345
12 The Lapse Function ......Page 349
13.1 Higher Derivatives of the Exterior Optical Function ......Page 359
13.2 Derivatives of the Interior Optical Function ......Page 394
14 The Last Slice ......Page 419
15 The Matching ......Page 451
16 The Rotation Vectorfields ......Page 474
16.1 Estimates in the Exterior ......Page 476
16.2 Estimates in the Interior ......Page 492
17 Conclusions ......Page 499
Bibliography ......Page 521
