Evolution Problem in General Relativity

"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

• 1 Introduction.- 1.1 Generalities about Lorentz manifolds.- 1.2 The Einstein equations.- 1.3 Local existence for Einstein’s vacuum equations.- 1.4 Appendix.- 2 Analytic Methods in the Study of the Initial Value Problem.- 2.1 Local and global existence for systems of nonlinear wave equations.- 2.2 Weyl fields and Bianchi equations in Minkowski spacetime.- 2.3 Global nonlinear stability of Minkowski spacetime.- 2.4 Structure of the work.- 3 Definitions and Results.- 3.1 Connection coefficients.- 3.2 Bianchi equations in an Einstein vacuum spacetime.- 3.3 Canonical double null foliation of the spacetime.- 3.4 Deformation tensors.- 3.5 The definitions of the fundamental norms.- 3.6 The initial data.- 3.7 The Main Theorem.- 4 Estimates for the Connection Coefficients.- 4.1 Preliminary results.- 4.2 Proof of Theorem Ml.- 4.3 Proof of Theorem 4.2.1 and estimates for the zero and first derivatives of the connection coefficents.- 4.4 Proof of Theorem 4.2.2 and estimates for the second derivatives of the connection coefficients.- 4.5 Proof of Theorem 4.2.3 and control of third derivatives of the connection coefficients.- 4.6 Rotation tensor estimates.- 4.7 Proof of Theorem M2 and estimates for the D norms of the rotation deformation tensors.- 4.8 Appendix.- 5 Estimates for the Riemann Curvature Tensor.- 5.1 Preliminary tools.- 5.2 Appendix.- 6 The Error Estimates.- 6.1 Definitions and prerequisites.- 6.3 The error terms ?2.- 6.4 Appendix.- 7 The Initial Hypersurface and the Last Slice.- 7.1 Initial hypersurface foliations.- 7.2 The initial hypersurface connection estimates.- 7.3 The last slice foliation.- 7.4 The last slice connection estimates.- 7.5 The last slice rotation deformation estimates.- 7.6 The extension argument.- 7.7 Appendix.- 8 Conclusions.- 8.1 The spacetimenull infinity.- 8.2 The behavior of the curvature tensor at the null-outgoing infinity.- 8.3 The behavior of the connection coefficients at the null-outgoing infinity..- 8.4 The null-outgoing infinity limit of the structure equations.- 8.5 The Bondi mass.- 8.6 Asymptotic behavior of null-outgoing hypersurfaces.- Reference.