The Evolution Problem in General Relativity

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.