The Evolution Problem in General Relativity

This monograph examines the global aspects of the problem of evolution equations in general relativity. Central to the text is a new self-contained proof of an extremely important concept: the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). The text 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 related concepts and results, some well-established, others new, unfold along the way. Features of the work include: a presentation in chapter one of the basic notions of the differential geometry used throughout the text; methods introduced for proving global existence results, stressing the role of symmetries; the concept of double null foliation of spacetime; full decay estimates for the electromagnetic and Weyl fields; and two chapters on the Cauchy problem in general relativity. Principal topics include: introduction of the Einstein vacuum equations and initial data sets; basic features of the initial value problem in general relativity; and review of local and global existence results and uniqueness.