On local and global aspects of the Cauchy problem in general relativity

In this paper we review some of the recent mathematical progress concerning the initial value problem formulation of general relativity. It is not our intention, however, to give an exhaustive presentation of all recent results on this topic, but rather to discuss some of the most promising mathematical techniques, which have been advanced in connection with the general Cauchy problem, in the absence of any special symmetries. Moreover, for the sake of simplicity and coherence, we restrict ourselves to the Einstein vacuum equations in the asymptotically flat regime. Our main goal is to discuss the main mathematical methods behind the various local existence and uniqueness results, as well as those used in the proof of global nonlinear stability of the Minkowski space. We also present an outline of a somewhat different and more transparent approach obtained by the authors in collaboration with Christodoulou. This relies, instead of the maximal foliation used by Christodoulou and Klainerman, on a double-null foliation. The new approach is fully adapted to domains of dependence and thus allows one to provide, directly, without having to rely on interior estimates, a proof of stability of 'null infinity' for large asymptotically flat initial data sets.