Francesco Nicolò

The Evolution Problem in General Relativity

Preface * Introduction * Analytic methods in the initial value problem * Definitions and results * Estimates for the connection coefficients * Estimates for the curvature tensor * The error estimates * The initial hypersurface and the last slice * Conclusions * Bibliography * Index

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.

Peeling properties of asymptotically flat solutions to the Einstein vacuum equations

We show that, under stronger asymptotic decay and regularity properties than those used in Christodoulou D and Klainerman S (1993 The Global Non Linear Stability of the Minkowski Space (Princeton Mathematical Series vol 41) (Princeton, NJ: Princeton University Press)) and Klainerman S and Nicolo F (2003 The Evolution Problem in General Relativity (Progress in Mathematical Physics vol 25) (Boston: Birkhauser)), asymptotically flat initial datasets lead to solutions of the Einstein vacuum equations which have strong peeling properties consistent with the predictions of the conformal compactification approach of Penrose. More precisely we provide a systematic picture of the relationship between various asymptotic properties of the initial datasets and the peeling properties of the corresponding solutions.

The Error Estimates

In this chapter we assume the spacetime K is foliated by a double null canonical foliation that satisfies the assumptions n n

Estimates for the Riemann Curvature Tensor

O leqslant epsilon_0 ,,D leqslant epsilon_0 ,

The Initial Hypersurface and the Last Slice

n n(6.0.1) n nand we make use of the inequality proved in Theorem M7 n n

Existence of finitely perturbed Friedmann models via the Cauchy problem

R leqslant cQ_K^{frac{1} {2}} .