Future Complete Vacuum Spacetimes
Abstract
In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of spatially compact variants of the $k=1$ FriedmannRobertsonWalker vacuum spacetime. We use a special gauge defined by constant mean curvature slicing and a spatial harmonic coordinate condition, and develop energy estimates through the use of the BelRobinson energy and its higher order generalizations. In addition to the smallness condition on the data, we need a topological constraint on the spatial manifold to exclude the possibility of a nontrivial moduli space of flat spacetime perturbations, since the latter could not be controlled by curvaturebased energies such as those of BelRobinson type. Our results also demonstrate causal geodesic completeness of the perturbed spacetimes (in the expanding direction) and establish precise rates of decay towards the background solution which serves as an attractor asymptotically.
 Publication:

The Einstein Equations and the Large Scale Behavior of Gravitational Fields
 Pub Date:
 2004
 DOI:
 10.48550/arXiv.grqc/0303045
 arXiv:
 arXiv:grqc/0303045
 Bibcode:
 2004eels.conf..299A
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 31 pages, latex, amsart