On "asymptotically flat" space-times with $G_{2}$-invariant Cauchy surfaces

Abstract

In this paper we study space-times which evolve out of Cauchy data $(Σ,{}^3g,K)$ invariant under the action of a two-dimensional commutative Lie group. Moreover $(Σ,{}^3g,K)$ are assumed to satisfy certain completeness and asymptotic flatness conditions in spacelike directions. We show that asymptotic flatness and energy conditions exclude all topologies and group actions except for a cylindrically symmetric $R^3$, or a periodic identification thereof along the $z$-axis. We prove that asymptotic flatness, energy conditions and cylindrical symmetry exclude the existence of compact trapped surfaces. Finally we show that the recent results of Christodoulou and Tahvildar-Zadeh concerning global existence of a class of wave-maps imply that strong cosmic censorship holds in the class of asymptotically flat cylindrically symmetric electro-vacuum space-times.