Weakly regular T-2-symmetric spacetimes. The future causal geometry of Gowdy spacetimes

摘要

We investigate the future asymptotic behavior of Gowdy spacetimes on T-3, when the metric satisfies weak regularity conditions, so that the metric coefficients (in suitable coordinates) are only in the Sobolev space H-1 or have even weaker regularity. The authors recently introduced this class of spacetimes in the broader context of T-2-symmetric spacetimes and established the existence of a global foliation by space like hypersurfaces when the time function is chosen to be the area of the surfaces of symmetry. In the present paper, we identify the global causal geometry of these spacetimes and, in particular, establish that weakly regular Gowdy spacetimes are future timelike geodesically complete. This result extends a theorem by Ringstrom for metrics with sufficiently high regularity. We emphasize that our proof of the energy decay is based on an energy functional inspired by the Gowdy-to-Ernst transformation. In order to establish the geodesic completeness property, we prove a higher regularity property concerning the metric coefficients along timelike curves and we provide a novel analysis of the geodesic equation for Gowdy spacetimes, which does not require high-order regularity estimates. Even when sufficient regularity is assumed, our proof provides an alternative and shorter proof of the energy decay and of the geodesic completeness property for Gowdy spacetimes. (C) 2015 Elsevier Inc. All rights reserved.