Fuchsian analysis of S2 × S1 and S3 Gowdy spacetimes

Abstract

The Gowdy spacetimes are vacuum solutions of the Einstein equations with two commuting Killing vectors having compact spacelike orbits with T3, S2 × S1 or S3 topology. In the case of T3 topology, Kichenassamy and Rendall have found a family of singular solutions which are asymptotically velocity dominated by construction. In the case when the velocity is between 0 and 1, the solutions depend on the maximal number of free functions. We consider the similar case with S2 × S1 or S3 topology, where the main complication is the presence of symmetry axes. The results for T3 may be applied locally except at the axes, where one of the Killing vectors degenerates. We use Fuchsian techniques to show the existence of singular solutions similar to the T3 case. We first solve the analytic case and then generalize to the smooth case by approximating smooth data with a sequence of analytic data. However, for the metric to be smooth at the axes, the velocity must be −1 or 3 there, which is outside the range where the constructed solutions depend on the full number of free functions. A plausible explanation is that in general a spiky feature may develop at the axis, a situation which is unsuitable for a direct treatment by Fuchsian methods.