Fuchsian analysis of S^2xS^1 and S^3 Gowdy spacetimes

TL;DR

This paper applies Fuchsian techniques to analyze singular solutions of Gowdy spacetimes with S^2xS^1 and S^3 topologies, extending previous work on T^3 topology and addressing complications at symmetry axes.

Contribution

It demonstrates the existence of singular solutions for S^2xS^1 and S^3 Gowdy spacetimes using Fuchsian methods, including the analytic case and approximation to smooth data.

Findings

Existence of singular solutions with specified topologies.

Solutions depend on the full set of free functions in certain cases.

Smoothness at axes requires specific velocity conditions.

Abstract

The Gowdy spacetimes are vacuum solutions of Einstein's equations with two commuting Killing vectors having compact spacelike orbits with T^3, S^2xS^1 or S^3 topology. In the case of T^3 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 zero and one, the solutions depend on the maximal number of free functions. We consider the similar case with S^2xS^1 or S^3 topology, where the main complication is the presence of symmetry axes. We use Fuchsian techniques to show the existence of singular solutions similar to the T^3 case. We first solve the analytic case and then generalise 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.