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

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.

arXiv commentLaTeX 2e, AMSLaTeX, a4wide, 25 pages

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• Smooth Gowdy symmetric generalized Taub-NUT solutions
• A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges
• Investigations of solutions of Einstein's field equations close to lambda-Taub-NUT
• A spectral solver for evolution problems with spatial S3-topology
• Fuchsian methods and spacetime singularities
• An exact smooth Gowdy-symmetric generalized Taub-NUT solution
• Self-gravitating fluid flows with Gowdy symmetry near cosmological singularities
• Wave equations with initial data on compact Cauchy horizons
• Non-genericity of the Nariai solutions: II. Investigations within the Gowdy class
• Localized big bang stability for the Einstein-scalar field equations
• Regularity of Cauchy horizons in S2xS1 Gowdy spacetimes
• A numerical algorithm for Fuchsian equations and fluid flows on cosmological spacetimes
• Stability of AVTD Behavior within the Polarized $T^2$-symmetric vacuum spacetimes
• Global properties of higher-dimensional cosmological spacetimes
• Stability of Asymptotic Behavior Within Polarised $T^2$-Symmetric Vacuum Solutions with Cosmological Constant