Data at the moment of infinite expansion for polarized Gowdy

Abstract

In a recent paper by Thomas Jurke, it was proved that the asymptotic behaviour of a solution to the polarized Gowdy equation in the expanding direction is of the form αln t + β + t−1/2ν + O(t−3/2), where α and β are constants and ν is a solution to the standard wave equation with zero mean value. Furthermore, it was proved that α, β and ν uniquely determine the solution. Here we wish to point out that given α, β and ν with the above properties, one can construct a solution to the polarized Gowdy equation with the above asymptotics. In other words, we show that α, β and ν constitute data at the moment of infinite expansion. We then use this fact to make the observation that there are polarized Gowdy spacetimes such that in the areal time coordinate, the quotient of the maximum and the minimum of the mean curvature on a constant time hypersurface is unbounded as time tends to infinity.