The polarized Gowdy equation

The polarized Gowdy solutions are the simplest solutions of the vacuum Einstein equations which are dynamical and not spatially homogeneous. Physically they represent a one-dimensional configuration of polarized gravitational waves propagating in a closed and otherwise empty universe. Here ‘closed’ means that space is compact. The simplest case is that in which space is a three-dimensional torus and here I will consider only that case. These solutions represent a simple model system for developing mathematical techniques for studying the Einstein equations. This system has accompanied me in my research for many years and now I want to take a minute to stand back and reflect on it.

The central equation involved is $P_{tt}+t^{-1}P_t-P_{\theta\theta}=0$. This equation is also known under the name Euler-Poisson-Darboux equation but this does not seem to have helped much in the study of Gowdy solutions. This is perhaps due to the fact that the side conditions (boundary and initial conditions) are different from those of interest in other applications. In the Gowdy case $P$ is assumed to be periodic in the spatial variable $\theta$ and initial data can be prescribed on a hypersurface $t=t_0$ for some positive real number $t_0$. The data are of the form $(P_0,P_1)$ where $P_0$ and $P_1$ are the restrictions of $P$ and $P_t$ to the initial hypersurface $t=t_0$. This is a linear hyperbolic equation and it follows from standard results that there is a unique smooth (i.e. $C^\infty$) solution corresponding to any smooth initial data set. Thus all solutions can be parametrized by data on this initial hypersurface.

The asymptotics of solutions in the limit $t\to 0$ are well understood. Any solution has an asymptotic expansion of the form $P(t,\theta)=-k(\theta)\log t+\omega(\theta)+R(t,\theta)$ where $R(t,\theta)=o(1)$ as $t\to 0$. Moreover $R$ has an asymptotic expansion of the form $R(t,\theta)\sim\sum_{j=1}^\infty (A_j(\theta)+B_j(\theta)\log t)t^j$. where all the coefficients $A_j$ and $B_j$ are determined uniquely in terms of $k$ and $\omega$. The asymptotic expansions may be differentiated term by term as often as desired. Conversely, given smooth functions $k$ and $\omega$ there is a solution $P$ for which the leading terms in the asymptotic expansion are exactly these functions. Thus $k$ and $\omega$ can be used to parametrize all solutions just as well as $P_0$ and $P_1$. After I wrote this I realized that published proofs of the statement about prescribing $k$ and $\omega$ apparently only cover the case where $k$ is everywhere positive. This restriction is not hard to remove using known techniques.

What about the limit $t\to\infty$? It was proved by Thomas Jurke that any solution has an asymptotic expansion of the form $P(t,\theta)=A\log t+B+t^{-1/2}\nu(t,\theta)+R(t,\theta)$ where $A$ and $B$ are constants, $\nu$ satisfies the flat space wave equation $\nu_{tt}=\nu_{\theta\theta}$ and $R(t,\theta)=O(t^{-3/2})$ as $t\to\infty$. It was proved by Hans Ringström that given constants $A$ and $B$ and a solution $\nu$ of the flat-space wave equation there is a unique solution for which the leading terms in the asymptotic expansion are given by exactly these objects. In this way a third parametrization of the solutions is obtained. To mirror what is known for the limit $t\to 0$ it would be good to have an asymptotic expansion for the remainder $R$ arising in the limit $t\to\infty$. An expansion of this type has apparently never been derived.

Supposing that a full expansion at late times had been obtained could it then be said that we knew essentially everything about solutions of the polarized Gowdy equations? I am not sure since I think there might still be some kind of intermediate asymptotics to be discovered.