Electrifying Gowdy

In a previous post I wrote about the polarized Gowdy equations. If the condition of polarization is dropped the full Gowdy equations are obtained. They form a system of two semilinear wave equations:

\displaystyle P_{tt}+t^{-1}P_t-P_{\theta\theta}=e^{2P}(Q_t^2-Q_\theta^2),

\displaystyle Q_{tt}+t^{-1}Q_t-Q_{\theta\theta}=-2(P_tQ_t-P_\theta Q_\theta).

These equations are very similar to those for a wave map from two-dimensional Minkowski space to the hyperbolic plane with metric dP^2+e^{2P}dQ^2 with the difference of the singular term involving t^{-1}. (It fact it is known that these equations can be given a formulation in terms of a wave map. The relevant wave map is on the domain with metric -dt^2+d\theta^2+t^2 dx^2 and is asssumed to be independent of the additional coordinate x.) The boundary conditions are as in the polarized case. Given initial data for t=t_0 there exists a unique solution on the time interval (0,\infty), as was proved by Vincent Moncrief in 1981.

The asymptotics for t\to 0 is significantly more complicated than in the polarized case. There is a large class of solutions with rather simple asymptotics. Their asymptotic form in the limit t\to 0 is:

P(t,\theta)=-k(\theta)\log t+\phi(\theta)+\ldots,Q(t,\theta)=q(\theta)+t^{2k(\theta)}\psi(\theta)+\ldots.

In 1998 Satyanad Kichenassamy and I proved the existence of solutions of this type for prescribed functions k, \phi, q and \psi, subject to the condition 0<k(\theta)<1. I will say no more about he positivity condition on k here but instead concentrate on the inequality k<1. The quantity k is sometimes called the asymptotic velocity and the condition k<1 the low velocity condition. It turns out that this is not just a technical restriction. It has been shown by Hans Ringström that the quantity -tP_t(t,\theta) always has a limit as t\to 0. Call its modulus the asymptotic velocity v_\infty (\theta). In the low velocity case described above it coincides with k(\theta). More generally, when it is allowed to exceed one other phenomena occur. They are associated to the phenomenon of spikes. At a low velocity point P_\theta/P converges to a smooth limit as t\to 0. A spike is a value of \theta where P_\theta/P becomes unbounded. In general v_\infty has a discontinuity at a point of this type. These phenomena have been analysed in detail under a genericity assumption by Ringström. He used these results in his proof of strong cosmic censorship for Gowdy spacetimes. There are also subtle phenomena which occur in the limit t\to\infty. (For more information about many of the results mentioned in this post see the papers on Ringström’s web page.)

The Gowdy spacetimes are solutions of the Einstein vacuum equations. They can be generalized by adding an electromagnetic field. Recently Ernesto Nungesser completed his diploma thesis under my supervision on the subject of these electromagnetic generalizations of Gowdy solutions. He and I just produced a paper where the results of his thesis are described and extended somewhat. One key element of this work is that there is a class of polarized ‘electro-Gowdy’ solutions which can be defined by symmetry conditions. Another is that there is a change of variables which transforms the polarized electro-Gowdy equations to the full (i.e. non-polarized) Gowdy equations. This allows the extensive results available on Gowdy solutions to be applied to the electromagnetic case. On the analytical level nothing happens but the geometrical interpretation of the variables is different in the two cases. It turns out that in this way strong cosmic censorship can be proved for polarized electro-Gowdy spacetimes. As discussed in the paper, much less is known in the non-polarized electro-Gowdy case .

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