On the T
3-Gowdy Symmetric Einstein-Maxwell Equations |
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Authors: | Hans Ringström |
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Institution: | 1. Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1, D-14476, Golm, Germany 2. Institutionen f?r matematik, KTH, Lindstedtsv?gen 25, S-10044, Stockholm, Sweden
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Abstract: | Recently, progress has been made in the analysis of the expanding direction of Gowdy spacetimes. The purpose of the present
paper is to point out that some of the techniques used in the analysis can be applied to other problems. The essential equations
in the case of the Gowdy spacetimes can be considered as a special case of a wider class of variational problems. Here we
are interested in the asymptotic behaviour of solutions to this class of equations. Two particular members arise when considering
the T3-Gowdy symmetric Einstein-Maxwell equations and when considering T3-Gowdy symmetric IIB superstring cosmology. The main result concerns the rate of decay of a naturally defined energy. A subclass
of the variational problems can be interpreted as wave map equations, and in that case one gets the following picture. The
non-linear wave equations one ends up with have as a domain the positive real line in Cartesian product with the circle. For
each point in time, the wave map can thus be seen as a loop in some Riemannian manifold. As a consequence of the decay of
the energy mentioned above, the length of the loop converges to zero at a specific rate.
Communicated by Sergiu Klainerman
submitted 14/02/05, accepted 21/04/05 |
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Keywords: | |
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