Title: On the expanding direction of Gowdy vacuum spacetimes
Abstract:
We consider a class of spacetimes for which the essential part of
Einstein's equations can be written as a wave map equation. The
domain is not the standard one, but the target is hyperbolic space.
One ends up with a 1+1 non-linear wave equation, where the space
variable belongs to the circle and the time variable belongs to the
positive real numbers. In this talk, we discuss the asymptotics of
solutions to these equations as
time tends to infinity. For each point in time, the solution
defines a loop in hyperbolic space, and the first result is that
the length of this loop tends to zero as time tends to infinity.
In other words, the solution in some sense
becomes spatially homogeneous. However, the asymptotic behaviour need
not be similar to that of spatially homogeneous solutions to the
equations. The orbits of such solutions are either a point
or a geodesic in the hyperbolic plane.
In the non-homogeneous case, one gets the following
asymptotic behaviour in the
upper half plane (after applying an isometry of hyperbolic space if
necessary). i) The solution converges to a point. ii) The solution
converges to the origin on the boundary along a straight line (which
need not be perpendicular to the boundary). iii) The solution goes to
infinity along a curve y=const. iv) The solution oscillates
around a circle inside the upper half plane. Thus, even though the
solutions become spatially homogeneous in the sense that the spatial
variations die out, the asymptotic behaviour may be radically
different from anything observed for spatially homogeneous solutions
of the equations. This analysis can then be applied to draw
conclusions concerning the associated class of spacetimes. For
instance, one obtains the leading order behaviour of the functions
appearing in the metric, and one can conclude future causal geodesic
completeness.
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Ringstrm's lecture
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Copyright 29.VIII.02 by P.T.Chrusciel, A.Chopin, and H.Ringstrm