In Gowdy spacetimes only a finite number of bounces are to be expected and the behaviour is eventually monotone (no more bounces). There is only one essential spatial dimension due to the symmetry and so large derivatives in general occur at isolated values of the one interesting spatial coordinate. Of course, these correspond to surfaces in space when the symmetry directions are restored. The existence of Gowdy solutions showing features of this kind has been proved in . This was done by means of an explicit transformation that makes use of the symmetry. Techniques should be developed which can handle this type of phenomenon more directly and more generally.
The formation of spatial structure calls the BKL picture into question (cf. the remarks in ). The basic assumption underlying the BKL analysis is that spatial derivatives do not become too large near the singularity. Following the argument to its logical conclusion then indicates that spatial derivatives do become large near a dense set of points on the initial singularity. Given that the BKL picture has given so many correct insights, the hope that it may be generally applicable should not be abandoned too quickly. However, the problem represented by the formation of spatial structure shows that at the very least it is necessary to think carefully about the sense in which the BKL picture could provide a good approximation to the structure of general spacetime singularities.
|Theorems on Existence and Global Dynamics for the
Alan D. Rendall
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