9 Further Results8 Structure of General Singularities8.2 Inhomogeneous solutions

8.3 Formation of localized structure 

Numerical calculations and heuristic methods such as those used by BKL lead to the conclusion that, as the singularity is approached, localized spatial structure will be formed. At any given spatial point the dynamics is approximated by that of a spatially homogeneous model near the singularity, and there will in general be bounces (cf. Section  8.1). However, there will be exceptional spatial points where the bounce fails to happen. This leads to a situation in which the spatial derivatives of the quantities describing the geometry blow up faster than these quantities themselves as the singularity is approached. In general spacetimes there will be infinitely many bounces before the singularity is reached, and so the points where the spatial derivatives are large will get more and more closely separated as the singularity is approached.

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 [221]. 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 [28]). 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.

9 Further Results8 Structure of General Singularities8.2 Inhomogeneous solutions

image Theorems on Existence and Global Dynamics for the Einstein Equations
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
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