### 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 [310]. This was done by means of an explicit
transformation that makes use of the symmetry.

The formation of spatial structure calls the BKL picture into question (cf. the remarks in [45]). 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. It should be kept in mind that the fact that certain derivatives become large
does not necessarily mean that they have a large effect on the dynamics (cf. the discussion
in [14]).