  We solve for the cosmological perturbations in a five-dimensional background
consisting of two separating or colliding boundary branes, as an expansion in
the collision speed V divided by the speed of light c. Our solution permits a
detailed check of the validity of four-dimensional effective theory in the
vicinity of the event corresponding to the big crunch/big bang singularity. We
show that the four-dimensional description fails at the first nontrivial order
in (V/c)^2. At this order, there is nontrivial mixing of the two relevant
four-dimensional perturbation modes (the growing and decaying modes) as the
boundary branes move from the narrowly-separated limit described by
Kaluza-Klein theory to the well-separated limit where gravity is confined to
the positive-tension brane. We comment on the cosmological significance of the
result and compute other quantities of interest in five-dimensional
cosmological scenarios.
