### 2.5 Rules for deriving the wall forms from the Lagrangian – Summary

We have recalled above that the generic behavior near a spacelike singularity of the system with
action (2.1) can be described at each spatial point in terms of a billiard in hyperbolic space. The action for
the billiard ball reads, in the gauge ,
where we recall that in the BKL-limit (proper time ), and is the metric in the
space of the scale factors,
introduced in Equation (2.15) above. As stressed there, this metric is flat and of Lorentzian signature.
Between two collisions, the motion is a free, geodesic motion. The collisions with the walls are controlled by
the potential , which is a sum of sharp wall potentials. The walls are hyperplanes and can be
inferred from the Lagrangian. They are as follows:

- Gravity brings in the symmetry walls
with , and the curvature wall
- Each -form brings in an electric wall
and a magnetic wall

We have written here only the (potentially) relevant walls. There are other walls present in the
potential, but because these are behind the relevant walls, which are infinitely steep in the
BKL-limit, they are irrelevant. They are relevant, however, when trying to exhibit the symmetry in a
complete treatment where the BKL-limit is the zeroth order term in a gradient expansion yet to be
understood [47].

The scalar product dual to the scalar product in the space of the scale factors is

for two linear forms , .
These recipes are all that we shall need for investigating the regularity properties of the billiards
associated with the class of actions Equation (2.1).