### 2.4 Dynamics as a billiard in hyperbolic space

The second step in the BKL-limit is to take the sharp wall limit of the potentials. This leads to the billiard picture. It is crucial here that the coefficients in front of the dominant walls are all positive. Again, just as for the first step, this limit has not been fully justified. Only heuristic, albeit convincing, arguments have been put forward.

The idea is that as one goes to the singularity, the exponential potentials get sharper and sharper and can be replaced in the limit by the corresponding -function, denoted for short and defined by for and for . Taking into account the facts that for all , as well as that some walls can be neglected, one finds that the Hamiltonian becomes in the sharp wall limit

with

The description of the motion of the scale factors (at each spatial point) is easy to give in that limit. Because the potential walls are infinite (and positive), the motion is constrained to the region where the arguments of all -functions are negative, i.e., to

In that region, the motion is governed by the kinetic term , i.e., is a geodesic for the metric in the space of the scale factors. Since that metric is flat, this is a straight line. In addition, the constraint , which reduces to away from the potential walls, forces the straight line to be null. We shall assume that the time orientation in the space of the scale factors is such that the straight line is future-oriented ( in the future).

It is easy to check that all the walls appearing in Equation (2.41), collectively denoted , are timelike hyperplanes. This is because the squared norms of all the ’s are positive,

Explicitly, one finds

Because the potential walls are timelike, they have a non-empty intersection with the forward light cone in the space of the scale factors. When the null straight line representing the evolution of the scale factors hits one of the walls, it gets reflected according to the rule [43]

where is the velocity vector (tangent to the straight line). This reflection preserves the time orientation since the hyperplanes are timelike and hence belong to the orthochronous Lorentz group where or according to whether there is no or one dilaton. The conditions define the “symmetry” or “centrifugal” walls, the conditions define the “curvature” or “gravitational” walls, the conditions define the “electric” walls, while the conditions define the “magnetic” walls.

The motion is thus a succession of future-oriented null straight line segments interrupted by reflections against the walls, where the motion undergoes a reflection belonging to . Whether the collisions eventually stop or continue forever is better visualized by projecting the motion radially on the positive sheet of the unit hyperboloid, as was done first in the pioneering work of Chitre and Misner [31138] for pure gravity in four spacetime dimensions. We recall that the positive sheet of the unit hyperboloid , , provides a model of hyperbolic space (see, e.g., [146]).

The intersection of a timelike hyperplane with the unit hyperboloid defines a hyperplane in hyperbolic space. The region in hyperbolic space on the positive side of all hyperplanes is the allowed dynamical region and is called the “billiard table”. It is never compact in the cases relevant to gravity, but it may or may not have finite volume. The projection of the motion of the scale factors on the unit hyperboloid is the same as the motion of a billiard ball in a hyperbolic billiard: geodesic arcs in hyperbolic space within the billiard region, interrupted by collisions against the bounding walls where the motion undergoes a specular reflection.

When the volume of the billiard table is finite, the collisions with the potential walls never end (for generic initial data) and the motion is chaotic. When, on the other hand, the volume is infinite, generic initial data lead to a motion that ultimately freely runs away to infinity. This is non-chaotic. For more information, see [135170]. An interesting criterion for chaos (equivalent to finite volume of hyperbolic billiard region) has been given in [111] in terms of illuminations of spheres by point sources.