Let us first briefly review some of the salient features encountered so far in the analysis. In the following
we denote by the Lorentzian “scale factor”-space (or
-space) in which the billiard
dynamics takes place. Recall that the metric in
, induced by the Einstein–Hilbert action, is a
flat Lorentzian metric, whose explicit form in terms of the (logarithmic) scale factors reads
One has, in fact, more than just the walls. The theory provides these walls with a specific normalization
through the Lagrangian, which is crucial for the connection to Kac–Moody algebras. Let us therefore
discuss in somewhat more detail the geometric properties of the wall system. The metric, Equation (5.12),
in scale factor space can be seen as an extension of a flat Euclidean metric in Cartesian coordinates, and
reflects the Lorentzian nature of the vector space
. In this space we may identify a pair of coordinates
with the components of a vector
, with respect to a basis
of
, such that
It is important to note that it is the wall forms that the theory provides, as arguments of the
exponentials in the potential, and not just the hyperplanes on which these forms vanish. The scalar
products between the wall forms are computed using the metric in the dual space
, whose explicit
form was given in Section 2.5,
The crucial additional observation is that (for the “interesting” theories) the matrix associated with
the relevant walls
,
For this reason, one can usefully identify the space of the scale factors with the Cartan subalgebra of
the Kac–Moody algebra
defined by
. In that identification, the wall forms become the simple
roots, which span the vector space
dual to the Cartan subalgebra. The rank
of the algebra is equal to the number of scale factors
, including the dilaton(s) if any
(
). This number is also equal to the number of walls since we assume the billiard to be a
simplex. So, both
and
run from
to
. The metric in
, Equation (5.12
),
can be identified with the invariant bilinear form of
, restricted to the Cartan subalgebra
. The scale factors
of
become then coordinates
on the Cartan subalgebra
.
The Weyl group of a Kac–Moody algebra has been defined first in the space as the group
of reflections in the walls orthogonal to the simple roots. Since the metric is non degenerate,
one can equivalently define by duality the Weyl group in the Cartan algebra
itself (see
Section 4.7). For each reflection
on
we associate a dual reflection
as follows,
Thus, we have the following correspondence:
As we have also seen, the Kac–Moody algebra We thereby arrive at the following important result [45, 46
, 48
]:
Let denote the region in scale factor space to which the billiard motion is confined,
We may therefore make the crucial identification
which means that the particle geodesic is confined to move within the fundamental Weyl chamber of
We have learned that the BKL dynamics is chaotic if and only if the billiard table is of finite volume when
projected onto the unit hyperboloid. From our discussion of hyperbolic Coxeter groups in Section 3.5, we
see that this feature is equivalent to hyperbolicity of the corresponding Kac–Moody algebra. This leads to
the crucial statement [45, 46
, 48
]:
As we have also discussed above, hyperbolicity can be rephrased in terms of the fundamental weights
defined as
Let us return once more to the example of pure four-dimensional gravity, i.e., the original “BKL
billiard”. We have already found in Section 3 that the three dominant wall forms give rise to
the Cartan matrix of the hyperbolic Kac–Moody algebra [46
, 48
]. Since the algebra is
hyperbolic, this theory exhibits chaotic behavior. In this example, we verify that the Weyl chamber is
indeed contained within the lightcone by computing explicitly the norms of the fundamental
weights.
It is convenient to first write the simple roots in the -basis as follows¿
Since the Cartan matrix of is symmetric, the relations defining the fundamental weights are
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