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

where counts the number of physical spatial dimensions (see Section 2.5). The role of all other “off-diagonal” variables in the theory is to interrupt the free-flight motion of the particle, by adding walls in that confine the motion to a limited region of scale factor space, namely a convex cone bounded by timelike hyperplanes. When projected onto the unit hyperboloid, this region defines a simplex in hyperbolic space which we refer to as the “billiard table”.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

The walls themselves are then defined by hyperplanes in this linear space, i.e., as linear forms , for which , where is the basis dual to . The pairing between a vector and a form is sometimes also denoted by , and for the two dual bases we have, of course, We therefore find that the walls can be written as linear forms in the scale factors: We call wall forms. With this interpretation they belong to the dual space , i.e., From Equation (5.16) we may conclude that the walls bounding the billiard are the hyperplanes through the origin in which are orthogonal to the vector with components .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 ,

is a Cartan matrix, i.e., besides having 2’s on its diagonal, which is rather obvious, it has as off-diagonal entries non-positive integers (with the property ). This Cartan matrix is of course symmetrizable since it derives from a scalar product.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,

which is the reflection relative to the hyperplane . This expression can be rewritten (see Equation (4.59)), or, in terms of the scale factor coordinates , This is precisely the billiard reflection Equation (2.45) found in Section 2.4.Thus, we have the following correspondence:

As we have also seen, the Kac–Moody algebra is Lorentzian since the signature of the metric Equation (5.12) is Lorentzian. This fact will be crucial in the analysis of subsequent sections and is due to the presence of gravity, where conformal rescalings of the metric define timelike directions in scale factor space.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,

where the index runs over all relevant walls. On the algebraic side, the fundamental Weyl chamber in is the closed convex (half) cone given by We see that the conditions defining are equivalent, upon examination of Equation (5.22), to the conditions defining the billiard table .We may therefore make the crucial identification

which means that the particle geodesic is confined to move within the fundamental Weyl chamber of . From the billiard analysis in Section 2 we know that the piecewise motion in scale-factor space is controlled by geometric reflections with respect to the walls . By comparing with the dominant wall forms and using the correspondence in Equation (5.22) we may further conclude that the geometric reflections of the coordinates are controlled by the Weyl group in the Cartan subalgebra 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

Just as the fundamental Weyl chamber in can be expressed in terms of the fundamental weights (see Equation (3.40)), the fundamental Weyl chamber in can be expressed in a similar fashion in terms of the fundamental coweights: As we have seen (Sections 3.5 and 4.8), hyperbolicity holds if and only if none of the fundamental weights are spacelike, for all .

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

By solving these equations for we deduce that the fundamental weights are where in the last step we have written the fundamental weights in the -basis. The norms may now be computed with the metric in root space and become We see that and are timelike and that is lightlike. Thus, the Weyl chamber is indeed contained inside the lightcone, the algebra is hyperbolic and the billiard is of finite volume, in agreement with what we already found [46].http://www.livingreviews.org/lrr-2008-1 |
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