3.1 Preliminary example: The BKL billiard (vacuum gravity)
To illustrate the regularity of the gravitational billiards and motivate the mathematical developments
through an explicit example, we first compute in detail the billiard characterizing vacuum, gravity.
Since this corresponds to the case originally considered by BKL, we call it the “BKL billiard”. We show in
detail that the billiard reflections in this case are governed by the “extended modular group”
, which, as we shall see, is isomorphic to the hyperbolic Coxeter group .
3.1.1 Billiard reflections
There are three scale factors so that after radial projection on the unit hyperboloid, we get a billiard in
two-dimensional hyperbolic space. The billiard region is defined by the following relevant wall inequalities,
(symmetry walls) and
(curvature wall). The remarkable properties of this region from our point of view are:
- It is a triangle (i.e., a simplex in two dimensions) because even though we had to begin with 6
walls (3 symmetry walls and 3 curvature walls), only 3 of them are relevant.
- The walls intersect at angles that are integer submultiples of , i.e., of the form
where is an integer. The symmetry walls intersect indeed at sixty degrees () since the
scalar product of the corresponding linear forms (of norm squared equal to ) is , while the
gravitational wall makes angles of zero (, scalar product ) and ninety (, scalar
product ) degrees with the symmetry walls.
These angles are captured in the matrix of scalar products,
which reads explicitly
Recall from the previous section that the scalar product of two linear forms and is,
in a three-dimensional scale factor space,
where we have taken , and . The corresponding billiard
region is drawn in Figure 1.
||The BKL billiard of pure four-dimensional gravity. The figure represents the billiard region
projected onto the hyperbolic plane. The particle geodesic is confined to the fundamental region
enclosed by the three walls and , as
indicated by the numbering in the figure. The two symmetry walls and
intersect at an angle of , while the gravity wall intersects, respectively, at angles
and with the symmetry walls and . The particle has no direction
of escape so the dynamics is chaotic.
Because the angles between the reflecting planes are integer submultiples of , the reflections in the walls bounding the
obey the following relations,
The product is a rotation by and hence squares to one; the product is a rotation
by and hence its cube is equal to one. There is no power of the product that is equal to one,
something that one conventionally writes as
The group generated by the reflections , and is denoted , for reasons that will
become clear in the following, and coincides with the arithmetic group , as we will now show
(see also [75, 116, 107]).
3.1.2 On the group
The group is defined as the group of matrices with integer entries and
determinant equal to , with the identification of and ,
Note that although elements of the real general linear group have (non-vanishing) unrestricted
determinants, the discrete subgroup only allows for in order for the
inverse to also be an element of .
There are two interesting realisations of in terms of transformations in two
- One can view as the group of fractional transformations of the complex plane
Note that one gets the same transformation if is replaced by , as one should. It is an easy
exercise to verify that the action of when defined in this way maps the complex upper
onto itself whenever the determinant of is equal to . This is not the case, however,
- For this reason, it is convenient to consider alternatively the following action of ,
(), which does map the complex upper-half plane onto itself, i.e., which is such that
The transformation (3.14) is the composition of the identity with the transformation (3.11)
when , and of the complex conjugation transformation, with the
transformation (3.11) when . Because the coefficients , , , and are real,
commutes with and furthermore the map (3.11) (3.14) is a group isomorphism, so that
we can indeed either view the group as the group of fractional transformations (3.11), or
as the group of transformations (3.14).
An important subgroup of the group is the group for which , also
called the “modular group”. The translation and the inversion are
examples of modular transformations,
It is a classical result that any modular transformation can be written as the product
but the representation is not unique .
Let , and be the -transformations
to which there correspond the matrices
The ’s are reflections in the straight lines , and the unit circle ,
respectively. These are in fact just the transformations of hyperbolic space , and
described in Section 3.1.1, since the reflection lines intersect at , and degrees,
One easily verifies that and that . Since any transformation of
not in can be written as a transformation of times, say, and
since any transformation of can be written as a product of ’s and ’s, it follows that the
group generated by the 3 reflections , and coincides with , as announced
above. (Strictly speaking, could be a quotient of that group by some invariant
subgroup, but one may verify that the kernel of the homomorphism is trivial (see Section 3.2.5
below).) The fundamental domains for and are drawn in Figure 2. The
equivalence between and the Coxeter group has been discussed previously
in [75, 116, 107].
||The figure on the left hand side displays the action of the modular group
on the complex upper half plane . The two generators of
are and , acting as follows on the coordinate , i.e.,
as an inversion and a translation, respectively. The shaded area indicates the fundamental
domain for the action of on .
The figure on the right hand side displays the action of the “extended modular group”
on . The generators of are obtained by augmenting the generators
of with the generator , acting as on . The additional two
generators of then become: , and their actions on
are . The new generator corresponds to a reflection in the line
, the generator is in turn a reflection in the line , while the generator
is a reflection in the unit circle . The fundamental domain of is
, corresponding to half the fundamental domain of
. The “walls” and correspond, respectively, to the gravity
wall , the symmetry wall and the symmetry wall of Figure 1.