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3.1 Preliminary example: The BKL billiard (vacuum D = 4 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, D = 4 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” P GL (2,ℤ ), which, as we shall see, is isomorphic to the hyperbolic Coxeter group A++1.

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,

β2 − β1 > 0, β3 − β2 > 0 (3.1 )
(symmetry walls) and
1 2β > 0 (3.2 )
(curvature wall). The remarkable properties of this region from our point of view are:

These angles are captured in the matrix A = (Aij)i,j=1,2,3 of scalar products,

A = (α |α ), (3.4 ) ij i j
which reads explicitly
( 2 − 2 0) ( ) A = − 2 2 − 1 . (3.5 ) 0 − 1 2
Recall from the previous section that the scalar product of two linear forms F = F βi i and G = G βi i is, in a three-dimensional scale factor space,
∑ (∑ ) ( ∑ ) (F|G ) = F G − 1- F G , (3.6 ) i i 2 i i i i i
where we have taken α1(β ) ≡ 2β1, α2(β) ≡ β2 − β1 and α3(β) ≡ β3 − β2. The corresponding billiard region is drawn in Figure 1View Image.
View Image

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 1 2 1 α1(β) = 2β = 0,α2(β) = β − β = 0 and 3 2 α3(β ) = β − β = 0, as indicated by the numbering in the figure. The two symmetry walls α2(β) = 0 and α3(β ) = 0 intersect at an angle of π∕3, while the gravity wall α1(β ) = 0 intersects, respectively, at angles 0 and π∕2 with the symmetry walls α (β) = 0 2 and α (β) = 0 3. The particle has no direction of escape so the dynamics is chaotic.

Because the angles between the reflecting planes are integer submultiples of π, the reflections in the walls bounding the billiard region6,

(γ|αi ) si(γ) = γ − 2(α-|α-)αi = γ − (γ|αi)αi, (3.7 ) i i
obey the following relations,
2 3 s1s3 = s3s1 ↔ (s1s3) = 1, (s2s3) = 1. (3.8 )
The product s1s3 is a rotation by 2π∕2 = π and hence squares to one; the product s2s3 is a rotation by 2π ∕3 and hence its cube is equal to one. There is no power of the product s1s2 that is equal to one, something that one conventionally writes as
∞ (s1s2) = 1. (3.9 )

The group generated by the reflections s 1, s 2 and s 3 is denoted A++ 1, for reasons that will become clear in the following, and coincides with the arithmetic group P GL (2,ℤ), as we will now show (see also [75Jump To The Next Citation Point116Jump To The Next Citation Point107Jump To The Next Citation Point]).

3.1.2 On the group P GL (2,ℤ )

The group P GL (2,ℤ ) is defined as the group of 2 × 2 matrices C with integer entries and determinant equal to ±1, with the identification of C and − C,

GL-(2,ℤ-) P GL (2,ℤ ) = ℤ . (3.10 ) 2
Note that although elements of the real general linear group GL (2,ℝ ) have (non-vanishing) unrestricted determinants, the discrete subgroup GL (2,ℤ ) ⊂ GL (2,ℝ ) only allows for det C = ±1 in order for the inverse C− 1 to also be an element of GL (2,ℤ ).

There are two interesting realisations of PGL (2,ℤ) in terms of transformations in two dimensions:

An important subgroup of the group P GL (2,ℤ) is the group P SL (2,ℤ ) for which ad − cb = 1, also called the “modular group”. The translation T : z → z + 1 and the inversion S : z → − 1∕z are examples of modular transformations,

(1 1) (0 − 1) T = , S = . (3.15 ) 0 1 1 0
It is a classical result that any modular transformation can be written as the product
m1 m2 mk T ST S ⋅⋅⋅ST , (3.16 )
but the representation is not unique [4].

Let s 1, s 2 and s 3 be the P GL (2,ℤ )-transformations

s1 : z → −z¯, s2 : z → 1 − ¯z, (3.17 ) 1- s3 : z → ¯z,
to which there correspond the matrices
(1 0) (1 − 1) (0 1) s1 = , s2 = , s3 = (3.18 ) 0 − 1 0 − 1 1 0
The s i’s are reflections in the straight lines x = 0, x = 1∕2 and the unit circle |z| = 1, respectively. These are in fact just the transformations of hyperbolic space s1, s2 and s3 described in Section 3.1.1, since the reflection lines intersect at 0, 90 and 60 degrees, respectively.

One easily verifies that T = s2s1 and that S = s1s3 = s3s1. Since any transformation of P GL (2,ℤ ) not in P SL (2,ℤ) can be written as a transformation of PSL (2,ℤ ) times, say, s1 and since any transformation of PSL (2,ℤ ) can be written as a product of S’s and T’s, it follows that the group generated by the 3 reflections s1, s2 and s3 coincides with PGL (2, ℤ), as announced above. (Strictly speaking, P GL (2,ℤ ) 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 P GL (2,ℤ) and PSL (2,ℤ ) are drawn in Figure 2View Image. The equivalence between P GL (2,ℤ) and the Coxeter group ++ A 1 has been discussed previously in [75Jump To The Next Citation Point116Jump To The Next Citation Point107Jump To The Next Citation Point].

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Figure 2: The figure on the left hand side displays the action of the modular group P SL (2,ℤ ) on the complex upper half plane ℍ = {z ∈ ℂ |ℑz > 0}. The two generators of P SL (2,ℤ ) are S and T, acting as follows on the coordinate z ∈ ℍ : S(z) = − 1∕z; T(z) = z + 1, i.e., as an inversion and a translation, respectively. The shaded area indicates the fundamental domain 𝒟P SL (2,ℤ) = {z ∈ ℍ | − 1∕2 ≤ ℜz ≤ 1∕2; |z| ≥ 1} for the action of PSL (2,ℤ ) on ℍ. The figure on the right hand side displays the action of the “extended modular group” P GL (2,ℤ ) on ℍ. The generators of P GL (2,ℤ ) are obtained by augmenting the generators of P SL (2,ℤ ) with the generator s1, acting as s1(z) = − ¯z on ℍ. The additional two generators of P GL (2,ℤ) then become: s2 ≡ s1 ∘ T ; s3 ≡ s1 ∘ S, and their actions on ℍ are s2(z) = 1 − ¯z; s3(z ) = 1 ∕¯z. The new generator s1 corresponds to a reflection in the line ℜz = 0, the generator s2 is in turn a reflection in the line ℜz = 1∕2, while the generator s 3 is a reflection in the unit circle |z| = 1. The fundamental domain of P GL (2,ℤ ) is 𝒟P GL (2,ℤ) = {z ∈ ℍ |0 ≤ ℜz ≤ 1∕2; |z| ≥ 1}, corresponding to half the fundamental domain of P SL (2,ℤ ). The “walls” ℜz = 0,ℜz = 1∕2 and |z| = 1 correspond, respectively, to the gravity wall α1(β ) = 0, the symmetry wall α2(β ) = 0 and the symmetry wall α3(β ) = 0 of Figure 1View Image.

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