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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.5 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 Θ ∞-function, denoted for short Θ and defined by Θ (x) = 0 for x < 0 and Θ (x) = + ∞ for x > 0. Taking into account the facts that aΘ (x) = Θ (x) for all a > 0, as well as that some walls can be neglected, one finds that the Hamiltonian becomes in the sharp wall limit

∫ d sharp H = d xℋ , (2.40 )
ℋsharp = K + ∑ Θ (− 2s (β )) + ∑ Θ(− 2α (β)) ji ijk ∑ i<j i⁄=j,i⁄=k,j∑⁄=k + Θ (− 2ei⋅⋅⋅i(β)) + Θ (− 2mi ⋅⋅⋅i (β )), (2.41 ) i <i< ⋅⋅⋅<i 1 p i <i< ⋅⋅⋅<i 1 p+1 1 2 p 1 2 p+1
where sji(β) = βj − βi. See [48Jump To The Next Citation Point] for more information.

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 Θ-functions are negative, i.e., to

sji(β ) ≥ 0 (i < j), αijk(β) ≥ 0, ei1⋅⋅⋅ip(β) ≥ 0, mi1 ⋅⋅⋅ip+1(β ) ≥ 0. (2.42 )
In that region, the motion is governed by the kinetic term K, 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 ℋ = 0, which reduces to K = 0 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 (g → 0 in the future).

It is easy to check that all the walls appearing in Equation (2.41View Equation), collectively denoted FA (β) ≡ FA μβμ = 0, are timelike hyperplanes. This is because the squared norms of all the FA’s are positive,

∑ ( )2 (∑ )2 ( )2 (FA |FA) = ∂FA-- − --1--- ∂FA-- + ∂FA-- > 0. (2.43 ) i ∂ βi d − 1 i ∂βi ∂φ
Explicitly, one finds
(sji|sji) = 2, (αijk|αijk) = 2, ( ) p(d − p − 1 ) λ(p) 2 (ei1⋅⋅⋅ip|ei1⋅⋅⋅ip) = ------------+ ------, (2.44 ) d − 1 ( 4 )2 p(d − p − 1 ) λ(p) (mi1⋅⋅⋅ip+1|mi1⋅⋅⋅ip+1) = ------------+ ------. d − 1 4

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]

v νF vμ → vμ − 2-----Aν-F μA, (2.45 ) (FA |FA)
where v 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 ↑ O (k,1) where k = d − 1 or d according to whether there is no or one dilaton. The conditions sji = 0 define the “symmetry” or “centrifugal” walls, the conditions α = 0 ijk define the “curvature” or “gravitational” walls, the conditions ei1⋅⋅⋅ip = 0 define the “electric” walls, while the conditions mi1⋅⋅⋅ip+1 = 0 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 O↑(k,1). 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 ∑ ∑ (βi)2 − ( βi)2 + φ2 = − 1, ∑ βi > 0, 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.


  1. The task of determining the billiard region is greatly simplified by the observation that some walls are behind others and are thus not relevant. For instance, it is clear that if β2 − β1 > 0 and β3 − β2 > 0 , then β3 − β1 > 0. Among the symmetry wall conditions, the only relevant ones are βi+1 − βi > 0, i = 1,2, ⋅⋅⋅ ,d − 1. Similarly, a wall of any given type can be written as a positive combination of the walls of the same type with smallest values of the indices i of the β’s and the symmetry walls (e.g., the electric wall condition 2 β > 0 for a 1-form with zero dilaton coupling can be written as β1 + (β2 − β1) > 0 and is thus a consequence of β1 > 0 and β2 − β1 > 0). Finally, one also verifies that in the presence of true p-forms (0 < p < d − 1), the gravitational walls are never relevant as they can be written as combinations of p-form walls with positive coefficients [49Jump To The Next Citation Point].
  2. It is interesting to determine the spatially homogeneous models that reproduce asymptotically the correct billiard limit. It is clear that in order to do so, homogeneous cosmological models need only contain the relevant walls. It is not necessary that they yield all the walls. Which homogeneity groups are acceptable depends on the system at hand. We list here a few examples. For vacuum gravity in four spacetime dimensions, the appropriate homogeneous models are the so-called Bianchi VIII or IX models. For vacuum gravity in higher dimensions, the structure constants of the homogeneity group must fulfill the conditions of [60] and the metric must include off-diagonal components (see also [58]). In the presence of a single p-form and no dilaton (0 < p < d − 1), the simplest (Abelian) homogeneity group can be taken [44Jump To The Next Citation Point].

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