In this second part of the book, we develop the billiard description of the BKL behavior. We show how, in the BKL limit, the equations of motion of systems containing gravity can be recast at each spatial point as equations of motion for a billiard ball moving in a region of hyperbolic space. The emerging billiards that capture the motion near a spacelike singularity are called “cosmological billiards.” We start with pure gravity in four space-time dimensions. The inclusion of matter fields, and the extension to higher dimensions, are considered in Chapter 6.

In most papers devoted to cosmological billiards, index conventions different from those used so far are adopted. These different conventions are that Greek indices are space-time indices while Latin indices are spatial indices. Thus, in four space-time dimensions, α, β, · · · = 0, 1, 2, 3, and a, b, · · · = 1, 2, 3. Furthermore, indices in non-coordinate frames are not distinguished with a bar from coordinate indices. In order to ease the reading of the cosmological billiard literature, we shall switch to these different conventions. No confusion should arise as the context is always clear.

The derivation of the billiard description of the BKL limit given in this second part of the book was originally worked out in a collaboration of the second author with Thibault Damour and Hermann Nicolai [45, 46, 47, 49, 51]. One of its main ingredients is the use of the Iwasawa decomposition of the metric for handling the influence of the off-diagonal metric components on the dynamics of the scale factors. Chapters 5 and 6 are largely based on the ideas developed in those works, and in particular draw very much from the review [51], from which we adopt in particular the notations.

Hamiltonian Form of the Action

The cosmological billiard picture is based on the Hamiltonian formalism, which we first briefly review. For more information, we refer the reader to the original papers [65, 5] as well as to the book [133], Chapter 21.