The Quantum Structure of Space and Time (293 pages)

78 The Quantum Structure of Space and Time

cussing the structure of generic cosmological singularities. The basic idea is that,
near a spacelike singularity, the time derivatives are expected to dominate over
spatial derivatives. More precisely, BKL found that spatial derivatives introduce
terms in the equations of motion for the metric which are similar to the “walls”
of a billiard table [3]. In an Hamiltonian formulation [4], [5] where one takes as
basic gravitational variables the logarithms of the diagonal components of the met-
ric, say pa, these walls are Toda-like potential walls, i.e. exponentials of linear
combinations of the p’s, say exp -2w(p), where w(p) = C, w,pa. To each wall is
therefore associated a certain linear form in p space, w(p) = Ca w,pa, and also a
corresponding hyperplane C, wapa = 0. Ref. [2] found that the set of leading walls

wi(p) entering the cosmological dynamics of SUGRAll or type-I1 string theories

could be identified with the Weyl chamber of the hyperbolic Kac-Moody algebra

El0 [6], i.e. the set of hyperplanes defined by the simple roots ai(h) of Elo. Here

h parametrizes a generic element of a Cartan subalgebra (CSA) of E10, and the
index i labels both the leading walls and the simple roots. [i takes r values, where
r denotes the rank of the considered Lie algebra. For Elo, T = 10.1 Let us also note
that, for Heterotic and type-I string theories, the cosmological billiard is the Weyl
chamber of another rank-10 hyperbolic Kac-Moody algebra, namely BElo.

The appearance of El0 in the BKL behaviour of SUGRAll revived an old sug-

gestion of B. Julia [7] about the possible role of El0 in a one-dimensional reduction

of SUGRA11. A posteriori, one can see the BKL behaviour as a kind of spon-

taneous reduction to one dimension (time) of a multidimensional theory. Note,

however, that it is essential to consider a generic inhomogeneous solution (instead

of a naively one-dimensionally reduced one) because the wall structure comes from

the sub-leading ( 8, << &) spatial derivatives.

Gradient expansion versus height expansion of the Elo/K(Elo) coset

model

Refs. [l, 81 went beyond the leading BKL analysis of Ref. [2] by including the

first three “layers” of spatial gradients modifying the zeroth-order free billiard dy-

namics defined by keeping only the time derivatives of the (diagonal) metric. This

gradient expansion [5] can be graded by counting how many leading wall forms wi(p)

are contained in the exponents of the sub-leading potential walls associated to these

higher-order spatial gradients. As further discussed below, it was then found that

this counting could be related (up to height 29 included) to the grading defined by

the height of the roots entering the Toda-like Hamiltonian walls of the dynamics

defined by the motion of a massless particle on the coset space Elo/K(Elo), with

action

Here, wSYm = !j(u + wT)(= P in [S]) is the symmetric part of the “velocity”

v = (dg/dt)g-’ of a group element g(t) running over Elo. The transpose operation