100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

50 H. Nicolai3.1. BKL dynamics and gravitational billiardsA remarkable and most important development in theoretical cosmologywas the analysis of spacelike (cosmological) singularities in Einstein’s theoryby Belinskii, Khalatnikov and Lifshitz (abbreviated as ‘BKL’ in theremainder), and their discovery of chaotic oscillations of the spacetime metricnear the initial singular hypersurface; 47 see also 48,49,50,51,52 . There is alarge body of work on BKL cosmology, see 53,54,55 for recent reviews and extensionsof the original BKL results. In particular, there is now convincingevidence for the correctness of the basic BKL picture both from numericalanalyses (see e.g. 56,57 ) as well as from more rigorous work 58,59,60,61 . Ithas also been known for a long time that the chaotic oscillations of themetric near the singularity can be understood in terms of gravitationalbilliards, although there exist several different realizations of this description,cf. 51,54,55 and references therein. The one which we will adopt here,grew out of an attempt to extend the original BKL results to more generalmatter coupled systems, in particular those arising in superstring and Mtheory 62,63,24,64,65 . It is particularly well suited for describing the relationbetween the BKL analysis and the theory of indefinite Kac Moody algebras,which is our main focus here, and which we will explain in the followingsection. See also 66,67 for an alternative approach.We first summarize the basic picture, see 65 for a more detailed exposition.Our discussion will be mostly heuristic, and we shall make no attemptat rigorous proofs here (in fact, the BKL hypothesis has been rigorouslyproven only with very restrictive assumptions 59,57,60,61,68 , but there is sofar no proof of it in the general case). Quite generally, one considers a bigbang-likespace-time with an initial singular spacelike hypersurface ‘located’at time t = 0. It is then convenient to adopt a pseudo-Gaussian gauge forthe metric (we will leave the number of spatial dimensions d arbitrary forthe moment)ds 2 = −N 2 dt 2 + g ij dx i dx j (31)and to parametrize the spatial metric g ijdreibein, θ a (a one form) din terms of a frame field, ord∑g ij dx i ⊗ dx j = θ a ⊗ θ a (32)a=1d The summation convention is in force for the coordinate indices i, j, . . . , but suspendedfor frame indices a, b, . . . .