Cosmological Singularities, Einstein Billiards and Lorentzian Kac-Moody Algebras

The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of BelinskiiKhalatnikov-Lifshitz and technically simplified by the use of the ArnowittDeser-Misner Hamiltonian formalism) that the asymptotic behaviour, as one approaches the singularity, of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. For certain Einstein-matter systems, notably for pure Einstein gravity in any spacetime dimension D and for the particular Einstein-matter systems arising in String theory, the billiard tables describing asymptotic cosmological behaviour are found to be identical to the Weyl chambers of some Lorentzian Kac-Moody algebras. In the case of the bosonic sector of supergravity in 11 dimensional spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac-Moody group E10, and there exists some evidence of a correspondence between the general solution of the Einstein-threeform system and a null geodesic in the infinite dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of E10.