COSMOLOGICAL SINGULARITIES, EINSTEIN BILLIARDS AND LORENTZIAN KAC–MOODY ALGEBRAS

Abstract

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 Belinskii–Khalatnikov–Lifshitz and technically simplified by the use of the Arnowitt–Deser–Misner Hamiltonian formalism) that the asymptotic behavior, 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 behavior 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-three-form system and a null geodesic in the infinite dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of E10.