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Poincare, Relativity, Billiards and Symmetry

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      Abstract

      This review is made of two parts which are related to Poincar\'e in different ways. The first part reviews the work of Poincar\'e on the Theory of (Special) Relativity. One emphasizes both the remarkable achievements of Poincar\'e, and the fact that he never came close to what is the essential conceptual achievement of Einstein: changing the concept of time. The second part reviews a topic which probably would have appealed to Poincar\'e because it involves several mathematical structures he worked on: chaotic dynamics, discrete reflection groups, and Lobachevskii space. This topic is the hidden role of Kac-Moody algebras in the billiard description of the asymptotic behaviour of certain Einstein-matter systems near a cosmological singularity. Of particular interest are the Einstein-matter systems arising in the low-energy limit of superstring theory. These systems seem to exhibit the highest-rank hyperbolic Kac-Moody algebras, and notably E(10), as hidden symmetries.

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      E10 and a "small tension expansion" of M Theory

      A formal ``small tension'' expansion of D=11 supergravity near a spacelike singularity is shown to be equivalent, at least up to 30th order in height, to a null geodesic motion in the infinite dimensional coset space E10/K(E10) where K(E10) is the maximal compact subgroup of the hyperbolic Kac-Moody group E10(R). For the proof we make use of a novel decomposition of E10 into irreducible representations of its SL(10,R) subgroup. We explicitly show how to identify the first four rungs of the E10 coset fields with the values of geometric quantities constructed from D=11 supergravity fields and their spatial gradients taken at some comoving spatial point.
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        Chaos in the Mixmaster Universe

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          E_(10), BE_(10) and Arithmetical Chaos in Superstring Cosmology

          It is shown that the never ending oscillatory behaviour of the generic solution, near a cosmological singularity, of the massless bosonic sector of superstring theory can be described as a billiard motion within a simplex in 9-dimensional hyperbolic space. The Coxeter group of reflections of this billiard is discrete and is the Weyl group of the hyperbolic Kac-Moody algebra E\(_{10}\) (for type II) or BE\(_{10}\) (for type I or heterotic), which are both arithmetic. These results lead to a proof of the chaotic (``Anosov'') nature of the classical cosmological oscillations, and suggest a ``chaotic quantum billiard'' scenario of vacuum selection in string theory.
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            Author and article information

            Journal
            20 January 2005
            hep-th/0501168
            Custom metadata
            35 pages, 3 figures, invited talk given at the Solvay Symposium on Henri Poincare (ULB, Brussels, 8-9 October 2004)
            hep-th astro-ph gr-qc

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