Poincare, Relativity, Billiards and Symmetry

Abstract

This review is made of two parts which are related to Poincaré in different ways. The first part reviews the work of Poincaré on the Theory of (Special) Relativity. One emphasizes both the remarkable achievements of Poincaré, 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 Poincaré 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.

arXiv comment35 pages, 3 figures, invited talk given at the Solvay Symposium on Henri Poincare (ULB, Brussels, 8-9 October 2004)

References in corpus (4)

• The Dynamics of General Relativity
• E10 and SO(9,9) invariant supergravity
• IIB supergravity and E10
• Hyperbolic Kac Moody Algebras and Einstein Billiards

Cited by in corpus (7)

• Poincaré, the Dynamics of the Electron, and Relativity
• Integrability of Supergravity Billiards and the generalized Toda lattice equation
• The general pattern of Kac Moody extensions in supergravity and the issue of cosmic billiards
• Cosmic Billiards with Painted Walls in Non-Maximal Supergravities: a worked out example
• Kac--Moody groups and automorphic forms in low dimensional supergravity theories
• On the origin of Special Relativity
• Einstein's impact on the physics of the twentieth century