![]() This important work has opened the way to many further fruitful investigations in theoretical cosmology. They provided convincing evidence that the generic solution of the dynamical Einstein equations, in the vicinity of a spacelike singularity, exhibits the following remarkable properties:Ĭosmological billiards and hidden symmetries of gravity In the late 1960’s, Belinskii, Khalatnikov and Lifshitz (“BKL”) gave a general description of spacelike singularities in the context of the four-dimensional vacuum Einstein theory. Indeed, careful investigations of the field equations in this extreme regime has revealed interesting and unexpected symmetry properties of gravity. Furthermore, analyzing general relativity close to such singularities also provides important information on the dynamics of gravity within the regime where it breaks down. Although it is expected that spacetime singularities will ultimately be resolved in a complete quantum theory of gravity, understanding their classical structure is likely to shed interesting light and insight into the nature of the mechanisms at play in the singularity resolution. However, their exact nature is still far from being well understood. It has been realized long ago that spacetime singularities are generic in classical general relativity. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity. An explicit example is provided for the case of the hyperbolic algebra E 10, which is conjectured to be an underlying symmetry of M-theory. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. ![]() Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. We review the intimate connection between (super-)gravity close to a spacelike singularity (the “BKL-limit”) and the theory of Lorentzian Kac-Moody algebras.
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