Abstract
We review the intimate connection between (super-)gravity close to a spacelike singularity (the “BKL-limit”) and the theory of Lorentzian Kac-Moody algebras. 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. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. 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. 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. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.
Highlights
It has been realized long ago that spacetime singularities are generic in classical general relativity [91]
These connections appear for the cases at hand because in the BKL-limit, can the equations of motion be reformulated as dynamical equations for billiard motion in a region of hyperbolic space, and this region possesses unique features: It is the fundamental Weyl chamber of some Kac–Moody algebra
We thereby arrive at the following important result [45, 46, 48]: The dynamics of theories coupled to gravity can in the BKL-limit be mapped to a billiard motion in the Cartan subalgebra h of a Lorentzian Kac–Moody algebra g
Summary
It has been realized long ago that spacetime singularities are generic in classical general relativity [91]. In the late 1960’s, Belinskii, Khalatnikov and Lifshitz (“BKL”) [16] gave a general description of spacelike singularities in the context of the four-dimensional vacuum Einstein theory. 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:. The solution exhibits strong chaotic properties of the type investigated independently by Misner [137] and called “mixmaster behavior” This chaotic behavior is best seen in the hyperbolic billiard reformulation of the dynamics due to Chitre [31] and Misner [138] (for pure gravity in four spacetime dimensions)
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