It has been realized long ago that spacetime singularities are generic in classical general relativity [91]. However, their exact nature is still far from being well understood. 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. Furthermore, analyzing general relativity close to such singularities also provides important information on the dynamics of gravity within the regime where it breaks down. Indeed, careful investigations of the field equations in this extreme regime has revealed interesting and unexpected symmetry properties of gravity.

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 spatial points dynamically decouple, i.e., the partial differential equations governing the dynamics of the spatial metric asymptotically reduce, as one goes to the singularity, to ordinary differential equations with respect to time (one set of ordinary differential equations per spatial point).
- 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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