Global dynamics of the mixmaster model

TL;DR

This paper rigorously analyzes the asymptotic behavior of vacuum Bianchi models near singularities, confirming the chaotic oscillations in types VIII and IX and clarifying conditions for unbounded curvature.

Contribution

It provides a mathematical proof of the asymptotic dynamics of Bianchi models, especially the oscillatory behavior of types VIII and IX near singularities, and links solutions to the BKL map.

Findings

Singularity is velocity dominated for types other than VIII and IX.

Unbounded Kretschmann scalar in most cases, except known smooth extensions.

Infinite oscillations occur near the singularity in certain solutions.

Abstract

The asymptotic behaviour of vacuum Bianchi models of class A near the initial singularity is studied, in an effort to confirm the standard picture arising from heuristic and numerical approaches by mathematical proofs. It is shown that for solutions of types other than VIII and IX the singularity is velocity dominated and that the Kretschmann scalar is unbounded there, except in the explicitly known cases where the spacetime can be smoothly extended through a Cauchy horizon. For types VIII and IX it is shown that there are at most two possibilities for the evolution. When the first possibility is realized, and if the spacetime is not one of the explicitly known solutions which can be smoothly extended through a Cauchy horizon, then there are infinitely many oscillations near the singularity and the Kretschmann scalar is unbounded there. The second possibility remains mysterious and it is left open whether it ever occurs. It is also shown that any finite sequence of distinct points generated by iterating the Belinskii-Khalatnikov-Lifschitz mapping can be realized approximately by a solution of the vacuum Einstein equations of Bianchi type IX.