The mixmaster universe: A chaotic Farey tale

TL;DR

This paper conclusively demonstrates that the mixmaster universe exhibits chaos by identifying a fractal strange repellor using observer-independent methods, resolving previous debates about its chaotic nature.

Contribution

The study introduces observer-independent, fractal techniques to prove the chaos in the mixmaster universe, providing clear, gauge-independent evidence of its chaotic dynamics.

Findings

Identification of a fractal strange repellor in the mixmaster universe.

Quantification of chaos via Lyapunov dimension, entropy, and multifractal analysis.

Confirmation that the mixmaster universe's chaos is gauge independent.

Abstract

When gravitational fields are at their strongest, the evolution of spacetime is thought to be highly erratic. Over the past decade debate has raged over whether this evolution can be classified as chaotic. The debate has centered on the homogeneous but anisotropic mixmaster universe. A definite resolution has been lacking as the techniques used to study the mixmaster dynamics yield observer dependent answers. Here we resolve the conflict by using observer independent, fractal methods. We prove the mixmaster universe is chaotic by exposing the fractal strange repellor that characterizes the dynamics. The repellor is laid bare in both the 6-dimensional minisuperspace of the full Einstein equations, and in a 2-dimensional discretisation of the dynamics. The chaos is encoded in a special set of numbers that form the irrational Farey tree. We quantify the chaos by calculating the strange repellor's Lyapunov dimension, topological entropy and multifractal dimensions. As all of these quantities are coordinate, or gauge independent, there is no longer any ambiguity--the mixmaster universe is indeed chaotic.