Mixmaster universe: A chaotic Farey tale

Neil J. Cornish and Janna J. Levin
Phys. Rev. D 55, 7489 – Published 15 June 1997
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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 six-dimensional minisuperspace of the full Einstein equations and in a two-dimensional discretization 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.

  • Received 21 August 1996

DOI:https://doi.org/10.1103/PhysRevD.55.7489

©1997 American Physical Society

Authors & Affiliations

Neil J. Cornish

  • DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, England

Janna J. Levin

  • Center for Particle Astrophysics, UC Berkeley, 301 Le Conte Hall, Berkeley, California 94720-7304

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Issue

Vol. 55, Iss. 12 — 15 June 1997

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