1. Introduction

The Eimtcm equaiions are shown to exhibit rormal chaotic behaviour that can be characterized by invariants of non-linear dynamics. An overview of new concept' in dynamical systems theory is provided. The Mixmaster universe is studied as a dynamical system in an appropriate phase space, a Poincare retur, mapping is constructed for the system and a smooth invariant measure is calculated. Several dynamical invariants can then t>c calculated for the Mixmaster model, including its metric entropy. Various results in the metric theory of numbers are employed to calculate other aspects of the chaotic behaviour. Perturbations of the Mixmaster return mapping and the rate of approach to the equilibrium measure are also considered The Mixmaster model is shown to be a Bernoulli system and the Hamiltonian formulation of Misner used to display the connection tviwccn solutions to Einstein's equations and geodesic flows in hyperbolic Riemannian space. We describe the source of chaotic behaviour in the Mixmaster model, the classification of homogeneous solutions to the Einstein equations by reference to the presence of chaos, gravitational surbulence. universal behaviour in Einstein's equations and a possible description of Penrose's gravitational entropy.

Until only recently it was believed that the presence of random' behaviour in a deterministic mathematical system always derived from prescribing random initial data, introducing stochastic forcing, or exciting a very large number of degrees of freedom. A series of detailed studies have revealed that although any of these contingencies are sufficient prerequisites for the onset of random or chaotic behaviour in a dynamical system, none are in fact necessary [1]. Very simple dynamical systems, notably iterated maps of the unit interval [2], with regular initial data, no stochastic forcing and a minimal number of degrees of freedom, exhibit behaviour which is for all practical purposes completely unpredictable.

models which exhibit chaotic behaviour. We shall not be concerned with randomicity associated with either quantum effects or the presence cf (^auchv horizons [3] in space-time.

First, we should indicate how it is possible for a deterministic system to be unpredictable or 'chaotic': consider a simple difference equation [4] which rotates points around the circumference of a circle (fig. i);

 

 

This discrete mapping is completely deterministic. If we know the initial position if P at 0O precisely we will 2ho know its subsequent position precisely after any specified number of iterations. However, suppose we approach (i.l) from a more realistic or 'experimental' perspective. If (1.1) were describing i real physical system the n our initial condition, always possesses some small uncertainty. * 0. After the map has been iterated n times this small initial uncertainty will expand exponentially to fill a portion of the available phase space (circumference) of angular extent 86„ where, from (1.1)

 

 

For sufficiently large n any finite initial uncertainty, however small, will eventually expand under the action of the mapping to fill the entire phase space available to it. Although the mapping (I.I) is completely deterministic, any finite uncertainty in the initial datum will render the output complete^ unpredictable after a sufficient number of iterations because of the sensitive dependence on initial conditions.

The aim of this article is to show how the Einstein equations exhibit chaotic behaviour of this sort and derive a number of invariants that characterize the 'entropy' or complexity of the behaviour 5] One of the motivations for these investigations is Penrose's corjecture [6] 'hat the striking an atop between the relations governing black hole mechanics and the laws of equilibrium thermodynamics might extend into cosmology where non-stationary gravitational fields are encountered. Penrose suggested that some function of the Weyl conformal curvature might provide a viable candidate for the

gravitational entropy' of a cosmological space-time. This Weyl portion of the space-time curvature describes tidal' forces which, while stretching bodies in some directions, <quash them in others so no overall change in volume results. Some pieces of the Weyl curvature have no Newtonian counterpart. By contrast, the Ricci portion of the curvature is associated with a simultaneous crushing in all directions and is driven by the matter content of space-time. The matter-filled Friedmann Universe (3] displays only the Ricci part of the curvature whilst the vacuum solution of Kasner [7) contains only the Weyl curvature and, in fact, that piece which does possess a Newtonian analogue.

The motivation for establishing a link between some formal 'entropy' function an . a geometrical aspect of space-time like the Weyl curvature is clear: If such a link could be forged it would impose a natural thermodynamic boundary condition upon the initial dynamical state of the Universe. The Universe would expand away from a virtually homogeneous and isotropic, low entropy, initial state where the Ricci curvature dominates the Weyl curvature. As the Universe ages, inhomogeneities and anisotropics would develop and the gravitational entropy gradually increase. If the space-time were to terminate at a second singularity this would be a high entropy state, strongly irregular and dominated by the Weyl portion of the curvature. A significant time asymmetry would be associated with the overall evolution of the Universe.

We shall describe how dynamical entropies can be calculated for cosmological space-times [5]. They have the properties envisaged by Penrose and are ultimately related to the non-Newtonian portion of the Weyl curvature. It is possible to evaluate these invariant dynamical entropies for all the spatially homogeneous cosmological solutions to Einstein's equations.

The concept of entropy we shall be interested in was introduced into the study of dynamical systems b Kolmogorov [8] in 1958. It provides a measure of the degree of disordet in the phase space of trajectories traced out by solutions to the system. Whereas thermal entropies give a logarithmic measure of the number of configurations admitted by a system as the number of degrees of freedom becomes infinite so dynamical entropies give a measure of the rate of generation of distinct solution trajectories in a system over arbitrarily long time intervals.

These invariants, and the systems in which they are non-zero, have recently been subjected to intensive and extensive analysis by both physicists and applied mathematicians [2,9], Chaotic systems appear to manifest latent order of a universal character [10] which, when quantified, may lead to a rigorous and predictive theory of hydrodynamic turbulence. Our investigations can be viewed as a description of 'gravitational turbulence'-the complex behaviour manifested by the general solution of the F.instein equations [11].

In section 2 we shall intioduce some of the concepts from dynamical systems theory which we will need to analyse the Einstein equations. These will be applied to homogeneous cosmological models in sections 3 and 4 and the results discussed with reference to possible future work in sections 5 and 6.