6,5. Gravitational entropy

it has been found that A„ converges to the aperiodic limit A® in geometric progression as n °°

 

where £ is a constant which measures the rate at which turbulent behaviour sets in. Alternatively, if

then

In uie introduction (section 1) we discussed the possibility that the isolation of dynamical entropies for certain cosmological models might provide a starting p-int for the quantitative investiga'ion of the 'gravitational entropy' mooted by Penrose [6]. The metric and topological entropies we have deter­mined for the homogeneous universes possess the properties envisaged by Penrose: they are zero in the Friedmann model and if a closed universe were to begin in a state close to Friedmann and evolve towards a second singularity, that second singularity would exhibit chaotic, Mixmaster behaviour and the metric entropy of the second singularity would be non-zero. However, the metric entropy wouu only take on a non-zero value after the first cycle to cycle exchange occurred. It is also interestinf " note that if dynamical entropies are connected with 'gravitational entropy' then ore has to investigate very general space-times like Mixmaster before any hint of it is revealed. AH the simplest and most-studied isotropic and anisotropic universe have zero dynamical entropies.

If we denote the equilibrium distribution by /eq and describe the behaviour close to equilibrium by a

form

 

then differentiate (6 38) with respect to t (noting / (df/dt) dr = 0 by conservation of particles) and substitute (6.39), we obtain

Now when the evolution is followed for short times        this is approximate!) related to the

equilibrium entropy, Seq, by

 

An obvious question to pose about the dynamical entropy of the Mixmaster is the elation it bears to a thermal entropy. This has ! een considered by Chi.ikov [100]. Thermal entropy. S, provides a description of the statistical state of a physical system ,n terms of a distribution function f(x. *) in a phase volume f

 

 

 

 

 

Now. since the metric entropy is the spatial average of the mean exponential divergence rate of trajectories in phase space, tiie integral of the rate of exponential approach to equilibrium, h{x). introduced in (6.39) satisfies /iM - J h dF and so

 

 

The metric entropy corresponds to the rate of increase in the thermodynamic entropy, or, the rate of information loss under evolution from given initial conditions. In principle, it is possible to relate the metric entropy of a cosmological model like Mixmaster to the 'thermal' entropy of the distribution of gravitational waves that generate the dynamical behaviour. It is still not clear how these concepts are related to Hawking's black hole entropy [6|. However, the Mixmaster Universe may also provide a description of the general internal structure of a black hole near its central singularity. If so, the evolution of the Hawking entropy during the period when the horizon is non-spherical and non- stationary may be influenced by the internal Mixmaster behaviour. These internal configurations could determine the black hole entropy when it is non-stationary.