Fractals and Symbolic Dynamics as Invariant Descriptors of Chaos in General Relativity

The study of dynamics in general relativity has been hampered by a lack of coordinate independent measures of chaos. Here I review a variety of invariant measures for quantifying chaotic dynamics in relativity that exploit the coordinate independence of fractal dimensions and symbolic entropies. 1 Time and chaos Historically, chaos theory was developed for Newtonian dynamics where time and space are absolute and the notion of a mechanical phase space is clear. In contrast, both space and time are dynamical and intermixed in general relativity. There is no such thing as the time direction. The fundamentally different role played by time in relativity and Newtonian mechanics manifests itself in the coordinate, or gauge, dependence of chaotic measures such as Lyapunov exponents 1. Lyapunov exponents quantify a system's sensitive dependence on initial conditions. If two initially close trajectories separate along a given eigendirection in phase space such that the separation ε(t) grows as ε(t) = ε 0 e λt , then λ represents the Lyapunov exponent along that direction. If λ > 0 for a set of trajectories with non-zero measure, the system is said to exhibit sensitive dependence on initial conditions with a characteristic chaotic, or Lyapunov, timescale T L = 1/λ. Unfortunately, this nice picture breaks down when applied to general relativity. Consider the allowed coordinate transformation t → ln τ. In terms of this time variable we find ε(τ) = ε 0 τ λ , which describes the standard power-law divergence of trajectories found in integrable system. In particular, the Lyapunov exponents in this coordinate system would all be zero. It should be mentioned that the Lyapunov exponents also depend on the choice of distance measure in phase space and are therefore variant under spatial coordinate transformations also. From the above discussion it is clear that standard coordinate dependent measures of chaos have to be either modified, abandoned or augmented in general relativity 2. This problem has now been solved in a series of papers 3,4,5 which introduced and illustrated the effectiveness of fractal dimensions and symbolic codings as invariant descriptors of chaos in general relativity. Central to both of these methods is the concept of a chaotic invariant set of orbits 6. It is interesting to note that the problem of describing chaos in quantum mechanics was also solved by focusing on periodic orbits 7. In this talk I review these developments and illustrate the methods by …