The Lyapunov characteristic exponent of two nearby trajectories, and , is defined by
where is a solution of the equations of motion in the phase space, is a connecting vector, is its length in the phase space and is an initial separation (the value of must be small; otherwise it is arbitrary and one usually scales to unity). The Kerr spacetime being stationary, we can measure the separations on the surfaces. One defines as the maximum value of with respect to variations of w and characterizes the chaoticity of the system by the value of . Positive values of indicate that neighbouring trajectories diverge exponentially in the course of time while negative corresponds only to polynomial divergence. The above definitions have been originally introduced within the framework of non-relativistic systems but they are directly applicable also to stationary systems in general relativity.
The maximum Lyapunov characteristic exponent is frequently determined numerically. This approach requires a careful choice of the integration scheme. In order to keep computational errors under control, one lets the trajectories evolve for a short interval of time, , after which w is rescaled back to unity (one denotes the norm of the connecting vector at the moment of the k-th rescaling). One can show (Benettin 1984) that the Lyapunov characteristic exponent corresponding to the original is given by
independently of the value of In addition, for almost all Again, one can conveniently set .
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999