TOWARD AN INVARIANT MEASURE OF CHAOTIC BEHAVIOR IN GENERAL-RELATIVITY

The ideas presented by Szydlowski and Lapeta [Phys. Lett. A 148 (1990) 239] are developed in the context of general relativity. We show the ineffectiveness of the classical criteria of the chaotic behaviour (Lyapunov exponents) in general relativity and in its cosmological applications. This is a simple consequence of general covariance of this theory. We investigate chaotic behaviour by reducing respective dynamical systems to geodesic flows on a Riemannian space. In our approach ''Lyapunov like exponents'' are invariant forms independently of any time-coordinate transformations. The criterion of the local instability of a geodesic flow and ''Lyapunov exponents'' are formulated in terms of the Ricci scalar R and other invariants of the Riemannian curvature tensor. Possible cosmological applications of the proposed formalism are also discussed.