Description of chaos in simple relativistic systems

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Abstract
Chaos is investigated in the context of general relativity and gravitation. We show how quantitative and global measures of chaos can be obtained from qualitative and local ones. After averaging—first, over all two-directions, and second, along the trajectory—the rate of separation of nearby trajectories (Lyapunov-like exponents) can be obtained. This gives us a tool to the invariant chaos description. The sign of the Ricci scalar serves as a criterion of the local instability in simple mechanical systems (systems with a natural Lagrange function). We also show how to reduce relativistic simple mechanical systems to the classical ones. Timelike and null geodesics in multi-black-hole cosmological spacetimes are considered. The role of relativistic systems in general relativity is emphasized. © 1996 The American Physical Society.