Lyapunov indices with two nearby trajectories in a curved spacetime

TL;DR

This paper compares three methods for computing invariant Lyapunov exponents in general relativity, introduces an invariant fast Lyapunov indicator, and applies it to analyze chaos in a static axisymmetric spacetime.

Contribution

It evaluates and compares three Lyapunov exponent calculation methods, proposes an invariant fast Lyapunov indicator, and demonstrates its effectiveness in identifying chaos in curved spacetime.

Findings

All three methods yield similar Lyapunov exponents.

M3 is simpler to implement than M2.

The invariant FLI effectively distinguishes chaotic from regular trajectories.

Abstract

We compare three methods for computing invariant Lyapunov exponents (LEs) in general relativity. They involve the geodesic deviation vector technique (M1), the two-nearby-orbits method with projection operations and with coordinate time as an independent variable (M2), and the two-nearby-orbits method without projection operations and with proper time as an independent variable (M3). An analysis indicates that M1 and M3 do not need any projection operation. In general, the values of LEs from the three methods are almost the same. As an advantage, M3 is simpler to use than M2. In addition, we propose to construct the invariant fast Lyapunov indictor (FLI) with two-nearby-trajectories and give its algorithm in order to quickly distinguish chaos from order. Taking a static axisymmetric spacetime as a physical model, we apply the invariant FLIs to explore the global dynamics of phase space of the system where regions of chaos and order are clearlyidentified.