Chaos in Static Axisymmetric Spacetimes I : Vacuum Case

TL;DR

This paper investigates the conditions under which chaos occurs in the motion of test particles in static axisymmetric vacuum spacetimes, identifying key criteria related to curvature and orbit tangles.

Contribution

It introduces two criteria for strong chaos in such spacetimes and analyzes their effectiveness in predicting chaotic behavior.

Findings

Weyl curvature is a sufficient condition for chaos.

Tangles of homoclinic orbits also indicate chaos.

Some particles exhibit chaos without the Weyl curvature criterion.

Abstract

We study the motion of test particle in static axisymmetric vacuum spacetimes and discuss two criteria for strong chaos to occur: (1) a local instability measured by the Weyl curvature, and (2) a tangle of a homoclinic orbit, which is closely related to an unstable periodic orbit in general relativity. We analyze several static axisymmetric spacetimes and find that the first criterion is a sufficient condition for chaos, at least qualitatively. Although some test particles which do not satisfy the first criterion show chaotic behavior in some spacetimes, these can be accounted for the second criterion.