CHAOS IN STATIC AXISYMMETRICAL SPACETIMES .1. VACUUM CASE

We study the motion of a test particle in static axisymmetric vacuum s pacetimes and discuss two criteria for strong chaos to occur: (i) a lo cal instability measured by the Weyl curvature, and (ii) a tangle of a homoclinic orbit, which is closely related to an unstable periodic or bit in general relativity. We analyse several static axisymmetric spac etimes 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 behaviour in some spaceti mes, these can be accounted for by the second criterion.