Chaos in Static Axisymmetric Spacetimes; 2, vacuum case

We examine the effect of local matter on the chaotic behavior of a relativistic test particle in non-vacuum static axisymmetric spacetimes. We find that the sign of the sectional curvature in the geodesic deviation equation defined by the Riemann curvature does not always become a good tool to judge the occurrence of chaos in the non-vacuum case. However, we show that the locally unstable region ( LU region ) defined by the Weyl curvature can provide information about chaos even in non-vacuum spacetime as well as in vacuum spacetime. Since the Weyl tensor affects only the shear part of the geodesic congruence, it works effectively to stretch some directions of geodesic congruence, which helps to cause the chaotic behavior of geodesics. Actually, the orbit moving around an unstable periodic orbit (UPO) becomes strongly chaotic if it passes through an LU region, which means that the LU region can be used as a good tool to know in which situation the chaos by homoclinic mixing occurs around a UPO