Stability and chaos around multipolar deformed bodies: A general relativistic approach

The exact solution to the Einstein equations that represents a static axially symmetric source deformed by an internal quadrupole is considered. By using the Poincare section method we numerically study the geodesic motion of test particles. For the prolate quadrupolar deformations we found chaotic motions contrary to the oblate case where only regular motion is found. We also study the metric that represents a rotating black hole deformed by a quadrupolar term. This metric is obtained as a two soliton solution in the context of Belinsky--Zakharov inverse scattering method. The stability of geodesics depends strongly on the relative direction of the spin of the center of attraction and the test particle angular momentum. The rotation does not alter the regularity of geodesic motions in the oblate case, i.e., the orbits in this case remain regular. We also employ the method of Lyapounov characteristic numbers to examine the stability of orbits evolving around deformed nonrotating centers of attraction. The typical time to observe instability of orbits is analyzed.