Influence of the gravitational radius on asymptotic behavior of the relativistic Sitnikov problem

The Sitnikov problem is a classical problem broadly studied in physics which can represent an illustrative example of chaotic scattering. The relativistic version of this problem can be attacked by using the post-Newtonian formalism. Previous work focused on the role of the gravitational radius lambda on the phase space portrait. Here we add two relevant issues on the influence of the gravitational radius in the context of chaotic scattering phenomena. First, we uncover a metamorphosis of the KAM islands for which the escape regions change insofar as lambda increases. Second, there are two inflection points in the unpredictability of the final state of the system when lambda similar or equal to 0.02 and lambda similar or equal to 0.028. We analyze these inflection points in a quantitative manner by using the basin entropy. This work can be useful for a better understanding of the Sitnikov problem in the context of relativistic chaotic scattering. In addition, the described techniques can be applied to similar real systems, such as binary stellar systems, among others.