Free Motion around Black Holes with Disks or Rings: Between Integrability and Chaos-V

The complete integrability of geodesic motion, the well-known feature of fields of isolated stationary black holes, can easily be "spoiled" by the presence of some additional sources (even if highly symmetric). In previous papers, we used various methods to show how free time-like motion becomes chaotic if the gravitational field of the Schwarzschild black hole is perturbed by that of a circular disk or ring, considering specifically the inverted first disk of the Morgan-Morgan counter-rotating family and the Bach-Weyl ring as additional sources. The present paper focuses on two new points. First, because the Bach-Weyl thin ring is physically quite unsatisfactory, we now repeat some of the analyses for a different, Majumdar-Papapetrou-type (extremally charged) ring around an extreme Reissner-Nordström black hole, and compare the results with those obtained before. We also argue that such a system is in fact more relevant astrophysically than it may seem. Second, we check numerically, for the latter system as well as for the Schwarzschild black hole encircled by the inverted Morgan-Morgan disk, how indicative the geometric (curvature) criterion is for the chaos suggested by Sota et al. We also add a review of the literature where the relevance of geometric criteria in general relativity (as well as elsewhere) has been discussed for decades.