Free motion around black holes with discs or rings: between integrability and chaos -- V

Abstract

Complete integrability of geodesic motion, the well known feature of the fields of isolated stationary black holes, can easily be "spoilt" by the presence of some additional source (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 disc or ring, considering specifically the inverted first disc of the Morgan-Morgan counter-rotating family and the Bach-Weyl ring as the additional sources. The present paper focuses on two new points. First, since the Bach-Weyl thin ring is physically quite unsatisfactory, we now repeat some of the analysis 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 disc, how indicative is the geometric (curvature) criterion for chaos suggested by Sota et al. (1996). 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.

Homoclinic orbit and the violation of chaos bound around the black hole with anisotropic matter fields

“…From the viewpoint of analysis, the number of constants of motion should be reduced by one at least. Then in the Poincare section, they will have non-trivial lines that show chaotic behavior [59,60,61]. We leave the study of particle motion with additional effects as future work.…”