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

## TL;DR

This paper investigates how the addition of discs or rings around black holes affects the integrability of geodesic motion, revealing conditions under which chaos emerges, with implications for astrophysical models.

## Contribution

It introduces analysis of a Majumdar-Papapetrou--type ring around an extremal Reissner-Nordstr"om black hole and compares it to previous models, enhancing understanding of chaos in black hole environments.

## Key findings

- Chaotic motion occurs when black holes are perturbed by rings or discs.
- Geometric curvature criteria can indicate the onset of chaos.
- Comparison shows different ring models influence chaos differently.

## 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-Nordstr\"om 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, Suzuki & Maeda (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.