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

Abstract

We continue the study of time-like geodesic dynamics in exact static, axially and reflection symmetric space-times describing the fields of a Schwarzschild black hole surrounded by thin discs or rings. In the previous paper, the rise (and decline) of geodesic chaos with ring/disc mass and position and with test particle energy was revealed on Poincaré sections, on time series of position or velocity and their power spectra, and on time evolution of the orbital 'latitudinal action'. In agreement with the KAM theory of nearly integrable dynamical systems and with the results observed in similar gravitational systems in the literature, we found orbits of very different degrees of chaoticity in the phase space of perturbed fields. Here we compare selected orbits in more detail and try to classify them according to the characteristics of the corresponding phase-variable time series, mainly according to the shape of the time-series power spectra, and also applying two recurrence methods: the method of 'average directional vectors', which traces the directions in which the trajectory (recurrently) passes through a chosen phase-space cell, and the 'recurrence-matrix' method, which consists of statistics over the recurrences themselves. All the methods proved simple and powerful, while it is interesting to observe how they differ in sensitivity to certain types of behaviour.

Vertical stability of circular orbits in relativistic razor-thin disks

“…The results of [11,12] point in the direction of the proof presented in this paper, whereas the results of [4,5,34,47] are mathematically inconsistent and the conclusions derived from them must be reformulated in light of the present approach. Since the stability criterion obtained and the adiabatic invariance analysis are both local, we can extend our formalism to black holes surrounded by razor-thin disks or rings [4,15,30,31,34], provided the metric outside the horizon has the form (5) and the orbit does not cross the horizon. In particular, the results obtained in [34,47] …”

Inducing Chaos by Breaking Axial Symmetry in a Black Hole Magnetosphere

“…In such a case, the electromagnetic field affects the geometry of the spacetime and as a particular consequence of this influence, we found that the system allows off-equatorial orbits even for neutral test particles. The interplay between the chaotic and ordered motion of test particles in the exactly given static field of a massive source encircled by a disk or a ring was also recently investigated (Suková & Semerák 2013;Semerák & Suková 2012. Nevertheless, in the following, we intend to focus on a more realistic model in which the rotation of the central object is included.…”

Recurrence analysis as a tool to study chaotic dynamics of extreme mass ratio inspiral in signal with noise

“…[10][11][12][13][14][15] In the above cases the Kerr black hole hypothesis 16 is assumed to hold for the SMBH. However, it has been shown that spacetimes diverging from the Kerr by a deformation parameter can also result in non-integrable systems.…”

“…Recurrence analysis and RQA have already been applied to detect the onset of chaos in various non-integrable relativistic systems. 13,14,[36][37][38][39] In these systems the RQA indicator DIV defined as an inverse value of the length of the longest diagonal line found in the RP…”