The motion of small spinning free test bodies is usually treated within the ‘pole–dipole’ approximation, which – in general relativity – leads to Mat

Mathisson—Papapetrou equations are solved numerically to obtain trajectories of spinning test particles in (the meridional section of) the Kerr space

Geodesic dynamics is regular in the fields of isolated stationary black holes. However, due to the presence of unstable periodic orbits, it easily be

The Mathisson-Papapetrou-Dixon (MPD) equations, providing the “pole-dipole” description of spinning test particles in general relativity, have to be supplemented by a condition specifying the worldline that will represent the history of the studied body. It has long been thought that the Mathisson-Pirani (MP) spin condition—unlike other major choices made in the literature—does not yield an explicit momentum-velocity relation. We derive here the desired (and very simple) relation and show that it is in fact equivalent to the MP condition. We clarify the apparent paradox between the existence of such a definite relation and the known fact that the MP condition is degenerate (does not specify a unique worldline), thus shedding light on some conflicting statements made in the literature. We then show how, for a given body, this spin condition yields infinitely many possible representative worldlines, and derive a detailed method how to switch between them in a curved spacetime. The MP condition is a convenient choice in situations when it is easy to recognize its “nonhelical” solution, as exemplified here by bodies in circular orbits and in radial fall in the Schwarzschild spacetime.

The dynamical system studied in previous papers of this series, namely a bound time-like geodesic motion in the exact static and axially symmetric space–time of an (originally) Schwarzschild black hole surrounded by a thin disc or ring, is considered to test whether the often employed ‘pseudo-Newtonian’ approach (resorting to Newtonian dynamics in gravitational potentials modified to mimic the black hole field) can reproduce phase-space properties observed in the relativistic treatment. By plotting Poincaré surfaces of section and using two recurrence methods for similar situations as in the relativistic case, we find similar tendencies in the evolution of the phase portrait with parameters (mainly with mass of the disc/ring and with energy of the orbiters), namely those characteristic to weakly non-integrable systems. More specifically, this is true for the Paczyński–Wiita and a newly suggested logarithmic potential, whereas the Nowak–Wagoner potential leads to a different picture. The potentials and the exact relativistic system clearly differ in delimitation of the phase-space domain accessible to a given set of particles, though this mainly affects the chaotic sea whereas not so much the occurrence and succession of discrete dynamical features (resonances). In the pseudo-Newtonian systems, the particular dynamical features generally occur for slightly smaller values of the perturbation parameters than in the relativistic system, so one may say that the pseudo-Newtonian systems are slightly more prone to instability. We also add remarks on numerics (a different code is used than in previous papers), on the resemblance of dependence of the dynamics on perturbing mass and on orbital energy, on the difference between the Newtonian and relativistic Bach–Weyl rings, and on the relation between Poincaré sections and orbital shapes within the meridional plane.

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 first paper of this series, the rise (and decline) of geodesic chaos with ring/disc mass and position and with test particle energy was revealed on Poincaré sections and on time series of position or velocity and their power spectra. In the second paper, we compared these results with those obtained by two recurrence methods, focusing on ‘sticky’ orbits whose different parts show different degrees of chaoticity. Here, we complement the analysis by using several Lyapunov-type coefficients which quantify the rate of orbital divergence. After comparing the results with those obtained by the previous methods, we specifically consider a system involving a black hole surrounded by a small thin disc or a large ring, having in mind the configuration which probably occurs in galactic nuclei. Within the range of parameters which roughly corresponds to our Galactic centre, we found that the black hole accretion disc does not have a significant gravitational effect on the dynamics of free motion at larger radii, while the inner circumnuclear molecular ring (concentrated above 1 pc radius) can only induce some irregularity in motion of stars (‘particles’) on smaller radii if its mass reaches 10 to 30 per cent of the central black hole (which is the upper estimate given in the literature), if it is sufficiently compact (which does not hold but maybe for its inner rim) and if the stars can get to its close vicinity. The outer dust ring between 60 and 100 pc appears to be less important for the geodesic dynamics in its interior.