1/f fluctuations in spinning-particle motions around Schwarzschild black hole

We study the properties of chaos in the motions of a spinning test particle in Schwarzschild spacetime. We characterize the chaos using the power spectrum of the time series of $z$ components of the particle’s position. It is found that the pattern of the power spectrum shows not only white noise but also $1\mathrm{/}f$-type fluctuation, depending on the value of the total angular momentum $J$ and the spin $S$ of the test particle. Therefore we succeed in classifying the chaotic motions, which have been classified as simply chaotic ones in former works, into the two distinct types. One is $1\mathrm{/}f$, and the other is white noise. Based on this classification, we plot, in the two-dimensional parameter space $(J,S)$, the phase diagram for the properties of the chaos. This phase diagram enables us in principle to guess the properties of the system $(J,S)$ by observing the dynamics of the test particle, even if the motion is chaotic. Furthermore, we detect that the origin of the $1\mathrm{/}f$ fluctuation is that the particle motion stagnates around regular orbits (tori), while traveling back and forth between them, which is called stagnant motion'' or sticky motion’’ in Hamiltonian dynamical systems. The point is that the difference of the property of the chaos or the power spectra is due to the topological structure of the phase space, which in turn is governed by the physical parameter set $(J,S)$ of the system. From this point of view, the chaos we found in this system is not always merely random.