Growth of resonances and chaos for a spinning test particle in the Schwarzschild background

TL;DR

This paper investigates how the spin of a small compact object affects its orbital dynamics around a black hole, revealing the presence of resonances and chaos, and explores how gravitational wave signals can indicate these complex behaviors.

Contribution

It demonstrates for the first time that resonances and chaos occur at astrophysical spin values and introduces a method to analyze resonance growth in spinning test particles.

Findings

Resonances and chaos are present even at astrophysical spin values.

Resonances caused by quadratic spin terms remain small in the small-mass-ratio limit.

Gravitational waveforms can distinguish between regular and chaotic orbital motion.

Abstract

Inspirals of stellar mass compact objects into supermassive black holes are known as extreme mass ratio inspirals. In the simplest approximation, the motion of the compact object is modeled as a geodesic in the space-time of the massive black hole with the orbit decaying due to radiated energy and angular momentum, thus yielding a highly regular inspiral. However, once the spin of the secondary compact body is taken into account, integrability is broken and prolonged resonances along with chaotic motion appear. We numerically integrate the motion of a spinning test body in the field of a non-spinning black hole and analyse it using various methods. We show for the first time that resonances and chaos can be found even for astrophysical values of spin. On the other hand, we devise a method to analyse the growth of the resonances, and we conclude that the resonances we observe are only caused by terms quadratic in spin and will generally stay very small in the small-mass-ratio limit. Last but not least, we compute gravitational waveforms by solving numerically the Teukolsky equations in the time-domain and establish that they carry information on the motion's dynamics. In particular, we show that the time series of the gravitational wave strain can be used to discern regular from chaotic motion of the source.